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INTERNATIONAL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2009. Mathematics Contest in Taiwan. Problems and Solutions

### Text of IMSO 2009 mathematics Olympiad

INTERNATIONAL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2009Mathematics Contest in Taiwan Name:111 1 School: 111111 Grade: 1111 Number: 11111

Short Answer: there are 12 questions, fill in the correct answers in the answer sheet. Each correct answer is worth 10 points. Time limit: 90 minutes. 1. Find the largest possible divisor of the three numbers, 480608, 508811 and 723217, so that the reminder is the same in each case. 2. In a small group of people it was found that all of the following relationships were present: father, mother, son, daughter, brother, sister, cousin, nephew, niece, uncle and aunt. What is the smallest group of people for which this is possible? 3. Seven cubes are glued together face to face as shown in the diagram. The volume of the solid formed in this way is 189 cm3. Find the surface area of the solid.

4. Jack said to Jim: If I give you 6 pigs for one horse, then you will own twice as many animals as I own. Dan said to Jack: If I give you 14 sheep for one horse, then you will own three times as many animals as I own. Jim said to Dan: If I give you 4 cows for one horse, then youll own six times as many animals as I own. How many animals in total do Jack, Jim and Dan own? 5. By adding brackets in various ways to the expression 13571113, what is the maximum number of different values which the expression can have? 6. Replace the asterisks with digits so that the multiplication below is correct: * * * * * * * * * What is the product? * * * * * * * 3 3 * * * * * * 2 0 * * 3 * * * * * * 3 7 *

0 9 *

1

7. Tom has a contract to dig out some foundations and it must be done in 30 days. His own machine, which he wishes to use as much as possible, would take 48 days to do all the work. He can hire a bigger machine which would do the complete job in 21 days, but it costs \$300 a day. There is only enough room for one machine at a time. What is the least number of days for which he will have to hire the larger machine? 8. Four different right-angled triangles all have sides which are of integral length and their perimeters are the same length. Find the smallest perimeter for which this is possible. 9. The diagram is of an irregular pentagon with all 5 of its diagonals drawn in. How many distinct triangles (not necessarily different) can be found, using only the lines (or parts of lines) shown in the diagram?

10. I have a rectangular picture whose edges are each an exact number of centimeters in length. At a quick glance it could be mistaken for a square, but it is not a square. It is placed inside a black border which is 3 cm wide all the way around the picture. The area of the border is exactly equal to the area of the picture. What is the area, in cm2, of the picture alone?

11. A combination lock on a safe needs a 6-letter sequence to open the safe. This is made from the letters A, B, C, D, E, F with none of them being used twice. Here are three guesses at the combination CBADF E AE D C B F E D FAC B In the FIRST guess only ONE letter is in its correct place. In the SECOND guess only TWO letters are in their correct places and those two correct places are not next to each other. In the THIRD guess only THREE letters are in their correct places. Each of the 6 letters is in its correct place once. What is the correct combination? 12. Given that ABCD is a square and the lengths EA, EB, EC are in the ratio EA:EB:EC=1:2:3, determine the size of the angle AEB, in degree.A a E 3a 2a B

D2

C

8. Find the smallest possible number that leaves a remainder of 1 when divided by 2, 3, 4, 5 or 6, and which can be divided by 7 exactly. ANS: 9. Each of the letters A, B, C, D, E is used to represent a single digit in these two statements.(Same letter = same digit.) A + B = C, C + D = EA (Note EA is a 2-digit number) What is the value of B + D ? ANS: 10. Ann, Ben and Carol each have some money. If Ann gave Ben \$30, then Ben would have twice as much as Ann. If Ben gave Carol \$30, then Carol would have twice as much as Ben. If Carol gave Ann \$30, then they would both have the same amount. How much money did Ann have? ANS:

\$

11. A square lawn has a path 1 m wide which goes around the outside of all the four edges. The area of the path is 40 m2. What is the area of the lawn?

m ANS: 12. Four consecutive odd numbers add up to a total of 80. What is the smallest of those four numbers.ANS:

2

13. A cube with an edge length of 10 cm is resting on a horizontal table. An insect starts crawling from the table at an angle of 30 degrees to the horizontal. How far will it have crawled on the cube by the time it gets to the top? ANS:

cm

14. On this diagram you may start at any square and move up or down or across (but NOT diagonally) into the next square. No square may be used twice. The digits in each square are written down in the order they are used to form a number. What is the largest number that can be made? 5 8 3 9 4 6 1 7 2 ANS:

15. A new monument is to be made in the shape of a cuboid. Only three of the faces are to be decorated. To allow for this: one face has to have an area of 48 m2; another is to have an area of 72 m2; and another of 96 m2. What will be the volume of the monument? ANS:

m3

16. Arrange the numbers 1 to 9, using each number only once and placing only one number in each cell so that the totals in both directions (vertically and horizontally) are the same. How many different sums are there?

ANS: 17. How many distinct squares (not necessarily different in size) can be traced out following only the lines of the grid drawn below?

ANS: 18. A 4-wheeler car has travelled 24,000 km and, in that distance, has worn out 6 tyres. Each tyre travelled the same distance. How far did each separate tyre travel? ANS:

km

19. A, B and C are three villages near to each other, shown in the diagram below, where the straight lines represent the only roads joining the villages. The figures give the distances in km between villages.B 8 7 C 12 A

A new fire station is to be built to serve all three villages. It is to be on a roadside at such a position that the greatest distance that the fire-engine has to travel along the roads in an emergency at one of the villages is as small as it can be. What is this smallest distance? ANS: 20. The diagonal of this 53 rectangle passes through 7 squares.

km

The diagonal of this 64 rectangle passes through 8 squares.

What is the number of squares passed through by the diagonal of a 3602009 rectangle? ANS:

INTERNATIONAL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOLS (IMSO) 2009Mathematics Contest (Second Round) in Taiwan, Essay Problems Name: School: Grade: Number:

Answer the following 10 questions, and show your detailed solution in the space provided after each question. Each question is worth 4 points. Time limit: 60 minutes. 1. There are two clocks. One of them gains 6 seconds in every hour, while the other loses 9 seconds in every hour. If they are both set to show the same time, and then set going, how long will it be before the time displayed on them is exactly 1 hour apart?

2. Replace the asterisks in 86**** with the digits 1, 2, 3 and 4. Using each of them once so that the six-digit number obtained is the largest possible number divisible by 132.

3. There are two isosceles triangles. They are equal in area. In both triangles all edges measure an exact number of cm, and the two edges of equal length are 13 cm. In one of them the third edge measures 10 cm. What is the length of the third edge of the other?

4. In a quadrilateral ABCD, BC is parallel to AD. E is the foot of the perpendicular from B to AD. Find BE if AB=17, BC=16, CD=25 and AD=44.A

B 17

16

C 25

E

44

D

5. Three different numbers from 1 to 10 were written on three cards. The cards were shuffled and dealt to three players. Each player got one card and wrote down the number of his card. Then the cards were collected and dealt again. After

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