Kvs Jr Maths OLympiad Papers

KVS Junior Mathematics Olympiad previous papers

5th KVS Junior Mathematics Olympiad (JMO) – 2001

M.M. 100 Time : 3 hours

Note : (i) Please check that there are two printed pages and ten question in

all.

(ii) Attempt all questions. All questions carry equal marks.

1. Fill in the blanks :

(a) If x + y = 1, x3 + y3 = 4, then x2 + y2 = ……..

(b) After 15 litres of petrol was added to the fuel tank of a car, the tank

was 75% full. If the capacity of the tank is 28 litres, then the

number of litres in the tank before adding the petrol was ……

(c) The perimeter of a rectangle is 56 metres. The ratio of its length to

width is 4:3. The length of the diagonal in metres is ……..

(d) If April 23 falls on Tuesday, then March 23 of the same year was a

……..

(e) The sum of the digits of the number 2200052004 is ….

2. (a) Arrange the following in ascending order :

25555, 33333, 62222

(b) Two rectangles, each measuring 3 cm x 7 cm,

are placed as in the adjoining figure :

Find the area of the overlapping

portion (shaded) in cm2.

3. (a) Solve : 3)x5(log)x35(log

10

310

(b) Simplify : )ac)(cb)(ba()ac)(cb)(ba(

acac

cbcb

baba

4. (a) Factorize :

(x-y)3+(y-z)3+(z-x)3

(b) If x2-x-1=0, then find the value of x3-2x +1

5. ABCD is a square. A line through B intersects CD produced at E, the side

AD at F and the diagonal AC at G.

B A

F

C D E

If BG = 3, and GF=1, then find the length of FE,

6. (a) Find all integers n such that (n2-n-1)n+2 = 1

(b) If x = baab4

, find the value of b2xb2x

a2xa2x

7. (a) Find all the positive perfect cubes that divide 99.

(b) Find the integer closest to 100 (12- 143 )

G

8. In a triangle ABC, BCA=90o. Points E and F lie on the hypotenuse AB

such that AE=AC and BF = BC. Find ECF.

A F

y E C B

9. An ant crawls 1 centimetre north, 2 centimetres west, 3 centimetres

south, 4 centimetres east, 5 centimetres north and so on, at 1 centimetre

per second. Each segment is 1 centimetre longer than the preceding one,

and at the end of a segment, the ant makes a left turn. In which direction

is the ant moving 1 minute after the start ?

10. Find the lengths of the sides of a triangle with 20, 28 and 35 as the

lengths of its altitudes.

x z



Note : (i)Please check that there are two printed pages and ten question

in all.

(ii)Attempt all questions. All questions carry equal marks.

1. Fill in the blanks

(a) Yash is carrying 100 hundred – rupee notes, 50 fifty –rupee notes, 20

twenty – rupee notes, 10 ten –rupee notes and 5 five-rupee notes. The

total amount of money he is carrying in Rupee, is ………….

(b) In a school, the ratio of boys to girls is 4:3 and the ratio of girls to

teachers is 8 :1. The ratio of students to teachers is ………

(c) The value of 2

5.015.0

is ………….

(d) (123456)2 + 123456 + 123457 is the square of ………

(e) The area of square is 25 square centimeters. In perimeter, in

centimeters, is …………….

2. (a) How many four digit numbers can be formed using the digits 1,2 only so

that each of these digits is used at least once ?

(b) Find the greatest number of four digits which when increased by 1 is

exactly divisible by 2, 3, 4, 5, 6 and 7.

3. (a) If f(x) = ax7 + bx5 + cx3 – 6, and f(-9) = 3, find f(9).

(b) Find the value of )1000(x)1002(x3)1000()1002()2002( 333

U

S R

4.(a) If x > 0 and 47x1x 4

4 , find the value of 33

x1x

(b) If 82x = 161-2x, find the value of 37x.

5. A train, after traveling 70 km from a station A towards a station B, develops

a fault in the engine at C, and covers the remaining journey to B at 43

of its

earlier speed and arrives at B 1 hour and 20 minutes late. If the fault had

developed 35 km further on at D, it would have arrived 20 minutes sooner.

Find the speed of the train and the distance from A to B.

A C D B

6. The adjoining diagram shows a square PQRS with each side of length 10

cm. Triangle PQT is equilateral. Find the area of the triangle UQR.

P Q

T

A square of side – length 64 cm is given. A second square is obtained by

connecting the mid points of the sides of the first square (as shown in the

diagram). If the process of forming smaller inner squares by connecting the

mid points of the sides of the previous squares is continued, what will be the

side-length of the eleventh square, counting the original square as the first

square ?

7. Seven cubes of he same size are glued together face to face as shown in

the adjoining diagram. What is the surface area, in square centimeters, of

the solid if its volume is 448 cubic centimeters ?

8. Anil, Bhavna, Chintoo, Dolly and Eashwar play a game in which each is

either a FOX or a RABBIT. FOXES’ statements are always false and RABBITS’ statements are always true. Anil says that Bhavna is a RABBIT. Chintoo says that Dolly is a FOX. Eashwar says that Anil is not a FOX. Bhavna says that Chintoo is not a RABBIT. Dolly says that Eashwar and Anil are different kinds of animals. How many FOXES are there ? (Justify your answer).

10. The accompanying diagram is a road-plan of a city. All the roads go east-

west or north-south, with the exception of one shown. Due to repairs one road is

impassable at the point X, of all the possible routes from P to Q, there are

several shortest routes. How many such shortest routes are there ?

x

Q

P




in all.

(ii) Attempt all questions. All questions carry equal marks.

1. Fill in the blanks

(a) The digits of the number 2978 are arranged first in descending

order and then in ascending order. The difference between the

resulting two numbers is ……………

(b) Yash is riding his bicycle at a constant speed of 12 kilometers per

hour. The number of metres he travels each minute is ………..

(c) The square root of 35 x 65 x 91 is ………..

(d) The number 81 is 15% of …………..

(e) A train leaves New Delhi at 9.45 am and reaches Agra at 12.58

pm. The time taken in the journey, in minutes, is …………..

2. (a) Find the largest prime factor of 203203.

(b) Find the last two (tens’ and units’) digits of (2003)2003.

3. (a) Find the number of perfect cubes between 1 and 1000009

which are exactly divisible by 9.

(b) If x = 5 + 2 6 , find the value of

(i) x

1x (ii) 33

x1x

4. (a) Solve :

9x6x

5x2x

8x5x

4x1x

2

2

2

2

2

2

2

2

(b) Find the remainder when xxxxx 9254981 is divided by

.xx3

5. (a) OPQ is a quadrant of a circle and semicircles are drawn on

OP and OQ. Areas a and b are shaded. Find a/b.

(b) Assuming all vertical lines are parallel, all angles are right angles

and all the horizontal lines are equally spaced, what fraction of

figure is shaded ?

6. Alternate vertices of a regular hexagon are joined as shown. What

fraction of the total area of a hexagon is shaded ? (Justify your answer)

7. In a competition consisting of 30 problems Neeta was given 12 points

for each correct solution, and 7 points were subtracted from her score

for each incorrect solution problems not attempted contributed 0

points to the score find the number of problems attempted correctly by

Neeta.

8. A cube with each edge of lengths 4 units is painted green on all the

faces. The cube is then cut into 64 unit cubes. How many of these

small cubes have (i) 3 faces painted (ii) 2 faces painted (iii) one face

painted (iv) no face painted.

9. Let PQR be an equilateral triangle with each side of length 3 units. Let

U, V, W, X Y and Z divide the sides into unit lengths. Find the ratio

of the area U, W, X, Y and Z divide the sides into unit lengths. Find

V. Y

Q W X R

the ratio of the area U W X Y (shaded) to the area of the whole

triangle PQR.

P

U Z

10. Five houses P, Q, R, S and T are situated on the opposite side of a

street from five other houses U, V, W, X and Y as shown in the diagram :

P. Q. R. .S .T

20 m

U. V. .W X. Y.

Houses on the same side of the street are 20 metres apart A postman is trying

to decide whether to deliver the letters using route PQRSTYXWVU or

route PUQVRWSXTY, and finds that the total distance is the same in

each case. Find the total distance in metres.

20 m




in all.

(ii)Attempt all questions. All questions carry equal marks

1. Fill in the blanks:

(a) If four times the reciprocal of the circumference of a circle equals

the diameter of the circle, then the area of the circle is …………..

(b) If 0x4

x41 2 then

x2 equals………

(c) If a=1000, b=100, c=10, and d=1, then

(a+b+c-d) + (a+b-c+d) + (a-b+c+d)+(-a+b+c+d) is equal to …….

(d) When the base of a triangle is increased by 10% and the altitude to

the base is decreased by 10%, the change in area is ……….

(e) If the sum of two numbers is 1, and their product is 1, then the sum

of their cubes is ……………

2. (a) If x = 82log2

8log find the value of log3x.

(b) If 824

yx

x

and 24339

y5

yx

find the value of x-y.

3. (a) Find the number of digits in the number 22005 x 52000 when

written in full.Find the remainder when 22005 is divided by 13.

4. (a) A polynomial p (x) leaves a remainder three when divided by x –

1 and a remainder five when divided by x-3. Find the remainder when

p(x) is divided by (x-1) (x-3).

(b) Find two numbers, both lying between 60 and 70, each of which is

exactly divides 243-1.

5. In triangle ABC the medians AM and CN to the sides BC and AB,

respectively intersect in the point O.P is the mid-point of side AC, and

MP intersects CN in Q. If the area of triangle OMQ is 24 cm2, find the

area of triangle ABC.

6. The base of a pyramid is an equilateral triangle of side length 6 cm. The

other edges of the pyramid are each of length 15 cm. Find the volume of

the pyramid.

7. Chords AB and CD of a circle (see figure) intersect at E and are

perpendicular to each other segments AE. EB and ED are of lengths 2cm,

6cm and 3cm respectively. Find the length of the diameter of the circle.

C

A 2 E 6 B 3

D

8. Three men A, B and C working together, do a job in 6 hours less time

than A alone, in 1 hour less time than B alone, and in one half the time

needed by C when working alone. How many hours will be needed by A

and B working together, to do the job ?

9. Pegs are put on a board 1 unit apart both horizontally and vertically. A

rubber band is stretched over 4 pegs as shown in the figure forming a

quadrilateral. Find the area of the quadrilateral in square units.

. . . . .

. . . . .

. . . . .

. . . . .

10. The odd positive integers 1, 3, 5, 7 ….. are arranged in five columns continuing with the pattern shown on the right. Counting from the left, in which column (I, II, III, IV or V) does the number 2005 appear ? (Justify your answer)

I II III IV V 1 3 5 7

15 13 11 9 17 19 21 23 31 29 27 25 33 35 37 39 47 45 43 41 49 51 53 55 . . . . . . . . . . . .




in all.


1) a,b,c are three distinct real numbers and there are real numbers x,y such

that a3+ax+y = 0, b3+bx=y = 0, c3+cx+y = o, so that a+b+c = 0.

2) The triangles A,B,C has CA = CB. P is a point on the circum circle between

A and B.(and on the opposite side of the line AB to C). D is the foot of the

perpendicular from C to PB. So that PA + PB = 2 PD.

3) Given reals x,y with (x2+ y2) /(x2 – Y2) + (x2-y2)/(x2+y2) = K. find the value

of ( x8+ y8) /(x8 – Y8) + (x8-y8)/(x8+y8) in terms of K.

4) In a traingle ABC right angled at B, a point P is taken on the side AB such

that AP = h and BP = b and AC = y such that h + y = b +d. Prove that

h = bd/(2b+d).

5) P is a point in the traingle ABC. Lines are drawn through P parallel to the

Sides of the traingle. The areas of the three resulting traingles with a vertex

at P have areas 4, 9 and 49. What is the area of the triangle ABC?

6) A lotus plant in a pool of water is ½. Cubit above the water level. When

propelled by air, the lotus sinks in the pool 2 cubits away from its position.

Find the depth of the water in the pool?

7) Let C1 be any point on the side AB of triangle ABC. Join C1C. The lines

through A and B paralle to CC1 meet BC and AC produced at A1 and B1

respectively. Prove that 1/AA1 + 1/BB1 = 1/CC1.

8) The triangle ABC has angle B = 900. When it is rotated about AB it gives a

cone of volume 800 ∏ Cubic Units When it is rotated about BC it gives a

cone of volume 1920∏ Cubic Units . Finmd the length of AC.

9) A number when divided by 7, 11 and 13 (the prime factors of 1001)

Successively leave the remainders 6,10 and 12 respectively. Find the

Remainder if the number is divided by 1001.

10) Two candles of the same height are lighted together. First one gets burned

up completely in three hours while the second in 4 hours. At what point of

time the length of the second candle will be doubled the length of the first

candle ?




in all.


1. Solve

| x-1 | + | x | + | x + 1 | = x + 2

2. Find the greatest number of four digits which when divided by 3,

5, 7, 9 leaves remainders 1, 3, 5, 7 respectively.

3. A printer numbers the pages of a book starting with 1. He uses

3189 digits in all. How many pages does the book have ?

4. ABCD is a parallelogram. P, Q, R and S are points on sides AB, BC, CD

and DA respectively such that AP=DR. If the area of the parallelogram is

16 cm2, find the area of the quadrilateral PQRS.

5. ABC is a right angle triangle with B = 90o. M is the mid point of AC and

BM = 117 cm. Sum of the lengths of sides AB and BC is 30 cm. Find

the area of the triangle ABC.

6. Solve : xa

)xa()xa()xa()xa(

7. Without actually calculating, find which is greater :

31 11 or 17 14

8. Show that there do not exist any distinct natural numbers a, b, c, d such

that

a3 + b3 = c3 + d3 and a + b = c + d

9. Find the largest prime factor of :

312 + 212 – 2.66

10. If only downward motion along lines is allowed, what is the total number

of paths from point P to point Q in the figure below ?

P

Q

11th KVS Maths Olympiad Contest – 2008 M.M. 100 Time : 3 hours

Note : (i)Please check that there are two printed pages and ten question in all.

(ii)Attempt all questions. All questions carry equal marks 1) Find the value of S = 12 – 22 + 32 – 42 + ……………….-982 + 992

2) Find the smallest multiple of ‘15’ such that each digit of the multiple is either ‘0’ or

‘8’.

3) At the end of year 2002. Ram was half as old as his grandfather. The sum of

years in which they were born is 3854. What is the age of Ram at the end of year

2003?

4) Find the area of the largest square, which can be inscribed in a right angle

triangle with legs ‘4’ and ‘8’ units.

5) In a Triangle the length of an altitude is 4 units and this altitude divides the

opposite side in two parts in the ratio 1:8. Find the length of a segment parallel to

altitude which bisects the area of the given triangle.

6) A number ‘X’ leaves the same remainder while dividing 5814, 5430, 5958. What

is the largest possible value of ‘X’?

7) A sports meet was organized for four days. On each day, half of existing total

medals and one more medal was awarded. Find the number of medals awarded

on each day.

8) Let ΔABC be isosceles with ABC = ACB = 780. Let D and E be the points

on sides AB and AC respectively such that BCD = 240 and CBE = 510. Find

the angle BED and justify your result.

9) If , and are the roots of the equation. (x - a) (x - b) (x - c) + 1 = 0.

Then show that a, b and c are the roots of the equation

( - x) ( - x) ( - x) + 1 = 0.

10) A 4 x 4 x 4 wooden cube is painted so that one pair of opposite faces is blue,

one pair green and one pair red. The cube is now sliced into 64 cubes of side 1

unit each.

(i) How many of the smaller cubes have no painted face?

(ii) How many of the smaller cubes have exactly one painted face?

(iii) How many of the smaller cubes have exactly two painted faces?

(iv) How many of the smaller cubes have exactly three painted faces?

(v) How many of the smaller cubes have exactly one face painted blue and

one face painted green ?

13th KVS Maths Olympiad Contest – 2010




1) Let a,b,c are real numbers a not equal to zero such that a and 4a +3b+2c

have the same sign. Show that the equation ax2 + bx +c = 0 can not have

both roots in the interval (1,2) .

2) a) find all the intgers which are equal to 11 times the sum of their digits.

b) Prove that 33 )52()52( is a rational number.

3) A circle centered at A with radius 1 unit and another circle centered at B

with radius 4 units touch each other externally. A third circle is drawn to

touch the first two circles and one of their common external tangents as

shown in the figure. What is the radius of the third circle?

4) Given three non co-linear points A,H and G. Construct a triangle with A as

vertex, H as orthocenter and G as centroid.

5) A triangle ABC, angle A is twice the angle B. Prove that a2 = b(b+c) where

a, b and c are the sides opposite to angles A,B and C respectively.

6) The equation x2+px+q = 0, where p and q are integers, has rational roots.

Prove that the roots are integers.

7) Triangle ABC is right triangle with angle C is 900. Measure of ABC =

600,and AB = 10 units. Let P be a point chosen randomly inside triangle

ABC.Extend BP to meet Ac at D. What is the probability that BD >5 2 .

8) Prove that the sum of hypotenuse and the altitude of a right angled triangle

dropped on the hypotenuse exceed the half perimeter of the triangle.

9) How many times is digit zero is written when llisting all numbers from

1 to 3333?

10) Find out the remainder when x+x9+x25+x49+x81 is divided by x3-x?

14th KVS Maths Olympiad Contest – 2011 M.M. 100 Time : 3 hours



1) Show that for any natural number ‘n’ the fraction 314421

nn is in its lowest

term.

2) a) Factorize: x6+5x3+8

b) Prove that 3a4-4a3b+b4 0, for all real numbers a and b.

3) M is any point on the minor arc BC of a circum circle of an equilateral

triangle ABC. Prove that AM = BM+CM.

4) Solve the inequality, 411 xx

5) a) Find the square root of 472)1(23 2 xxx ,x>4

b) Given real natural numbers x,y and z are such that x+ y + z = 3,

x2+y2+z2 = 5, x3+y3+z3 = 7. Find the value of x4+y4+z4 ?

6) In the triangle given each side is of length 4 units. If the length PQ is 1 unit

and TQ is perpendicular to PR, find the ratio of areas of triangle PQT and

the quadrilateral QRST.

7) Prove that for any natural number ‘n’ , the expression

A = 2903n – 803n – 464n+ 261n is divisible by 1897.

8) Find the number of odd integers between 30,000 and 80,000

in which no digit is repeated?

9) Find all the integers which are equal to 11 times the sum

of their digits?

10) Prove that in any triangle ABC is one angle is 1200 , the triangle

formed by the feet of angle bisectors is a right angled.

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Kvs Jr Maths OLympiad Papers