121381706-Math-Kangaroo-Practice-Problems-Grades-1-8.pdf

Math Kangaroo

http://www.mathkangaroo.org/2010page/Clark/clark/pdb/#Past Contests

Grade 01-02

Problem Kangur_2005_02_1 (3 pts) http://www.mathkangaroo.org

In the enchanted garden of the Green King, there are apple trees that produce golden apples.

Every day, 5 golden apples become ripe on each tree, and at the end of each day they fall

from the trees. Today, the Green Gardener has picked up 20 ripe apples that fell under the

trees last night. How many enchanted trees are there in the garden?

A) 4

B) 5

C) 6

D) 7

E) 8


Alma, Maria, Anne and Michael had 2 apples each. Each one ate one apple. How many

apples do they now have altogether?

A) 1

B) 2

C) 4

D) 6

E) 8


Only one digit from 1 to 9 is repeated three times in this drawing.

The rest of the digits are repeated twice. Which digit is repeated three times?

A) 9

B) 8

C) 3

D) 4

E) 7

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How many different digits can you see in the picture below?

A) 3

B) 4

C) 5

D) 6

E) 9


What number is hidden under the question mark in the picture below (on the last car)?

A) 7

B) 4

C) 0

D) 1

E) 2


5 - 4 + 3 - 2 + 1 = ?

A) 1

B) 2

C) 3

D) 4

E) 0


When Ann was born, Michael was 4. Now Ann is 3 years old. How old is Michael?

A) 1

B) 6

C) 7





D) 8

E) 10


How many blocks were used to build the figure shown in the picture?

A) 7

B) 12

C) 13

D) 14

E) 16


Which four beads below need to be added to this string:


How many more square tiles do we need to put on the kitchen floor to cover all of it? (See the

picture.)

A) 12

B) 10

C) 9

D) 6

E) 4


Helga is climbing stairs in such a way that she goes up 2 steps at a time. She is standing on

the third step now. On which step will she be after she moves up 3 times?

A) 9

B) 1

C) 6





D) 5

E) 8


Some pages are missing from an open book. On the left page you can see page number 12

and on the right page you can see page number 15. How many pages are missing?

A) 1

B) 2

C) 3

D) 4

E) 5


One hen lays one egg a day. In how many days will two hens lay 6 eggs?

A) 1

B) 3

C) 6

D) 9

E) 10


There are two horses, one duck, one fish, an eagle, and a boy in a private garden. How many

legs do they have altogether?

A) 10

B) 12

C) 14

D) 16

E) 18


Anne has some apples. Maria has 2 apples more than Anne. Altogether they have 8 apples.

How many apples does Anne have?

A) 3

B) 5

C) 6

D) 7

E) 10







Which figure is next in this sequence:


Monika is 2 years old and Karl is 4. How old will Monika be when Karl is 11?

A) 6

B) 7

C) 8

D) 9

E) 10


During the race, right before the finish line, I passed the runner who won the third place.

What place did I win?

A) 1

B) 2

C) 3

D) 4

E) 5


There are three weights on the scales in the left picture: 1kg, 4 kg, 2 kg and just one weight of

1 kg on the scales in the right picture. What is the weight of the fruit in the basket?

A) 7 kg

B) 6 kg

C) 5 kg

D) 3 kg

E) 2 kg


Which set of signs +, - needs to be used to make this expression true?





A) +, -, +

B) -, +, -

C) +, +, -

D) +, +, +

E) -, +, +


The sum of two digits, one from inside the square and one from outside the square is greater

than 10. How many such pairs can you make?

A) 19

B) 11

C) 6

D) 24

E) 18


What number is covered by ? in the last picture below?

A) 1

B) 2

C) 3

D) 4

E) 5


Hans fills the square table with numbers in such a way that the sum of the numbers in each

column is 15 and the sum of the numbers in each row is 15 and the sum on each diagonal is

15. What number will he put in place of ? ?

A) 1

B) 3

C) 2




D) 6

E) 9


A train has four cars in four colors: red, green, white and yellow. The green car is not the first

nor the last. The yellow car is not next to the white car nor next to the red car. The first car is

white. What is the order of the cars in that train?

A) white, green, red, yellow

B) white, yellow, green, red

C) green, yellow, red, white

D) red, white, green, yellow

E) white, red, green, yellow

Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©


Grade 03-04


A butterfly sat down on a correctly solved problem. What number did it cover up?

A) 250

B) 400

C) 500

D) 910

E) 1800


At noon, the minute hand of a clock is in the following position:

What will the position of the minute hand be after 17 quarters of an hour?

A) B) C) D) E)


Joan bought some cookies, each of which costs 3 dollars. She gave the salesperson 10 dollars,

and received 1 dollar as change. How many cookies did Joan buy?

A) 2

B) 3

C) 4

D) 5

E) 6


After the trainer's first whistle, the monkeys at the circus formed 4 rows. There were 4

monkeys in each row. After the second whistle, they rearranged themselves into 8 rows. How

many monkeys were there in each row after the second whistle?





A) 1

B) 2

C) 3

D) 4

E) 5


Eva lives with her parents, her brother, one dog, two cats, two parrots, and four fish. What is

the total number of legs that they have altogether?

A) 22

B) 24

C) 28

D) 32

E) 40


John has a chocolate bar consisting of square pieces 1 cm x 1 cm in size. He has already eaten

some of the corner pieces (see the picture). How many pieces does John have left?

A) 66

B) 64

C) 62

D) 60

E) 58


Two traffic signs mark the bridge in my village (see the picture below). These signs indicate

the maximum vehicle width and the maximum vehicle weight allowed on the bridge. Which

one of the following trucks is allowed to cross that bridge?

A) It is 315 cm wide and it weights 4400 kg.

B) It is 330 cm wide and it weights 4250 kg.

C) It is 325 cm wide and it weights 4400 kg.




D) It is 330 cm wide and it weights 4200 kg.

E) It is 325 cm wide and it weights 4250 kg.


Each of seven boys has paid the same amount of money for a trip. The total sum of what they

paid is a three digit number, which can be written as 3 0. What is the middle digit of this

number?

A) 3

B) 4

C) 5

D) 6

E) 7


What is the smallest possible number of children in a family where each child has at least one

brother and at least one sister?

A) 2

B) 3

C) 4

D) 5

E) 6


Out of all five numbers below, I chose one. The number is even and all of its digits are

different. The hundreds digit is double the ones digit. The tens digit is greater than the

thousands digit. Which number did I choose?

A) 1246

B) 3874

C) 4683

D) 4874

E) 8462


A square piece of paper has been cut into three pieces. Two of them are shown in the picture:

Which of the pieces below is the third one?





A) B) C) D) E)


An elevator cannot carry more than 150 kg. Four friends weigh 60 kg, 80 kg, 80 kg, and 80

kg, respectively. What is the least number of trips necessary to carry the four friends to the

highest floor?

A) 1

B) 2

C) 3

D) 4

E) 7


Ala has 24 dollars, Barb has 66 dollars, and Sophia has as many dollars more than Ala as she

has less than Barb. How many dollars does Sophia have?

A) 33

B) 35

C) 42

D) 45

E) 48


There are eight kangaroos in the cells of the table (see the picture). What is the least number

of the kangaroos that need to be moved to the empty cells so that there would be exactly two

kangaroos in any row and in any column of the table?

A) 4

B) 3

C) 2

D) 1

E) 0





Greg needs to bring four full sacks of sand from the river to a house located at the other end

of the village. Unfortunately, on his way through the village, half of the sand spills out of the

sack through a hole. How many trips does Greg need to make from the river to the house in

order to bring the required amount of sand?

A) 4

B) 5

C) 6

D) 7

E) 8


During a Kangaroo camp, Adam solved five problems every day, and Brad solved two

problems daily. After how many days did Brad solve as many problems as Adam solved in 4

days?

A) After 5 days

B) After 7 days

C) After 8 days

D) After 10 days

E) After 20 days Problems 5 points each


There were 9 pieces of paper. Some of them were cut into three pieces. As a result, there are

15 pieces of paper now. How many pieces of paper were cut?

A) 2

B) 3

C) 4

D) 5

E) 6


Using 6 matches, only one rectangle with a perimeter of 6 matches can be made (see the

picture). How many different rectangles with a perimeter of 14 matches can be made using 14

matches?

A) 2

B) 3

C) 4





D) 6

E) 12


A picture frame was constructed using pieces of wood which all have the same width. What

is the width of the frame if the inside perimeter of the frame is 8 decimeters less than its

outside perimeter?

A) 1 dm

B) 2 dm

C) 4 dm

D) 8 dm

E) It depends on the size of the picture


In a trunk there are 5 chests, in each chest there are 3 boxes, and in each box there are 10 gold

coins. The trunk, the chests, and the boxes are locked. At least how many locks need to be

opened in order to take out 50 coins?

A) 5

B) 6

C) 7

D) 8

E) 9


The figure shows a rectangular garden with dimensions of 16 m by 20 m. The gardener has

planted six identical flowerbeds (they are gray in the diagram). What is the perimeter of each

of the flowerbeds?




A) 20 m

B) 22 m

C) 24 m

D) 26 m

E) 28 m


Mike chose a three-digit number and a two-digit number. The difference of these numbers is

989. What is their sum?

A) 1001

B) 1010

C) 2005

D) 1000

E) 1009


Five cards are laying on the table in the order: 5, 1, 4, 3, 2 as shown in the top row of the

picture. They need to be placed in the order shown in the bottom row. In each move, any two

cards may be switched. What is the least number of moves that need to be made?

A) 2

B) 3

C) 4

D) 5

E) 6


Which of the cubes has the plan shown in the picture below?




A) B) C) D) E)



2001+ 2002 + 2003 + 2004 + 2005 =

A) 1,015

B) 5,010

C) 10,150

D) 11,005

E) 10,015


Marek was 4 years old when his sister was born. Today he blew out all 9 candles on his

birthday cake. What is the difference between Marek's and his sister's age today?

A) 4 years

B) 5 years

C) 9 years

D) 13 years

E) 14 years


The picture below shows a road from town A to town B (indicated by solid line) and a detour

(marked by a dash line) caused by renovation of the section CD. How many kilometres

longer is the road from town A to town B because of the detour now?

A) 3 km

B) 5 km

C) 6 km

D) 10 km

E) This cannot be calculated.






Which of the results below is not identical to the difference 671 - 389?

A) 771 - 489

B) 681 - 399

C) 669 - 391

D) 1871 - 1589

E) 600 - 318


There were some birds sitting on the telegraph wire. At one moment, 5 of them flied away

and after some time, 3 birds came back. At that time there were 12 birds sitting on the wire.

How many birds were there at the very beginning?

A) 8

B) 9

C) 10

D) 12

E) 14


Which numbers are inside a rectangle and inside a circle but not inside a triangle at the same

time?

A) 5 and 11

B) 1 and 10

C) 13

D) 3 and 9

E) 6, 7 and 4


Buildings on Color Street are numbered from 1 to 5 (see the picture). Each building is

colored with one of the following colors: blue, red, yellow, pink, and green. It is known that:

- The red building neighbours with the blue one only.

- The blue building is between the red one and the green one.




What is the color of the building numbered with 3?

A) Blue

B) Red

C) Yellow

D) Pink

E) Green


How many white squares need to be shaded so that the number of shaded squares equals

exactly to half of the number of white squares?

A) 2

B) 3

C) 4

D) 6

E) It is impossible to calculate it.


Five identical sheets of a plastic rectangles were divided into white and black squares. Which

of the sheets from A) to E) has to be covered with the sheet to the right in order to get totally

black rectangle?

A) B) C) D) E)





The scales in the pictures had been balanced. There are pencils and a pen on the arms of the

scales. What is the weight of the pen in grams?

A) 6 g

B) 7 g

C) 8 g

D) 9 g

E) 10 g


I notice four clocks on the wall (see the picture). Only one of them shows correct time. One

of them is 20 minutes ahead, another is 20 minutes late, and the other is stopped. What is the

time at the moment?

A) 4:45

B) 5:05

C) 5:25

D) 5:40

E) 12:00


Ella brought a basket of apples and oranges for a birthday party. Guests ate half of the apples

and the third part of the oranges. In the basket remained:

A) Half of all fruits

B) More than half of all fruits

C) Less than half of all fruits



D) A third part of all fruits

E) Less than a third part of all fruits


Ania divided a certain number by 10 instead of multiplying it by 10. As a result she got 600.

What would the result have been if she hadn't made that mistake?

A) 6

B) 60

C) 600

D) 6,000

E) 60,000


Kathy found a book, which was lack of certain number of sheets. When she opened the book

she saw number 24 on the left side and number 45 on the right side. How many sheets

between those sides were missing?

A) 9

B) 10

C) 11

D) 20

E) 21


Eva is 52 days older than her girlfriend Ania. Eva had her birthday on Tuesday in March of

this year. On which day of the week will Ania celebrate her birthday this year?

A) Monday

B) Tuesday

C) Wednesday

D) Thursday

E) Friday


Into the squares of diagram numbers were placed so that the sum of the numbers in the first

row is equal to so that the sum of the numbers in the first row is equal to 3, the sum of the

numbers in the second row is equal to 8, and the sum of the numbers in the first column is

equal to 4. What is the sum of the numbers in the second column?

A) 4

B) 6

C) 7





D) 8

E) 11


The cube (see the picture) is colored with three colors so that every side of this cube is one

color and every two opposite sides are the same color. From which of the patterns below this

kind of cube can be made?


Four square tiles were arranged in a way shown in the picture. The lengths of the sides of two

tiles are indicated in the picture. What is the length of the side of the largest tile?

A) 24

B) 56

C) 64

D) 81

E) 100


Girls and boys from Maria's and Mathew's class have formed a line. There are 16 students on

Maria's right, and Mathew is among them. There are 14 students on Mathew's left, and Maria

is among them. There are 7 students between Maria and Mathew. How many students are in

this class?

A) 37

B) 30

C) 23

D) 22

E) 16


The sum of the digits of the 10-digit number is 9.What is the product of the digits of this

number?





A) 0

B) 1

C) 45

D) 9 x 8 x 7 x ... x 2 x 1

E) 10


Out of 125 small, white and black cubes, the big cube was formed (see the picture). Every

two adjacent cubes have different colors. The vertices of the big cube are black. How many

white cubes does the big cube contain?

A) 62

B) 63

C) 64

D) 65

E) 68


A lottery-ticket was 4 dollars. Three boys: Paul, Peter, and Robert made a contribiution and

bought two tickets. Paul gave 1 dollar, Peter gave 3 dollars, and Robert gave 4 dollars. One of

the tickets they bought was worth 1000 dollars. Boys shared the award fairly, meaning,

proportionally to their contributions. How much did Peter receive?

A) 300

B) 375

C) 250

D) 750

E) 425


In three soccer games the Dziobak's team scored three goals and lost one. For every game

won the team gets 3 points, for a tie it gets 1 point, and for the game lost it gets 0 points. For

sure, the number of points the team earned in those three games was not equal to which of the

following numbers?

A) 7

B) 6

C) 5

D) 4

E) 3





In every white section of a diagram, the products of two numbers from grey sections - one

from above and one from the left - was placed (for example: 42 = 7 · 6 ). Some of these

products are represented by letters. Which two letters represent the same number?

A) L and M

B) T and N

C) R and P

D) K and P

E) M and S



The picture below shows the letter U drawn on grid paper. How many squares does the letter

U cover?

A) 10

B) 8

C) 11

D) 13

E) 12


What is the sum of 0 + 1 + 2 + 3 + 4 - 3 - 2 - 1 - 0?

A) 0

B) 2

C) 4




D) 10

E) 16


The first train car, right behind the engine, contains 10 boxes. In each of the other cars there

are twice as many boxes as in the car in front of it. How many boxes are there in the fifth car?

A) 100

B) 120

C) 140

D) 160

E) 180


Zosia is drawing kangaroos. The first one is blue, the next one green, followed by red, and

finally yellow, and then again blue, green, red, yellow, and so on, in the same order. What

color will the seventeenth kangaroo be?

A) Blue

B) Green

C) Red

D) Black

E) Yellow


In the teachers' lounge there are 6 tables with 4 chairs by each one, 4 tables with 2 chairs by

each, and 3 tables with 6 chairs by each. How many chairs are there in the lounge?

A) 40

B) 25

C) 50

D) 36

E) 44


In one of these pictures, there are three times as many hearts as other shapes. Which picture is

it?

A) B) C) D) E)






A rectangle with dimensions 7 x 4 was outlined on grid paper. How many squares will a

diagonal of this rectangle intersect?

A) 8

B) 9

C) 10

D) 11

E) 12


The figure presented in the picture, made with identical cubes, weighs 189 grams. How much

does one cube weigh?

A) 29 g

B) 25 g

C) 21 g

D) 19 g

E) 17 g


Peter wrote out consecutive natural numbers starting with 3 until he had written 35 digits.

What was the greatest number that Peter wrote?

A) 12

B) 22

C) 23

D) 28

E) 35


Anna fell asleep at 9:30 PM and woke up at 6:45 AM the next day. Her little brother Peter

slept 1 hour and 50 minutes longer. How long did Peter sleep?





A) 30 hr 5 min

B) 11 hr 35 min

C) 11 hr 5 min

D) 9 hr 5 min

E) 8 hr 35 min


A pattern, the beginning and the end of which is shown in the picture, is made up of

alternating black and white bars. There are 17 bars altogether. The bars on both ends are

black. How many white bars are there in the pattern?

A) 9

B) 16

C) 7

D) 8

E) 15


Jumping Kangaroo is practicing for the animal Olympics. His longest jump during the

training was 55 dm 50 mm long, but in the finals of the Olympics he won with a jump that

was 123 cm longer. How long was Jumping Kangaroo's longest jump during the Olympics?

A) 6 m 78 cm

B) 5 m 73 cm

C) 5 m 55 cm

D) 11 m 28 cm

E) 7 m 23 cm


Peter chose a certain number, then he subtracted 203 from it, then he added 2003 to that

difference. His final result was 20003. What number did Peter choose at the beginning?

A) 23

B) 17797

C) 18203

D) 21803

E) 22209


Barbara likes to add the digits showing the current time on her electronic watch (for example,

when the watch shows 21:17, she gets the number 11 as the result). What is the greatest sum

she can get?





(Hint: in some countries and sometimes in USA, instead of telling it is 1P.M., people say it is

13:00. When it is 2P.M. they say it is 14:00, and when it is 12A.M., they say it is 24:00. In

this problem 21:17 means 9:17P.M. Time expressed with this method is called military time

sometimes.)

A) 24

B) 36

C) 19

D) 25

E) 23


Mark said to his friends, "If I had picked twice as many apples as I picked, I would have 24

more apples than I have now." How many apples did Mark pick?

A) 48

B) 24

C) 42

D) 12

E) 36


Points A, B, C, D all of which lie on a straight line, are marked in the figure below. The

distance between points A and C is 10 m, between B and D is 15 m, and between A and D is

22 m. What is the distance between points B and C?

A) 1 m

B) 2 m

C) 3 m

D) 4 m

E) 5 m


There are 29 students in the class. 12 of the students have a sister and 18 of the students have

a brother. In this class, only Tania, Barbara, and Anna do not have any siblings. How many

students from this class have both a brother and a sister?

A) None

B) 1

C) 3

D) 4

E) 6






Peter has 11 pieces of paper. He cut some of them into three parts and now he has 29 pieces

of paper. How many pieces of paper did he cut?

A) 3

B) 2

C) 8

D) 11

E) 9


Peter bought 3 kinds of cookies: large, medium, and small. The large cookies cost 4 zlotys

each, the medium: 2 zlotys each, and the small: 1 zloty each. (A zloty is the Polish unit of

money.) Altogether, Peter bought 10 cookies and paid 16 zlotys. How many large cookies did

he buy?

A) 5

B) 4

C) 3

D) 2

E) 1


Christopher built a rectangular prism using red and blue cubes of identical size. The outer

walls of this prism are red but all the inner cubes are blue. How many blue cubes did

Christopher use in this construction?

A) 12

B) 24

C) 36

D) 40

E) 48


Jurek is planning to buy some basketballs. If he were to buy 5 balls, he would have 10 zlotys

left over, and if he were to buy 7 balls, he would have to borrow 22 zlotys. (A zloty is the

Polish unit of money.) How much does one basketball cost?




A) 11

B) 16

C) 22

D) 26

E) 32


Mark built a rectangular prism using 3 blocks, each of which is made up of 4 small cubes

connected in various ways. Two of the blocks are shown in the picture. Which is the third,

white block, of which only two sides are visible?

A) B) C) D) E)


From a square puzzle, two pieces, which made up the shaded region, were cut out (see the

figure). Which two of the pieces below are these?

A) 1 and 3

B) 2 and 4

C) 2 and 3

D) 1 and 4

E) 3 and 4





At the toy store, among other things, you can buy dogs, bears, and kangaroos. Three dogs and

two bears together cost as much as four kangaroos. For the same amount of money you can

buy one dog and three bears. Then:

A) A dog is twice as expensive as a bear.

B) A bear is twice as expensive as a dog.

C) The prices of a dog and of a bear are identical.

D) A bear is three times as expensive as a dog.

E) A dog is three times as expensive as a bear.



Which of the squares below should be put into the picture to the right, to get the symbol of

our competition?

A)

B)

C)

D)

E)


After we simplify 2 + 2 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + 2 what will be the result?



A) 0

B) 2

C) 4

D) 12

E) 20


Andrzej received three cars, four balls, three teddy bears, ten pens, two chocolate bars, and a

book for his birthday. How many items did he get in all?

A) 15

B) 17

C) 20

D) 23

E) 27


A square was divided into pieces (see the picture). Which of the following pieces does not

occur in this divided square?

A)

B)

C)

D)

E)





Julia, Kasia, Zuzanna, and Helena have their birthdays on March 1st, May 17th, July 20th,

and March 20th. Kasia and Zuzanna were born in the same month. Julia and Zuzanna were

born on the same day of a month. Which of the girls was born on May 17th?

A) Julia

B) Kasia

C) Zuzanna

E) Helena

E) It cannot be determined from the given informfation.


A human heart beats an average of 70 times per minute. On average how many times does it

beat during one hour?

A) 42,000

B) 7,000

C) 4,200

D) 700

E) 420


Quadrilateral ABCD is a square and its side is 10 cm long. Quadrilateral ATMD is a

rectangle and its shorter side is 3 cm. What is the difference between the sum of the lengths

of all the sides of the square and the sum of the lengths of all the sides of the rectangle?

A) 14 cm

B) 10 cm

C) 7 cm

D) 6 cm

E) 4 cm


Which of the figures below (see the picture) couldn't be made with folding a rectangular sheet

just once?




A)

B)

C)

D)

E)


Houses on the street where John lives are numbered from 1 to 24. How many times does the

digit 2 appear in the numbering of those houses?

A) 2

B) 4

C) 8

D) 16

E) 32


There are six identical oranges on one scale of the balance and two identical melons on the

other scale. After we put one melon on the scale with the oranges, the scales will be balanced.

How many oranges weigh as much as one melon?

A) 2

B) 3

C) 4

D) 5

E) 6





This picture below is a sketch of a castle. Which of the lines below does not belong to the

sketch?

A)

B)

C)

D)

E)


We add 17 to the smallest two-digit number and then we divide the sum by the largest one-

digit number. What is the result?

A) 3

B) 6

C) 9

D) 11

E) 27


In a certain ancient country the numbers: one, ten, and sixty were expressed with the

following symbols:

one, ten, sixty.

Using those symbols people were writing down other numbers, for example the number 22

was written as

Which of the following notations represents the number 124 ?

A)

B)

C)



D)

E)


A face of a clock was divided into four parts. The sums of the numbers in each of those parts

are consecutive E numbers. Which of the following pictures satisfies this rule?

A)

B)

C)

D)

E)


Klara and Zosia had 60 matches altogether. Klara took as many matches as she needed to

build a triangle, each side 6 matches long. Zosia used the remaining matches to build a

rectangle, which had one side equal to 6 matches. How many matches long is each of the

longer sides of this rectangle?

A) 9

B) 12

C) 15

D) 18

E) 30





Three kangaroos: Miki, Niki, and Oki participated in a competition. Jumping at the same

speed, they jumped along the lines you can see in the picture. Only one of the following

sentences A, B, C, D and E is true. Which one?

A) Miki and Oki finished at the same time.

B) Niki finished first.

C) Oki finished last.

D) All kangaroos finished at the same time.

E) Miki and Niki finished at the same time.


Each boy: Mietek, Mirek, Pawel, and Zbyszek has exactly one of the following animals: a

cat, a dog, a gold fish, and a canary-bird. Mirek has a pet with fur. Zbyszek has a pet with

four legs. Pawel has a bird, and Mietek and Mirek don't like cats. Which of the following

sentences is not true?

A) Zbyszek has a dog.

B) Pawel has a canary.

C) Mietek has a golden fish.

D) Zbyszek has a cat.

E) Mirek has a dog.


Marysia leaves her house at 6:55 and arrives at school at 7:32. Zosia needs 12 minutes less

than Marysia to get to school. Yesterday Zosia showed up at school at 7:45. What time did

she leave her house?

A) At 7:07

B) At 7:20

C) At 7:25

D) At 7:30

E) At 7:33


Robert had a certain number of identical cubes. He glued a tunnel using half of his blocks

(see Picture 1). With some of the remaining cubes he formed a pyramid (see Picture 2). How




many blocks were not used to build those structures?

A) 34

B) 28

C) 22

D) 18

E) 15


Daughter is 3 years old, and her mother is 28 years older than the daughter. How many years

later will the mother be three times older than her daughter?

A) 9

B) 12

C) 10

D) 1

E) 11


A conductor wanted to make a trio consisting of a fiddler, a pianist, and a drummer. He had

to choose one of two fiddlers, one of two pianists, and one of two drummers. He decided to

try each of the possible trios. How many attempts did he have to make?

A) 3

B) 4

C) 8

D) 24

E) 25


One medal can be cut out from a golden square plate. If four medals are made from four

plates, the remaining parts of those four plates can be used to make one more plate. What is

the largest number of medals that could be formed when 16 plates are used?

A) 17

B) 19

C) 20

D) 21

E) 32





Twenty eight students from the fourth grade competed in the math competition. Each student

earned a different number of points. The number of students who received more points than

Tomek is two times smaller than the number of students who had less points than Tomek. In

which position did Tomek finish that competition?

A) 6th

B) 7th

C) 8th

D) 9th

E) 10th


An odometer in a car shows the number 187569 of passed kilometers. This number consists

of all different digits. After passing how many kilometers will the odometer show a number

consisting of all different digits again?

A) After 777 km

B) After 12,431 km

C) After 431 km

D) After 21 km

E) After 11 km




Grade 05-06


A butterfly sat down on a correctly solved problem. What number did it cover up?

2005 + 205 = 3500 -

A) 1295

B) 1190

C) 1390

D) 1195

E) 1290


Together, Anna and Olla have ten pieces of candy. Olla has two more pieces of candy than

Anna. How many pieces of candy does Olla have?

A) 8

B) 7

C) 6

D) 5

E) 4


There are eight kangaroos in the diagram (see the picture). What is the least number of

kangaroos that have to be moved to the empty boxes in order to have two kangaroos in each

row and each column?

A) 0

B) 1

C) 2

D) 3

E) 4






Eva lives with her parents, a brother, a dog, two cats, two parrots, and four gold fish. How

many legs do they have altogether?

A) 40

B) 32

C) 28

D) 24

E) 22


2005 x 100 + 2005 =

A) 2005002005

B) 20052005

C) 20072005

D) 202505

E) 22055


An ant is walking from point A to point B on a cube along the indicated path. The edge of the

cube is 12 cm long. How far does the ant need to travel?

A) 40 cm

B) 48 cm

C) 50 cm

D) 60 cm

E) 36 cm


On a shelf, there are 24 balls in three colors: white, red and brown. of them are white, and

of the rest of the balls are red. How many of them are brown?

A) 4

B) 5

C) 6




D) 7

E) 8


There are five cards on the table, labeled with numbers 1 to 5 as shown in the top row. One

move consists of switching two cards. How many moves do you need to make so that the

cards are arranged in the way shown in the bottom row?

A) 2

B) 4

C) 1

D) 3

E) 5


Tom picked a natural number and multiplied it by 3. Which number CANNOT be the result

of this multiplication?

A) 987

B) 444

C) 204

D) 105

E) 103


How many hours is half of a third part of a quarter of 24 hours?

A)

B)

C) 1

D) 2

E) 3


Eva cut a paper napkin into 10 pieces. She then also cut one of the pieces into 10 pieces. She

repeated this process two more times. Into how many pieces did she cut the napkin?





A) 27

B) 30

C) 37

D) 40

E) 47


Mowgli usually walks from home to the beach, and returns on an elephant. It takes him 40

minutes altogether. One day he traveled on the elephant from home to the beach and back,

which took him 32 minutes. How much time would he need to travel the same distance on

foot?

A) 24 min

B) 42 min

C) 46 min

D) 48 min

E) 50 min


A rectangular garden with an area of 30 m² was divided into three rectangular sections of

flowers, vegetables, and strawberries (some of the dimensions are shown in the diagram).

What is the area of the vegetable section, if the flower part has an area of 10 m²?

A) 4 m²

B) 6 m²

C) 8 m²

D) 10 m²

E) 12 m²


Grandpa suggested dividing all peanuts between the family members in the following way:

one person would get 5 kilos, two people would get 4 kilos each, four people would get 2

kilos each, two people would get 1.5 kilo each, and one person would not get any nuts.

Grandma suggested dividing the peanuts equally among all of the family members. For how

many people would the division suggested by Grandma be better than the one suggested by

Grandpa?




A) 3

B) 4

C) 5

D) 6

E) 7


How many two digit numbers are there, which can be expressed only by using different odd

digits?

A) 15

B) 20

C) 25

D) 30

E) 50


Which of the cubes below represents the plan of the cube shown to the right?


Sum of five consecutive natural numbers is equal to 2005. What is the greatest number

among them?

A) 401

B) 403

C) 404

D) 405

E) 2001


What is the number of all divisors of number 100?





A) 3

B) 6

C) 7

D) 8

E) 9


The frame of a rectangular painting was made out of wooden pieces of the same width. What

is the width of those pieces if the outer perimeter of the frame is 8 decimeters longer than the

inner perimeter?

A) 4dm

B) 2dm

C) 1dm

D) 8dm

E) The width depends on the dimensions of the painting.


How many more triangles than squares are shown in the picture?

A) 4 more

B) 2 more

C) 1 more

D) 5 more

E) 3 more


There are five containers in a treasure chest, in each container there are three boxes and in

each box there are 10 golden coins. The treasure chest, the containers, and the boxes are all

locked. How many locks do you need to open to get 50 coins?




A) 5

B) 7

C) 9

D) 6

E) 8


What number should replace x, if we know that the number in the circle in the upper row is

the sum of the numbers from the two circles right below it.

A) 32

B) 50

C) 55

D) 82

E) 100


In a two-digit number, a is the tens digit and b is the ones digit. Which of the conditions

below ensures that the number will be divisible by 6?

A) a + b = 6

B) b = 6 a

C) b = 5 a

D) b = 2 a

E) a = 2 b


A wooden cube with the length of its side equal to 3 dm was painted with 0.25 kg of paint.

The cube was then cut up into unit cubes (side length of 1 dm). How much paint is needed to

paint the unpainted sides of the little cubes?

A) 1.25 kg

B) 1 kg

C) 0.75 kg

D) 0.5 kg

E) 0.25 kg






Five circles have radii of the same length (see the picture). Four of them are touching the fifth

circle, and their centers are the vertices of a square. The ratio of the area of the shaded region

of the circles to the area of unshaded regions of the circles is:

A) 1 : 3

B) 1 : 4

C) 2 : 5

D) 2 : 3

E) 5 : 4


From noon until midnight, Wise Cat sleeps under a chestnut tree. From midnight until noon

he is awake telling stories. There is a note on that tree which says: "Two hours ago, Wise Cat

was doing the same thing that he will be doing in an hour". How many hours, out of 24 hours,

is the note true?

A) 6

B) 12

C) 18

D) 3

E) 21


Mark has 42 cubes with side length of 1 cm. He used them to construct a prism, the base of

which has a perimeter of 18 cm. What is the height of that prism?

A) 6 cm

B) 5 cm

C) 4 cm

D) 3 cm

E) 2 cm


On the board Peter wrote all the three-digit numbers that have the following properties: the

digits in each of the numbers are different, the first digit is the square of the quotient of the

second digit and the third digit. How many numbers did Peter write?

A) 1

B) 2




C) 3

D) 4

E) 8


Equilateral triangle ABC (all sides congruent) has an area equal to 1. A bigger triangle was

constructed out of 49 of these triangles (see the picture). The area of the shaded region is

equal to:

A) 20

B) 22.5

C) 23.5

D) 25

E) 32


Mary, Dorothy, Sylvia, Ella, and Kathy are sitting on a bench in the park. Mary is not sitting

on the farthest right side; Dorothy is not sitting the farthest to the left. Sylvia is not sitting the

farthest to the left nor the farthest to the right. Kathy is not sitting next to Sylvia, and Sylvia

is not sitting next to Dorothy. Ella is sitting to the right of Dorothy, but not necessarily next to

her. Which girl is sitting the farthest to the right?

A) It cannot be determined.

B) Dorothy

C) Sylvia

D) Ella

E) Kathy



How much is 1000 - 100 + 10 - 1?




A) 111

B) 900

C) 909

D) 990

E) 999


In each of the little squares Karolina places one of the digits: 1, 2, 3, 4. She makes sure that in

each row and each column each of these numbers is placed. In the figure below, you can see

the way of filling these squares. What number should she put in the square marked with an x?

1 x 2

4 1

3

2

A) 1

B) 2

C) 3

D) 4

E) Cannot be determined.


(10 · 100) · (20 · 80) =

A) 20,000 · 80,000

B) 2000 · 8000

C) 2000 · 80,000

D) 20,000 · 8000

E) 2000 · 800


How many hours is 360,000 seconds?

A) 3 hours

B) 6 hours

C) 8.5 hours

D) 10 hours

E) More than 90 hours.


What is the remainder when you divide 20042003 by 2004?





A) 0

B) 1

C) 2

D) 3

E) 2003


Five identical sheets of a plastic rectangles were divided into white and black squares. Which

of the sheets from A to E has to be covered with the sheet to the right in order to get totally

black rectangle?

A) B) C) D) E)


Which of the following numbers is not a factor of 2004?

A) 3

B) 4

C) 6

D) 8

E) 12


The three members of a rabbit family ate 73 carrots altogether during a week. The father ate

five carrots more than the mother. Their son ate 12 carrots. How many carrots did mother eat

in that week?

A) 27

B) 28

C) 31

D) 33

E) 56






Nine bus stops are equally spaced along a bus route. The distance between the first stop and

the third one is 600 m. How long is the bus route?

A) 1800 m

B) 2100 m

C) 2400 m

D) 2700 m

E) 3000 m


What is the value of the expression 1 - (2 - (3 - (4 - 5) ) ) equal to?

A) 0

B) -3

C) -9

D) 3

E) 9


You are given two identical puzzle pieces and you are not allowed to turn them over. Which

figure cannot be made out of these two pieces?


Karol folds a sheet of paper in a half and then repeats this four more times. Then he makes a

hole in the folded paper. How many holes does the sheet of paper have after unfolding?

A) 6

B) 10

C) 16

D) 20

E) 32






The different figures represent different digits. Find the digit corresponding to the square.

A) 9

B) 8

C) 7

D) 6

E) 5


The weight of 3 apples and 2 oranges is 255 g. The weight of 2 apples and 3 oranges is 285 g.

Each apple weighs the same and each orange weighs the same. What is the combined weight

of 1 apple and 1 orange?

A) 110 g

B) 108 g

C) 105 g

D) 104 g

E) 102 g


Tomek, Romek, Andrzej, and Michal said the following about a certain number: Tomek:

"This number is equal to 9"; Romek: "This number is prime."; Andrzej: "This number is

even."; Michal: "This number is equal to 15." Only one statement given either by Romek or

Tomek is true, as well as only one statement given by either Andrzej or Michal is true. What

number is it?

A) 1

B) 2

C) 3

D) 9

E) 15


What is the smallest number of the little squares that have to be shaded in order to get at least

one axis of symmetry of the figure below?




A) 1

B) 2

C) 3

D) 4

E) 5


One corner of a cube was cut off. Which of the figure below represents the pattern of the

cube after unfolding it?


Four snails: Fin, Pin, Rin, and Tin are moving along identical rectangular tiles. The shape and

length of each snail's trip is shown below. How many decimeters has snail Tin gone?

Snail Fin has gone 25 dm.

Snail Pin has gone 37 dm.

Snail Rin has gone 38 dm.

Snail Tin has gone ? dm

A) 27 dm

B) 30 dm

C) 35 dm

D) 36 dm

E) 40 dm


The Island of Turtles has an unusual weather system: Mondays and Wednesdays are rainy,

Saturdays are foggy and the other days are sunny. A group of tourists would like to go on a

44-day long vacation to the island. Which day of the week should be the first day of their

vacation in order to enjoy the most of the sunny days?




A) Monday

B) Wednesday

C) Thursday

D) Friday

E) Tuesday


The sum of two natural numbers is equal to 77. If the first number is multiplied by 8 and the

second by 6, then those products are equal. What is the larger of these numbers?

A) 23

B) 33

C) 43

D) 44

E) 54


What is the number of all divisors of 2 · 3 · 5 · 7 ?

A) 4

B) 14

C) 16

D) 17

E) 210


Ella and Ola had 70 mushrooms altogether. of Ella's mushrooms are brown and of

Ola's mushrooms are white. How many mushrooms did Ella have?

A) 27

B) 36

C) 45

D) 54

E) 10


There are 11 fields in the picture. Number 7 is written in the first field and number 6 in the

ninth field. What number has to be placed in the second field so that the sum of the numbers

from every three consecutive fields is equal to 21?

A) 7

B) 8

C) 6





D) 10

E) 21


The square below was divided into small squares. What part of the area of the shaded figure

is the area of the figure that is not shaded?

A)

B)

C)

D)

E)


In a CD store two CDs have the same price. The price of the first CD was reduced by 5% and

the price of the other one was increased by 15%. After this change the prices of the two CDs

differed by $6.00. How much is the cheaper CD now?

A) $1.50

B) $6.00

C) $28.50

D) $30.00

E) $34.50


In the little squares of a big square the consecutive natural numbers are placed in a way

shown in the figure. Which of the numbers below cannot be placed in the square with letter

x?




A) 128

B) 256

C) 81

D) 121

E) 400


Ania divided number by 3. What is the number of zeros in the quotient?

A) 670

B) 669

C) 668

D) 667

E) 665


Imagine that you have 108 red balls and 180 green balls. The balls have to be packed in

boxes in such a way that every box contains the same number of balls and there are balls of

only one color in every box. What is the smallest number of boxes that you need?

A) 288

B) 36

C) 18

D) 8

E) 1


During a competition in the Kangaroo Summer Camp in Zakopane students were given 10

problems to solve. For each correct answer a student was given 5 points and for each

incorrect one the student was loosing 3 points. Everybody solved all the problems. Mathew

got 34 points, Philip got 10 points and John got 2 points. How many problems did they

answer correctly all together?

A) 17

B) 18

C) 15

D) 13

E) 21


A right triangle with legs of length 6 cm and 8 cm was cut out of a paper and then folded

along a straight line. Which of the numbers below can express the area of the resulting

polygon?





A) 9 cm²

B) 12 cm²

C) 18 cm²

D) 24 cm²

E) 30 cm²



Which of the following numbers is greatest?

A) 2 + 0 + 0 + 3

B) 2 x 0 x 0 x 3

C) ( 2 + 0 ) x ( 0 + 3 )

D) 20 x 0 x 3

E) ( 2 x 0 ) + ( 0 x 3 )


Zosia is drawing flowers of different colors. The first flower is blue, then white, red, yellow,

and again blue, white, red, yellow, and so on in the same order. What is the color of the

twenty ninth flower drawn by Zosia?

A) Blue

B) White

C) Red

D) Pink

E) Yellow


How many integers are there on the number line between the numbers 2.09 and 15.3?

A) 13

B) 14

C) 11

D) 12

E) Infinitely many


The least positive integer which, is divisible by 2, 3, and 4, is:

A) 1

B) 6

C) 12





D) 24

E) 36


Two of the numbers located on the two circles (see the picture) are represented by letters A

and B. The sum of the numbers on each circle is equal to 55. What number is represented by

letter A?

A) 9

B) 10

C) 13

D) 16

E) 17


Tomek has 9 bills worth 100 zlotys each, 9 bills worth 10 zlotys each, and 10 coins worth 1

zloty each. How much money does Tomek have? (a zloty [zl] is a monetary unit in Poland)

A) 1,000 zl

B) 991 zl

C) 9, 910 zl

D) 9,901 zl

E) 99, 010 zl


A square with the length of side equal to x consists of a square with an area of 81 cm², two

rectangles with areas of 18 cm² each, and a small square. What is the value of x?

A) 2 cm

B) 7 cm

C) 9 cm




D) 10 cm

E) 11 cm


The value of the expression is equal to:

A) 2003

B)

C) 3

D)

E) 6009


Basia likes to add the digits that indicate the actual time on her electronic watch (for example,

when the watch shows 21:17, she gets the sum equal to 11). What is the greatest sum she can

get? (Hint: in some countries and sometimes in USA, instead of telling it is 1P.M., people say

it is 13:00. When it is 2P.M. they say it is 14:00, and when it is 12A.M., they say it is 24:00.

In this problem 21:17 means 9:17P.M. Time expressed with this method is called military

time sometimes.)

A) 24

B) 36

C) 19

D) 25

E) 28


The picture shows Clown Jan dancing on two balls and a cube. The radius of the lower ball is

6 dm, and the radius of the upper ball is three times shorter. The edge of the cube is 4 dm

longer than the radius of the upper ball. At what height is Jan dancing?

A) 14 dm

B) 20 dm

C) 22 dm




D) 24 dm

E) 28 dm


Let AC = 10 m, BD = 15 m, AD = 22 m (see the figure below). What is length of segment

BC is equal to?

A) 1 m

B) 2 m

C) 3 m

D) 4 m

E) 5 m


How many shortest distances along the edges of the cube are there that connect vertex A with

the opposite vertex B?

A) 4

B) 6

C) 3

D) 12

E) 16


From a square puzzle two pieces are cut out. These two pieces made the shaded region, (see

the figure). Among the four figures below, which are these two pieces?




A) 1 and 4

B) 2 and 4

C) 2 and 3

D) 1 and 3

E) 3 and 4


We add two different numbers chosen from the numbers: 1, 2, 3, 4, 5. How many different

sums can we get?

A) 5

B) 6

C) 7

D) 8

E) 9


The figure in the picture consists of 7 squares. Square A has the greatest area, and square B -

the smallest area. The lengths of two of the squares are given. How many B squares will it

take to fill up square A completely?

A) 16

B) 25

C) 36

D) 49

E) It is impossible.


A certain bar code consists of 17 black bars. A white bar divides each two black bars. The

first bar and the last bar in the code are black. There are two kinds of black bars: wide and

narrow. The number of white bars is 3 more than the number of wide black bars. How many

narrow black bars are there in this bar code?

A) 1

B) 2

C) 3




D) 4

E) 5


Ewa has 20 balls of four colors: yellow, green, blue, and black. 17 of them are not green, 5

are black, and 12 are not yellow. How many blue balls does Ewa have?

A) 3

B) 4

C) 6

D) 7

E) 8


There are 17 trees on one side of the street on Tomek's way from his house to school. One

day Tomek marked these trees with white chalk in the following way: on the way from his

house to the school he marked every other tree, starting with the first one. On his way back

home he marked every third tree, starting with the first one. How many trees were not

marked?

A) 4

B) 5

C) 6

D) 7

E) 8


Today the date is 3.20.2003 and the time is 20:03 (8:03 P.M.) What will be the date after

2003 minutes?

A) 3.21.2003

B) 3.22.2003

C) 3.23.2003

D) 4.21.2003

E) 4.22.2003


What is the digit of ones in the number 20032003

?

A) 7

B) 1

C) 9

D) 5

E) 3







With how many zeros does the product of the consecutive natural numbers from 1 to 50 end?

A) 5

B) 10

C) 12

D) 20

E) 50


The square ABCD consists of a white square and four shaded rectangles. Each of the

rectangles has a perimeter of 40 cm. What is the area of square ABCD?

A) 100 cm²

B) 200 cm²

C) 160 cm²

D) 400 cm²

E) 80 cm²


We have six segments with lengths: 1, 2, 3, 2001, 2002, 2003. In how many ways can we

select three of these segments to build a triangle?

A) 1

B) 3

C) 5

D) 6

E) 10


Piotrek is writing the numbers from 0 to 109 into a five-column table using a rule which is

easy to understand (see the picture below). Which of the pieces below can not be filled in




with numbers to fit Piotrek's table?


In the figure, the beginning part of the path from point A to point B is shown. How long is the

whole path?

A) 10,200 cm

B) 2,500 cm

C) 909 cm

D) 10,100 cm

E) 9,900 cm


At 3:00 o'clock the minute hand and the hour hand make a right angle. What will the measure

of the angle between these hands be after 10 minutes?

A) 90°

B) 30°

C) 80°

D) 60°

E) 35°





In the addition, every square stands for a certain digit, every triangle stands for another

specific digit, and every circle denotes yet another digit. What is the sum of the numbers

represented by the square and the circle?

A) 6

B) 7

C) 8

D) 9

E) 13


The shaded figure at the picture consists of five identical isosceles right triangles (see the

figure below). What is the area of the shaded figure?

A) 20 cm²

B) 25 cm²

C) 35 cm²

D) 45 cm²

E) 60 cm²


Red and green dragons lived in a cave. Every red dragon had 6 heads, 8 legs, and 2 tails.

Every green dragon had 8 heads, 6 legs, and 4 tails. There were 44 tails altogether, and there

were 6 less green legs than red heads. How many red dragons lived in the cave?

A) 6

B) 7

C) 8

D) 9

E) 10


Ania has 9 crayons in a box. At least one of them is blue. From every 4 crayons at least two

are of the same color, and from every 5 crayons at most three are of the same color. How

many blue crayons are in this box?




A) 2

B) 3

C) 4

D) 1

E) 5



The number 2002 read from left to right and from right to left is the same. Which number

from the numbers below does not have this property?

A) 1991

B) 2323

C) 2112

D) 2222

E) 4334


The picture below is a sketch of a castle. Which of the lines below does not belong to the

sketch?

A)

B)

C)

D)

E)


Mr. and Mrs. Kowalski have three daughters. Each of them has two brothers. How many

children does the Kowalski family have?




A) 9

B) 7

C) 6

D) 5

D) 11


In which number below is the square of the tens digit equal to the triple of the sum of the

digits of hundreds and ones?

A) 192

B) 741

C) 385

D) 138

E) 231


What is the product 22 · 22000 · 2 equal to? (· denotes multiplication)

A) 24000

B) 22002

C) 22003

D) 24002

E) 24001


On which string is the number of black hearts equal to two thirds of the number of all the

hearts on that string?

A)

B)

C)

D)

E)


Which of the numbers below is the greatest? (· denotes multiplication, : denotes division)





A) 10 · 0.001 · 100

B) 0.01 : 100

C) 100 : 0.01

D) 10,000 · 100 : 10

E) 0.1 · 0.01 · 10,000


What is the area of the figure in the picture below?

A) 43

B) 88

C) 58

D) 30

E) 15


The area of a certain rectangle is equal to 1 m². What is the area of a triangle that was cut off

from that rectangle along the line connecting the midpoints of the two adjacent sides?

A) 33 dm²

B) 25 dm²

C) 40 dm²

D) 3,750 cm²

E) 1,250 cm²


We subtracted the smallest three-digit number with all different digits from the greatest three-

digit number with all different digits. What is the result?

A) 864

B) 885

C) 800

D) 899

E) Other number






Figures I, II, III and IV are squares. The perimeter of square I is equal to 16 m, and the

perimeter of square II is equal to 24 m. What is the perimeter of square IV ?

A) 56 m

B) 60 m

C) 64 m

D) 72 m

E) 80 m


One medal can be cut out from a golden square plate. If four medals are made from four

plates, the remaining parts of those four plates can be used to make one more plate. What is

the largest number of medals that could be formed when 64 plates are used?

A) 85

B) 64

C) 80

D) 84

E) 100


Rectangle ABCD (see the picture) is built out of 24 little squares with the length of each side

equal to 1. What is the area of triangle ALM?

A) 5

B) 6

C) 7

D) 8

E) Other





In the picture below the coordinates of points A and B were indicated. What are the

coordinates of points C and D if AB = 2 BC, BC = 2 CD ?

A) 24 and 32

B) 24 and 28

C) 24 and 26

D) 22 and 24

E) 22 and 23


Mirek has 9 sticks with the lengths of 1 dm, 2 dm, 3 dm, 4 dm, 5 dm, 6 dm, 7 dm, 8 dm, 9

dm. With the sticks he builds triangles of which each side is built with one stick. How many

triangles with a side of 1 dm can be built with those sticks?

A) 6

B) 3

C) 2

D) 1

E) 0


How many convex angles with different measures are made by the rays with P as the starting

point (see the picture)?

A) 4

B) 6

C) 8

D) 10

E) 11


How many different three-digit numbers divisible by 25 can be made with the digits 0, 3, 5, 7

if the digits can be repeated?

A) 16

B) 9

C) 81




D) 64

E) 3


Each boy: Mietek, Mirek, Pawel, and Zbyszek has exactly one of the following animals: a

cat, a dog, a gold fish, and a canary-bird. Mirek has a pet with fur. Zbyszek has a pet with

four legs. Pawel has a bird, and Mietek and Mirek don't like cats. Which of the following

sentences is not true?

A) Zbyszek has a dog.

B) Pawel has a canary.

C) Mietek has a golden fish.

D) Zbyszek has a cat.

E) Mirek has a dog.


The next day after his birthday Jas said: "The day after tomorrow will be Thursday." On what

day of the week did Jas have his birthday?

A) On Monday

B) On Tuesday

C) On Wednesday

D) On Thursday

E) On Friday


In the picture below, the area of triangle ABD is equal to 15, the area of triangle ABC is

equal to 12 and the area of triangle ABE is equal to 4. What is the area of pentagon ABCED?

A) 19

B) 31

C) 23

D) 27

E) 35


The weight of each possible pair of boys from a group of 5 was recorded. The following

results were obtained: 90 kg, 92 kg, 93 kg, 94 kg, 95 kg, 96 kg, 97 kg, 98 kg, 100 kg and 101

kg. What is the total weight of all five boys?





A) 225 kg

B) 230 kg

C) 239 kg

D) 240 kg

E) 250 kg


There are four congruent squares. In each of them the midpoints of the sides are indicated and

some regions with areas S1, S2, S3 and S4 are shaded. Which expression below is true?

A) S3 < S4 < S1 = S2

B) S3 < S1= S2 = S4

C) S3 < S1 = S4 < S2

D) S3 < S4 < S1 < S2

E) S4 < S3 < S1 < S2


You count from 1 to 100 and you clap when you say the multiples of number 3 and the

numbers that are not multiples of 3 but have 3 as the last digit. How many times will you clap

your hands?

A) 30

B) 33

C) 36

D) 39

E) 43


The cyclist went up the hill with the speed of 12 km/h and went down the hill with the speed

of 20 km/h. The ride up the hill took him 16 minutes longer than the ride down the hill. How

many minutes did the cyclist take to go down the hill?

A) 24

B) 40

C) 32

D) 16

E) 28






Symbols P, Q, R, S indicate the total weight of the figures drawn above them.

It is known that any two figures of the same shape have the same weight. If P < Q < R, then:

A) P < S < Q

B) Q < S < R

C) S < P

D) R < S

E) R = S


Ada has 14 gray balls, 8 white balls and 6 black balls in a bag. What is the least number of

the balls she has to take out of her bag having her eyes closed to make sure that she took at

least one ball of each color?

A) 23

B) 22

C) 21

D) 15

E) 9 A


A computer virus destroys computer memory. On the first day it destroyed of this

memory. On the second day it destroyed of the memory remaining after the first day; on

the third day it destroyed of the memory remaining after two days and on the fourth day it

destroyed of the memory remaining after three days. What part of all the computer

memory was left after those four days?

A)

B)

C)

D)

E)




What is the greatest value of the sum of the digits of the number made from the sum of the

digits of a three-digit number?

A) 9

B) 10

C) 11

D) 12

E) 18


In the chess competition 32 players were competing. The competition was taking place by

steps. In each step all the players were divided into groups of four. In each of these groups

every player played once with every other player. The two best players from the group went

to the next level and the two worst players were out of the competition. After the step in

which four last players played, the two best players were playing an additional final game.

How many games were played during the whole competition?

A) 49

B) 89

C) 91

D) 97

E) 181


A net with 32 hexagonal spaces in three rows was made out of matches (see the picture.)

How many matches were used to make this net?

A) 123

B) 124

C) 125

D) 120

E) 121





Grade 07-08


2005 · 5002 =

A) 1291

B) 102910

C) 10029010

D) 1000290010

E) 100002900010 ( · denotes multiplication)


How many hours are there in half of a third part of a quarter of a day?

A)

B)

C) 1

D) 2

E) 3


The edge of the cube is 12 cm long. The ant moves on the cube surface from point A to point

B along the path shown in the figure. Find the length of the ant's path.

A) 60cm

B) 50cm

C) 48cm

D) 40cm

E) It cannot be determined.


The sum of the volume of three pitchers and two bottles equals 16 liters. The volume of each

pitcher is two times greater than the volume of each bottle. What is the sum of the volume of

two pitchers and three bottles?

A) 12 liters

B) 13 liters





C) 14 liters

D) 16 liters

E) 17 liters


At our school, 50% of the students have bikes. Of the students who have bikes, 30% have

skateboards. What percent of the students at our school have both a bike and a skateboard?

A) 15

B) 20

C) 25

D) 40

E) 80


In triangle ABC, the measure of the angle at vertex A is three times the measure of the angle

at vertex B and half the measure of the angle at vertex C. What is the measure of the angle at

vertex A?

A) 30º

B) 36º

C) 54º

D) 60º

E) 72º


How many three-digit numbers are there in which all the digits are even?

A) 25

B) 64

C) 75

D) 100

E) 125


The diagram shows the floor plan of a room. Adjacent walls are perpendicular to each other.

Letters a and b represent the lengths of some the walls. What is the area of the room?





A) 2ab + a(b-a)

B) 3a(a+b) - a²

C) 3a²b

D) 3a(b-a) + a²

E) 3ab


In the diagram, the five circles have the same radii, and they touch as shown. The small

square joins the centres of the four outer circles. What is the ratio of the shaded area of all the

circles to the non-shaded area of all the circles?

A) 2 : 3

B) 1 : 3

C) 2 : 5

D) 5 : 4

E) 1 : 4


There was a certain number of crows sitting on trees in the garden. If there had been just one

crow sitting on each tree, then one crow would not have had a tree to sit on. However, if two

crows had been sitting on each tree, then there would not be any crows on one tree. How

many trees were there in the garden?

A) 2

B) 3

C) 4

D) 5

E) 6


Anna, Barbara, Teddy, and Wally went to a dance. They danced in pairs. Anna danced with

Teddy and with Wally. Barbara danced with Teddy, but she didn't dance with Wally. Decide

which statement is false.

A) Each of the girls danced with one of the two boys.

B) One of the two girls didn't dance with one of the two boys.

C) One of the boys danced with both girls.

D) Each of the boys danced with one of the two girls.

E) One of the boys didn't dance with any of the two girls.





A group of classmates was planning a trip. If each of them paid $14, then they would be $4

short to pay for the trip. On the other hand, if each of them paid $16, they would have $6

more than they needed. How much should each of the classmates contribute so they collect

the exact amount needed for the trip?

A) $14.40

B) $14.60

C) $14.80

D) $15.00

E) $15.20


Carla cut a sheet of paper into 10 pieces. Then she took one piece and cut it again into 10

pieces. She then repeated this three more times. How many pieces of paper did she have after

the last cutting?

A) 36

B) 40

C) 46

D) 50

E) 56


A doorman works according the following schedule: he works for 4 consecutive days and has

the fifth day off. Last Sunday he had the day off, and on Monday he started work according

to his schedule. After how many days, including that Monday, will he have a day off on

Sunday again?

A) 30

B) 36

C) 12

D) 34

E) 7


Two rectangles ABCD and DBEF are shown in the picture. What is the area of rectangle

DBEF?





A) 10 cm²

B) 12 cm²

C) 13 cm²

D) 14 cm²

E) 16 cm²


What is the measure of angle indicated in the picture?

A) 110º

B) 115º

C) 120º

D) 126º

E) 130º


From noon until midnight, Clever Cat sleeps under the oak tree, and from midnight until

noon he tells stories. There is a sign on the oak tree saying: "Two hours ago Clever Cat was

doing the same thing that he will be doing in an hour." How many hours a day is the

information given on the sign true?

A) 6

B) 12

C) 18

D) 3

E) 21


The diagram shows an equilateral triangle and a regular pentagon. What is the measure of

angle x?

A) 124º

B) 128º

C) 132º




D) 136º

E) 140º


Which of the numbers below is the sum of four consecutive whole numbers?

A) 15

B) 2000

C) 2002

D) 2004

E) 2005


Each pair vertices of a cube are connected with a segment. How many different points are

there that are the midpoints of these segments?

A) 8

B) 12

C) 16

D) 19

E) 28


Let's call the number of prime factors of a natural number n, the product of which is equal to

the given natural number n, "the length". For example, the length of 90 = 2x3x3x5 equals 4.

How many odd numbers less than 100 have the length of 3?

A) 2

B) 3

C) 5

D) 7

E) Other number.


In each of the four small squares shown in the picture, a different natural odd number less

than 20 was written. Only one of the statements below is true. Which one is it?

A)The sum of the numbers that are inscribed in the square equals 66.

B)The sum of the numbers that are inscribed in the square equals 12.

C)Product of all the numbers inscribed in the square equals 2005.





D)The product of the numbers on each diagonal equals 21.

E)The sum of the number on each diagonal equals 20.


The sequence of letters AGKNORU in alphabetical order corresponds to a sequence of

different digits placed in increasing order. What is the greatest number that corresponds with

the word KANGOUROU?

A) 987654321

B) 987654354

C) 436479879

D) 536479879

E) 597354354


Peter wrote down all three-digit numbers which have the following features: each number

consists of three different digits, and the first digit is equal to the second power of the

quotient of the second and the third digit. How many numbers did Peter write down?

A) 8

B) 4

C) 3

D) 2

E) 1


Five lines: l1, l2, l3, l4, l5 intersect point O and they are intersected by five other lines k1, k2,

k3, k4, k5 (see the picture). What is the sum of the measures of 10 shaded angles, shown in

the picture?

A) 300

B) 450




C) 360

D) 600

E) 720


There were 64 liters of juice in a barrel. Then, 16 liters of juice were dumped out and

replaced with 16 liters of water. After being mixed together, again 16 liters of the mixture

were dumped and replaced with 16 liters of water. After being mixed together, this was done

again, 16 liters of the mixture were dumped and replaced with water. How many liters of

juice are there in the mixture now?

A) 27

B) 24

C) 16

D) 30

E) 48


How many two-digit numbers with the ones digit greater than zero are there that are greater

than three times the number that is created out of these numbers by reversing its digits?

A) 6

B) 10

C) 15

D) 22

E) 33


ABC is a right triangle. AH is the height of the triangle and AK is a bisector of right angle A.

If the ratio CK : KB equals then what is the ratio of CH : HB?

A) 1 : 3

B) 1 : 9

C) 1 :

D) 1 : 6

E) 1 : 4






If the average of 10 different positive integers is 10, how large can the greatest of these

numbers be?

A) 91

B) 55

C) 50

D) 45

E) 10


A particle moves through the first quadrant of the figure as follows: during the first minute it

moves from the origin to (1,0). Then, it continues to follow the pattern indicated in the figure,

going back and forth between the positive x and y axes, moving one unit of distance parallel

to an axis in one minute. Which point will the particle reach after exactly 2 hours?

A) (10,0)

B) (1,11)

C) (10,11)

D) (2,10)

E) (11,11)



What is the value of the expression: 2004 - 200 · 4?

A) 400,800

B) 0

C) 1204

D) 1200

E) 2804


Tom has $147 and Stan has $57. How much money does Tom need to give to Stan, so that he

would have twice as much money left as Stan would have then?




A) $11

B) $19

C) $30

D) $45

E) $49


What is the remainder when dividing the sum: 2001 + 2002 + 2003 + 2004 + 2005 by 2004?

A) 1

B) 2001

C) 2002

D) 2003

E) 1999


In each of the little squares Karolina places one of the digits: 1, 2, 3, 4. She makes sure that in

each row and each column each of these numbers is placed. In the figure below, you can see

the way she started. In how many ways can she fill the square marked with an x?

1 x

4 1

3

2

A) None

B) 1

C) 2

D) 3

E) 4


What is the value of the expression: (1 - 2) - (3 - 4) - (5 - 6) - (7 - 8) - (9 - 10) - (11 - 12)?

A) -6

B) 0

C) 4

D) 6

E) 13






A section was made on a cube. On the net of the cube this section was indicated with a

perforated line (see the figure). What figure was made by the section?

A) Equilateral triangle

B) A rectangle but not a square

C) Right triangle

D) Square

E) Hexagon


By how much does the area of a rectagle increase if its length and the width are increased by

10% each?

A) 10%

B) 20%

C) 21%

D) 100%

E) 121%


What is the length of the diameter of the circle shown in the figure?

A) 18

B) 16

C) 10

D) 12

E) 14


An ice cream stand was selling ice cream in five different flavors. A group of children came

to the stand and each child bought two scoops of ice cream with two different flavors. If none

of the children chose the same combination of flavors and every such combination of flavors

was chosen, how many children were there?




A) 5

B) 10

C) 20

D) 25

E) 30


The number x was multiplied by 0.5 and the product was divided by 3. The result was

squared and 1 was added to it. The final result was 50. What was the value of number x?

A) 18

B) 24

C) 30

D) 40

E) 42


Alfonso the ostrich was training for the Head in the Sand Competition in the Animal

Olympiad. He put his head in the sand at 8:15 on Monday morning and reached his new

personal record by keeping it underground for 98 hours and 56 minutes. When did Alfonso

pull his head out of the sand?

A) On Thursday at 5:19 A.M.

B) On Thursday at 5:41 A.M.

C) On Thursday at 11:11 A.M.

D) On Friday at 5:19 A.M.

E) On Friday at 11:11 A.M.


Two semicircles with diameters AB and AD were inscribed in square ABCD (see the figure).

If |AB| = 2, then what is the area of the shaded region?




A) 1

B) 2

C)

D) 2

E)


If a and b are positive integers, neither of which is divisible by 10, and if a · b = 10,000 then

what is the sum a + b?

A) 1024

B) 641

C) 1258

D) 2401

E) 1000


There were more Thursdays than Tuesdays in the first of two consecutive years. Which day

of the week appeared the most in the second year, if neither of these years was a leap year?

A) Tuesday

B) Wednesday

C) Friday

D) Saturday

E) Sunday


Isosceles triangle ABC satisfies: |AB| = |AC| = 5, and angle BAC > 60°. The length of the

perimeter of this triangle is expressed with a whole number. How many triangles of that kind

are there?

A) 1

B) 2

C) 3

D) 4

E) 5


How many divisors does number 2 x 3 x 5 x 7 x 11 have?

A) 2310

B) 10

C) 5





D) 2004

E) 32


Tad has a large number of building blocks which are rectangular prisms with dimensions 1 x

2 x 3. What is the smallest number of blocks needed to build a solid cube?

A) 12

B) 18

C) 24

D) 36

E) 60


Each of 5 children wrote one of the numbers: 1, 2, 4 on the board. Then the written numbers

were multiplied. Which number can be the product of those numbers?

A) 100

B) 120

C) 256

D) 768

E) 2048


The average age of a grandmother, a grandfather and 7 grandchildren is 28. The average age

of 7 grandchildren is 15 years. How old is the grandfather, if he is 3 years older than the

grandmother?

A) 71

B) 72

C) 73

D) 74

E) 75


The equilateral triangle ACD is rotated counterclockwise around point A. What is the angle

of rotation when triangle ACD covers triangle ABC the first time?





A) 60°

B) 120°

C) 180°

D) 240°

E) 300°


There are at least two kangaroos in the enclosure. One of them said: "There are 6 of us here"

and he jumped out of the enclosure. Afterwards, every minute one kangaroo was jumping out

of the enclosure saying: "Everybody who jumped out before me was lying." This continued

until there were no kangaroos left in the enclosure. How many kangaroos were telling the

truth?

A) 0

B) 1

C) 2

D) 3

E) 4


Points A and B are placed on a line which connects the midpoints of two opposite sides of a

square with side of 6 cm (see the figure). When you draw lines from A and B to two opposite

vertices, you divide the square in three parts of equal area. What is the length of segment

AB?

A) 3.6 cm

B) 3.8 cm

C) 4.0 cm

D) 4.2 cm

E) 4.4 cm


Jack rides his bike from home to school uphill with average speed of 10 km/h. On the way

back home his speed is 30km/h. What is the average speed of his round trip?

A) 12 km/h

B) 15 km/h

C) 20 km/h

D) 22 km/h

E) 25km/h





John put magazines on a bookshelf. They have either 48 or 52 pages. Which one of the

following numbers cannot be the total number of pages of all the magazines on the

bookshelf?

A) 500

B) 524

C) 568

D) 588

E) 620


Inside the little squares of a big square the consecutive natural numbers were placed in a way

shown in the picture. Which of the following numbers cannot be placed in square x?

A) 128

B) 256

C) 81

D) 121

E) 400


In the figure there are 11 boxes. Number 7 was written in the first box and number 6 was

written in the ninth box. What was the number placed in the second field with the following

condition: the sums of each three consecutive numbers in the boxes are equal to 21?

A) 7

B) 10

C) 8

D) 6

E) 21


For each triple of numbers (a, b, c) another triple of numbers (b + c, c + a, a + b) was created.

This was called operation. 2004 such operations were made starting with numbers (1, 3, 5),

and resulting with numbers (x, y, z). What is the difference x - y equal to?





A) -2

B) 2

C) 4008

D) 2004

E) (-2)2004


Number 2004 is divisible by 12 and the sum of its digits is equal to 6. Altogether, how many

four-digit numbers have these two properties?

A) 10

B) 12

C) 13

D) 15

E) 18


Rings with dimensions shown in the figure were linked together, forming 1.7m long chain.

How many rings were used to create the chain?

A) 30

B) 21

C) 42

D) 85

E) 17


On each face of a cube a certain natural number was written, and at each vertex a number

equal to the product of the numbers on the three faces adjacent to that vertex was placed. If

the sum of the numbers on the vertices is 70 then what is the sum of the numbers on all the

faces of the cube?

A) 12

B) 35

C) 14

D) 10

E) Cannot be determined.






A segment of a length equal to 4 was divided with 4 points into segments of equal length.

How long is each segment?

A) 0.4

B) 1

C) 0.8

D) 0.5

E) 0.6


A square built of 16 small squares is cut with a line. What is the greatest number of the little

squares that the line can go through?

A) 3

B) 4

C) 6

D) 7

E) 8


When 29 is subtracted from the greatest 2-digit number and the difference is divided by the

smallest 2-digit number what is the result?

A) 11

B) 9

C) 7

D) 10

E) 6


If , then is equal to?

A) 9

B) 2

C) 10

D) 3

E) 15






Tomek saved 120 zl. One day he bought a present for his brother and he spent 1/3 of all his

money. The next day he bought a book for himself and he spent 1/4 of the remaining money.

How much money did he have left after shopping? (zl is a monetary unit in Poland like dollar

in USA)

A) 50 zl

B) 80 zl

C) 70 zl

D) 20 zl

E) 60 zl


A rectangular prism was made out of three blocks, each consisting of four cubes (see the

picture). Which of the blocks below has the same shape as the white block?


There are 17 trees on one side of the street on Tomek's way from his house to school. One

day Tomek marked these trees with white chalk in the following way: on the way from his

house to the school he marked every other tree, starting with the first one. On his way back

home he marked every third tree, starting with the first one. How many trees were not

marked?

A) 4

B) 5

C) 6

D) 7

E) 8


In triangle ABC : DA = DB = DC (see the figure). Then:





A) Angle ACB is obtuse.

B) Angle ACB is acute.

C) Angle ACB is right.

D) Angle ACB changes, depending on the lengths of the sides AC and BC.

E) This kind of triangle ABC does not exist.


There were 5 parrots in a pet store. The average price of each of them was 600 zl. One day,

the most beautiful parrot was sold. The average price of each of the remaining 4 birds was

500 zl.What was the price of the sold parrot? (zl is a monetary unit in Poland like dollar in

USA)

A) 100 zl

B) 200 zl

C) 550 zl

D) 600 zl

E) 1000 zl


What is the greatest number of inner right angles that a hexagon can have (not necessary a

convex hexagon)?

A) 2

B) 3

C) 4

D) 5

E) 6


How many integers are there, the squares of which are found between the numbers -100 and

100, inclusive?

A) 11

B) 22

C) 21

D) 20

E) 10


Five girls Ania, Beata, Celina, Dorota, and Ela, received the following assignment: they were

supposed to draw four segments and to count all the points of intersection of these segments.

After they finished, Ania said she counted 2 points, Beata 3 points, Celina 5 points, Dorota 6

points, Ela 7 points. Who made a mistake?

A) Ania

B) Beata





C) Celina

D) Dorota

E) Ela


A cube was made from the given configuration (see the figure). Which wall will be opposite

to the wall with the letter x?

A) a

B) b

C) c

D) d

E) e


A square piece of paper is folded twice and cut in the way you can see in the picture. How

will the piece of paper look after unfolding?


In how many ways can you express the number 2003 as a sum of two prime numbers?

A) 1

B) 2

C) 3

D) 4

E) This expression doesn't exist





Which of the following sums is equal to 2003?

A)162 + 262 + 322

B)162 + 262 + 332

C)152 + 272 + 322

D)172 + 252 + 332

E)172 + 242 + 342


An empty truck weighs 2000 kg. After the truck was loaded, freight made up 80% of the

weight of the loaded truck. At the first stop one fourth of the freight was unloaded. What

percent of the loaded truck's weight did the load make up after that? (Hint: freight = load.)

A) 20%

B) 25%

C) 55%

D) 60%

E) 75%


The combined capacity of a bottle and a glass is equal to the capacity of a pitcher. The

capacity of a bottle is equal to the combined capacity of a glass and a mug. The combined

capacity of three mugs is equal to the combined capacity of two pitchers. How many glasses

altogether have the capacity of one mug?

A) 3

B) 4

C) 5

D) 6

E) 7


Michal has 42 identical cubical blocks, each one with an edge of 1 cm. From all of these

blocks, he built a rectangular prism with a base perimeter equal to 18 cm. What is the height

of the prism which he built?

A) 1 cm

B) 2 cm

C) 3 cm

D) 4 cm

E) 5 cm







A fresh mushroom contains 90% water. How many kilograms of fresh mushrooms are needed

in order to make 1 kg of dried mushrooms containing 11% water?

A) 8.9

B) 79

C) 1.01

D) 8.18

E) 9.09


There are five men in a room. Each of them is either a liar that always lies or a knight that

always tells the truth. Each of them was asked the question: "How many liars are among

you?" The answers were: "one," "two," "three," "four," "five." How many liars were in that

room?

A) 1

B) 2

C) 3

D) 4

E) 5


In rectangle ABCD, points P, Q, R, S are respectively the midpoints of sides AB, BC, CD,

and DA. Let T be the midpoint of segment SR. What part of the area of rectangle ABCD is

the area of triangle PQT?

A)

B)

C)

D)

E)





There are six segments with lengths: 1, 2, 3, 2001, 2002, 2003. In how many ways can we

select three of these segments to build a triangle?

A) 1

B) 3

C) 5

D) 6

E) 10


Six consecutive points were indicated on a number line in this order: A, B, C, D, E, and F.

Regardless of the location of these points, if only AD = CF and BD = DF, then the following

equation is true:

A) AB = BC

B) BC = DE

C) BD = EF

D) AB = CD

E) CD = EF


In the picture four squares are shown and the lengths of their sides are indicated. What is the

difference between the combined area of the shaded regions and the combined area of the

black regions?

A) 25

B) 36

C) 44

D) 64

E) 0


Which of the following numbers, after it is multiplied by 768, yields a product that ends with

the largest number of zeros?

A) 7,500

B) 5,000

C) 3,125




D) 2,500

E) 10,000


A square was divided into 25 identical small squares (see the picture). What is the sum of the

measures of angles < MAN, < MBN, < MCN, < MDN, < MEN?

A) 300

B) 450

C) 600

D) 750

E) 900


How many natural numbers n have such a property that out of all the positive divisors of

number n, which are different from both 1 and n, the greatest one is 15 times greater than the

smallest one?

A) 1

B) 2

C) 3

D) There are no such numbers.

E) Infinitely many.


In a number with at least two digits, the last digit was deleted. The resulting number was n

times smaller than the previous one. What is the greatest possible value of n?

A) 9

B) 10

C) 11

D) 19

E) 20






How many natural numbers n have the property that the remainder of dividing 2003 by n is

equal to 23?

A) 22

B) 19

C) 13

D) 12

E) 36



This year the International Competition in Mathematics "Kangaroo" takes places on March

21st. How many prime numbers divide the number 21?

A) 2

B) 3

C) 4

D) 1

E) 21


Which of the fractions below is the greatest?

A) B) C) D) E)


You count from 1 to 100 and you clap while saying the multiples of the number 3 and the

numbers that are not the multiples of three but have 3 as the last digit. How many times will

you clap your hands?

A) 30

B) 33

C) 36

D) 39

E) 43


On July 1st in Newbury the sun will rise at 4:53 A.M. and set at 9:25 P.M. In the middle of

that period of time there is so called local noon. At what time will the local noon be in

Newbury on July 1st?





A) At 12:00 P.M.

B) At 12:39 P.M.

C) At 1:09 P.M.

D) At 4:32 P.M.

E) At 11:08 A.M.


Points K, L, M, N are the midpoints of the sides of rectangle ABCD and points O, P, R, S are

the midpoints of the sides of rhombus KLMN. What is the ratio of the area of shaded figure

to the area of rectangle ABCD?

A) B) C) D) E)


Ada has 7 gray balls, 4 white balls and 3 black balls in a bag. What is the least number of the

balls she has to take out of her bag, having her eyes covered, to make sure that she took out at

least one ball of each color?

A) 12

B) 11

C) 10

D) 4

E) 3


A certain charity organization decided to buy 2002 notebooks. The warehouse was selling

boxes of 24 notebooks. What is the least number of boxes that the warehouse should buy in

order to have 2002 notebooks, and by what number will the number of 2002 notebooks be

exceeded?

A) 83 boxes, 10 notebooks

B) 84 boxes, 10 notebooks

C) 83 boxes, 14 notebooks

D) 84 boxes, 16 notebooks

E) 84 boxes, 14 notebooks


Which of the expressions below cannot have the value of 2002, if a and b represent the

natural numbers?





A) 7a + 7b

B) 13a + 13b

C) 17a + 17b

D) 11(2a + 7b)

E) 28a + 14b


Each boy: Mietek, Mirek, Pawel and Zbyszek has exactly one of four animals: a cat, a dog, a

gold fish and a canary. Mirek has an animal with fur, Zbyszek has an animal with four legs,

Pawel has a bird and Mietek and Mirek don't like cats. Which of the statements below is not

true?

A) Zbyszek has a dog

B) Pawel has a canary

C) Mietek has a gold fish D) Zbyszek has a cat

E) Mirek has a dog


A basket of oranges costs 20 zloty, a basket of pears costs 30 zloty and a basket of kiwi fruits

costs 40 zloty. Eight baskets of these fruits were bought for 230 zloty. What is the largest

possible number of baskets of kiwi fruits that were bought?

A) 1

B) 2

C) 3

D) 4

E) 5


If a : b = 9 : 4 and b : c = 5 : 3 then (a - b) : (b - c) is equal to:

A) 4 : 1

B) 25 :8

C) 7:12

D) 5 : 2

E) It cannot be determined


Before going to a summer camp, the scouts from Torun packed provisions that were

sufficient for them for 30 days. At the last minute 15 scouts from Bydgoszcz wanted to go to

the summer camp with the scouts from Torun. Now the provisions already made were

sufficient for just 25 days provided the daily amount of food allotted for one scout would not

change. How many scouts from Torun were planning to go for that summer camp?

A) 15

B) 20





C) 55

D) 70

E) 75


There were 25% boys and 75% girls among all the students taking part in the school event.

Half of the boys and 20% of the girls, together 99 students, had blue eyes. How many

students were taking part in the school event?

A) 360

B) 340

C) 240

D) Other answer



Points P and Q are the centers of two outside tangent circles (see the picture.) The line going

through points P and Q intersects these circles at points A and B. If the area of rectangle

ABCD is 15 then what is the area of triangle PQT?

A) 4 B) C) D) 5 E) 2


The weight of each possible pair of boys from a group of 5 was recorded. The following

results were obtained: 90 kg, 92 kg, 93 kg, 94 kg, 95 kg, 96 kg, 97 kg, 98 kg, 100 kg and 101

kg. What is the total weight of the five boys?

A) 225 kg

B) 230 kg

C) 239 kg

D) 240 kg

E) 250 kg


Four daughters, Ania, Basia, Celina and Danusia, bought together a single present for their

dad. One of the girls hid the present. Mom asked which of them had done it. Ania and Basia

said: "I did not do it." Celina: "Danusia did it." Danusia: "Basia did it." It turned out that only

one of the girls lied. Who of them hid the present?





A) Ania

B) Basia

C) Celina

D) Danusia



In one country a part of the residents can speak English only, a part can speak French only

and the rest can speak both languages. It is known that 85% residents can speak English, 75%

can speak French. What percent of the residents of this country can speak both English and

French?

A) 50%

B) 57%

C) 25%

D) 60%

E) 40%


The symbols P, Q, R, S indicate the total weight of the figures drawn above them (see the

picture):

It is known that any two figures of the same shape have the same weight. If P < Q < R then:

A) P < S < Q

B) Q < S < R

C) S <P

D) R < S

E) R = S


Triangle ABC is an isosceles triangle, |AC| = |BC|, <ACB = 36°. Triangles BDA and EBD

are also isosceles triangles and |AB| = |AD|, |DE| = |DB|. What is the measure of <DEB?

A) 90°

B) 18°

C) 36°




D) 54°

E) 72°


Out of the net shown in the picture the cube was made. What is the greatest sum of the dots

on three sides with a common vertex?

A) 15

B) 14

C) 13

D) 12

E) Other answer


It is known that the positive whole number n is divisible by 21 and by 9. Which of the

answers below can be the number of divisors of the number n?

A) 3

B) 4

C) 5

D) 6

E) 7


In some of the segments of the rectangular diagram with the dimensions 2 x 9 (see the picture

below) there are coins. Each of the segment either has a coin in it or has a side common with

the segment containing a coin. What is the least number of coins that must be in this

diagram?

A) 5

B) 6

C) 7

D) 8

E) 9


In one month three Sundays were on even dates. What day of the week was the 20th day of

the month?





A) Monday

B) Tuesday

C) Wednesday

D) Thursday

E) Saturday


The front face of the clock cracked into three parts in such a way that in each part the sum of

the numbers indicating the hours was the same. Knowing that none of the lines along which

the crack happened divides the digits of the number we can say that:

A) 12 and 3 are not in the same part

B) 8 and 4 are in the same part

C) 7 and 5 are not in the same part

D) 11, 1 and 5 are in the same part

E) 2, 11 and 9 are in the same part


Following the teacher's assignment, the students were drawing two circles and three lines on

the piece of paper. After that each of them was counting the points of the intersection of these

lines. What is the biggest possible number of such points?

A) 18

B) 17

C) 16

D) 15

E) 14


A square piece of paper was folded into a pentagon in the following way: first we fold the

square in a way so that the vertices B and D would go into one point laying on the diagonal

AC (see picture 2), and then we fold the resulting quadrilateral in a way so that point C would

go into point A (see picture 3.) What is the measure of the angle with the question mark?

Picture 1 ... Picture 2 ... Picture 3




A) 104°

B) 106° 30'

C) 108°

D) 112° 30'

E) 114° 30'


A solid was made out of 112 identical cubes. The solid is a cube with three tunnels drilled

through it as you can see in the picture. After the glue dried the solid was dipped into a dish

with paint. How many little cubes have exactly one side painted?

A) 30

B) 26

C) 40

D) 48

E) 24


With the digits 1, 2, 3, 4 all possible four-digit numbers with all different digits were made.

What is the sum of all these numbers?

A) 55,550

B) 99,990

C) 66,660

D) 100,000

E) 98,760


In the picture DC = AC = 1 and CB = CE = 4. If the area of triangle ABC is equal to S then

what is the area of the quadrilateral AFDC?




A) B) C) D) E)


During math class the teacher wrote number 1 on the board and asked Tomek to write down

any other natural number. Then other students were coming up to the board and each of them

wrote a number that was the sum of all the numbers written before. At some point Piotr wrote

the number 1000. Which of the numbers below could not Tomek write?

A) 999

B) 499

C) 299

D) 249

E) 124



H T T P : / / M A T H K A N G A R O O F L O R I D A . B L O G S P O T . C O . U K /

M O N D A Y , M A R C H 2 , 2 0 0 9

Week 4 - Problem 4

Figures I, II, III and IV are squares. The circumference of square I is 16m and the circumference

of square II is 24m. Find the circumference of square IV.

A. 56m B. 60m C. 64m D. 72m E. 80m

Week 4 - Problem 3

The area of a rectangle equals 1. What is the area of the triangle, which is cut off from the

rectangle by the line connecting the midpoints of the two adjacent sides?

A. 1/3 B. 1/4 C. 2/5 D. 3/8 E. 1/8

S A T U R D A Y , F E B R U A R Y 2 8 , 2 0 0 9

Week 4 - Problem 2

The combination for opening a safe is a three – digit number made up of different digits. How

many different combinations can you make using only digits 1, 3, and 5?

A) 2 B) 3 C) 4 D) 5 E) 6

Week 4 - Problem 1

The numbers 34 and 142 have the same sum of their digits (3+4=7 and 1+4+2=7). What is the

first number greater than 2007 such that the sum of its digits is the same as the sum of the digits

of 2007?

A) 2016 B) 2115 C) 2008 D) 7002 E) 2070

M O N D A Y , F E B R U A R Y 2 3 , 2 0 0 9

Week 3 - Problem 6

The fraction

equals to

a) 2003 b) 1/3 c) 3 d) 5/2 e) None

Week 3 - Problem 5

Betty likes calculating the sum of the digits that she sees on her digital clock (for instance, if

the clock shows 21:17, then Betty gets 11). What is the biggest sum she can get if the clock is a

24-hour clock?

http://mathkangarooflorida.blogspot.co.uk/

http://mathkangarooflorida.blogspot.co.uk/2009/03/week-5-problem-4.html






http://4.bp.blogspot.com/_ukVQRanRu2w/Sa1meY973bI/AAAAAAAAABg/XVJitjfBcPA/s1600-h/squares.jpg

http://4.bp.blogspot.com/_ukVQRanRu2w/Sa1meY973bI/AAAAAAAAABg/XVJitjfBcPA/s1600-h/squares.jpg

A) 24 B) 36 C) 19 D) 25 E) another answer

S A T U R D A Y , F E B R U A R Y 2 1 , 2 0 0 9

Week 3 - Problem 4

Rachel opened her math book and found that the sum of the facing pages was 243. What pages

did she open to?

Week 3 - Problem 3

A math student interviewed 50 fifth graders. 41 students said they liked peanut butter

sandwiches, 35 liked jam sandwiches, and 30 liked both on their sandwiches. How many

students like neither?

T H U R S D A Y , F E B R U A R Y 1 9 , 2 0 0 9

Week 3 - Problem 2

Sophie draws kangaroos: a blue one, then a green, then a red, then a black, a blue, a green, a

red, a black, and so on…What color is the 29th kangaroo?

A) blue B) green C) red D) black E) it’s impossible to know

Week 3 - Problem 1

The sum of the numbers in each ring below should be 55. What is the value of A?

A) 9 B) 10 C) 13 D) 16 E) 17

W E D N E S D A Y , F E B R U A R Y 1 8 , 2 0 0 9

Challenge Problem

In the addition problem below, different letters represent different digits. What digit does A

represent?

A A

+ A A

____

C A B

Challenge Problem

http://mathkangarooflorida.blogspot.co.uk/2009/02/rachel-opened-her-math-book-and-found.html




http://mathkangarooflorida.blogspot.co.uk/2009/02/challenge-problem_18.html

http://mathkangarooflorida.blogspot.co.uk/2009/02/challenge-problem.html

Mr. Chin went to a store where he spent one-half of his money and then $14 more. He then went

to another store where he spent one-third of his remaining money and then $14 more. He then

had no money left. How much did he have when he entered the first store?

T U E S D A Y , F E B R U A R Y 1 7 , 2 0 0 9

Week 2 - Problem 6

Anna wrote a 2-digit number. Ben created a 4-digit number by copying Anna's number

twice. Then Anna divided Ben's number by her number. What was the result she got?

A) 100 B) 101 C) 1000 D) 1001 E) 10

Week 2 - Problem 5

Bill thought of an integer number. Nick multiplied this number either by 5 or by 6. John

added either 5 or 6 to Nick’s result. Last, Andrew subtracted either 5 or 6 from John’s result.

The final result obtained was 73. What was Bill’s number?

A) 10 B) 11 C) 12 D) 14 E) 15

M O N D A Y , F E B R U A R Y 1 6 , 2 0 0 9

Week 2 - Problem 4

Harry Potter let an owl out at 7:30 a.m. to deliver an important message to his friend Ron.

The owl delivered the envelope at 9:10 a.m. An owl flies 4 km in 10 minutes. What was

the distance between Harry and Ron?

A) 14 km B) 20 km C) 40 km D) 56 km E) 64 km

Week 2 - Problem 3

What is the 2007th letter in the sequence KANGAROOKANGAROOKANG… ?

A) K

B) A

C) N

D) R

E) O

Week 2 - Problem 2

There were 60 birds on three trees. At some moment 6 birds flew away from the first tree,

8 birds flew away from the second tree, and 4 birds flew away from the third tree. After that,

it turned out that the number of birds on each tree was the same. How many birds were there

on the second tree in the beginning?

A) 26 B) 24 C) 22 D) 21 E) 20

Week 2 - Problem 1

The square in the figure is a mini-sudoku: the numbers 1, 2, and 3 must be written in the cells so

that each of them appears in each row and in each column. Harry started to fill in the square. In

how many ways can he complete the task?

A) 1 B) 2 C) 3 D) 4 E) 5






http://mathkangarooflorida.blogspot.co.uk/2009/02/square-in-figure-is-mini-sudoku-numbers.html

http://4.bp.blogspot.com/_ukVQRanRu2w/SZnmHu4SIRI/AAAAAAAAAA4/RYKRtDBbmcI/s1600-h/sudoku.jpg

CHALLENGE PROBLEM

ABCD represents a four-digit number. The product of its digits is 70. What is the largest four-

digit number that ABCD can represent?

CHALLENGE PROBLEM

A prime number is a whole number, greater than 1, that is divisible only by itself and 1. Some

examples of prime numbers are 2,3,5,7,11, and 13. What is the largest prime number, P, such

that 9 times P is less than 400?

W E D N E S D A Y , F E B R U A R Y 1 1 , 2 0 0 9

Week 1 - Problem 5

The odometer of my car indicates 187569. All the digits of this number are different. After how

many more kilometres will this happen again?

A. 1 B. 21 C. 431 D. 12431 E. 13776

Week 1 - Problem 4

Robert made a tunnel using some identical cubes (fig.1). When he got bored, he rearranged the

tunnel into a complete pyramid (fig.2). How many cubes from the original tunnel did he not use

for the pyramid?

A. 34 B. 29 C. 22 D. 18 E. 15

Week 1 - Problem 3

From her window Karla looks at the wall of a house. There she can see the silhouette of a

rectangular flag flying in the wind. At five different moments she draws the silhouette. Which of

the 5 pictures cannot be right unless the flag is torn?

T H U R S D A Y , F E B R U A R Y 5 , 2 0 0 9

Week 1 - Problem 2

The face of a clock is cracked into 4 pieces. The sums within the parts are consecutive numbers.

Provided there is only one possible way to crack it, the face would look like:





http://3.bp.blogspot.com/_ukVQRanRu2w/SYubOCwlUvI/AAAAAAAAAAg/F6svWQfzdF8/s1600-h/Pr2.jpg



W E D N E S D A Y , F E B R U A R Y 4 , 2 0 0 9

Week 1 - Problem 1

ABCD is a square. Its side is equal to 10cm. AMTD is a rectangle. Its shorter side is equal to 3

cm. How many centimetres is the perimeter of the square ABCD larger than that of the rectangle

AMTD?

A.14 cm. B. 10 cm. C. 7 cm. D. 6 cm. E. 4 cm

http://kangaroo.math.ca/samples/workingbackward/index.html

Question 1

Marissa wrote her favorite number in the dark cloud and performed correctly several

calculations following the sequence in the diagram. What is Marissa’s favorite number?

Question 2

The three members of a rabbit family have altogether eaten some number of carrots for

breakfast.

Father woke up first and ate half of the carrots.

Mother woke up next and ate seven of the remaining carrots.

Son Bunny woke up last and ate the remaining four carrots.

How many carrots had the family eaten?

First off, here is a chart of the carrot eating that was going on:

http://mathkangarooflorida.blogspot.co.uk/2009/02/problems-for-thursday.html

http://kangaroo.math.ca/samples/workingbackward/index.html

http://1.bp.blogspot.com/_ukVQRanRu2w/SYuPwXNOS4I/AAAAAAAAAAU/D5MYRDsgapo/s1600-h/Pr1.jpg



An example problem

There were several birds on two trees.

First, five birds flew from the first tree to the second tree.

Second, one bird flew away from the second tree.

Third, four birds from the second tree flew to the first one.

In the end, there were six birds on each tree.

How many birds were on each tree at first?

The forward process

The first step in solving "backward" problems is converting the question which is usually

given in words into a diagram of symbols that will help you easily see what steps you

should take to get to the solution.

Here is the arrow diagram of what was taking place in this problem:

We see that, after the first step, the number of birds on the first tree decreased by 5,

while the number of birds on the second tree increased by 5. After the second step, the

number of birds on the second tree decreased by 1. On the third step, the number of

birds on the second tree decreased by 4, while the number of birds on the first tree

increased by 4 birds from the second tree that moved there (this step is illustrated on

the diagram by the slanted arrow. At the end, there are 6 birds on each tree.

Progressing backward

So we had drawn a symbol's chart and placed the problem operationsin it to produce this

diagram:

We now have to find the inverse operations for these operations and we will put them in

a diagram too. So for example where we have a "+4" arrow going into the top-right tree,

we will find the inverse (or opposite) which becomes a "-4" arrow going away from the

top-right tree. Applying similar ideas, we then get the following diagram below:

Everything together

Now, let us reverse the steps, one at a time, by doing the opposite (following the thick

arrows):

If we take the top tree for example, starting at "6" at the top-right corner, we follow the

backward arrows and perform: 6 - 4 = 2, 2 + 5 =7.

For the bottom tree, again starting from the bottom-right tree and following the

backward arrows, we will get: 6 + 4 = 10, 10 + 1 = 11, 11 - 5 = 6.

Thus there were 7 birds on the first tree and 6 birds on the second tree at the start!!!

http://www.topbananaeducation.org/blog

http://www.topbananaeducation.org/blog

Documents

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