MathDB

1

KVS IOQM 21 x in {1,2..,8}, у in {9,10,.,.,16}, z in {17,18,...,24} , x+y+z=36

A = \{1,2,3,4,5,6,7,8\}

,

B = \{9,10,11,12,13,14,15,16\}

and

C =\{17,18,19,20,21,22,23,24\}

. Find the number of triples

(x, y, z)

such that

x \in A, y \in B, z \in C

and

x + y + z = 36

.

20

1

KVS IOQM 20 two people travel around a circular track

Two people

A

and

B

start from the same place at the same time to travel around a circular track of length

100

m in opposite directions. First

B

goes more slowly than

A

until they meet then by doubling his rate he next meets

A

at the starting point. Let

d

m be the distance travelled by

B

before he met

A

for the first time after leaving the starting point. Find the integer closest to

d

.

22

1

KVS IOQM 22 sum r+r_1+ r_2, inradii in right triangle

Let

ABC

be a triangle with

\angle BAC = 90^o

and

D

be the point on the side

BC

such that

AD \perp BC

. Let

r, r_1

, and

r_2

be the inradii of triangles

ABC, ABD

, and

ACD

, respectively. If

r, r_1

, and

r_2

are positive integers and one of them is

5

, find the largest possible value of

r+r_1+ r_2

.

18

1

KVS IOQM 18 ratios and cevians

Let

D,E,F

be points on the sides

BC,CA,AB

of a triangle

ABC

, respectively. Suppose

AD, BE,CF

are concurrent at

P

. If

PF/PC =2/3, PE/PB = 2/7

and

PD/PA = m/n

, where

m, n

are positive integers with

gcd(m, n) = 1

, find

m + n

.

15

1

KVS IOQM 15 4 green and 8 red marbles around a circle

Ria has

4

green marbles and 8 red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is

\frac{p}{q}

where the positive integers

p,q

have no common factors other than

1

, what is

p+q

?

14

1

KVS IOQM 14 square inscribed in equilateral

Let

ABC

be an equilateral triangle with side length

10

. A square

PQRS

is inscribed in it, with

P

on

AB, Q, R

on

BC

and

S

on

AC

. If the area of the square

PQRS

is

m +n\sqrt{k}

where

m, n

are integers and

k

is a prime number then determine the value of

\sqrt{\frac{m+n}{k^2}}

.

19

1

KVS IOQM 19 folding a semicircle along a chord

A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is

4

units and the point of tangency divides the diameter in the ratio

7 :1

. If the length of the crease (the dotted line segment in the figure) is

\ell

then determine

\ell^2

. https://cdn.artofproblemsolving.com/attachments/5/6/63fed83742c8baa92d9e63962a77a57d43556f.png

12

1

KVS IOQM 12 x^2 + 2020x + n = 0, x^2 + mx + 2020 = 0

Let

A = \{m : m

an integer and the roots of

x^2 + mx + 2020 = 0

are positive integers

\}

and

B= \{n : n

an integer and the roots of

x^2 + 2020x + n = 0

are negative integers

\}

. Suppose

a

is the largest element of

A

and

b

is the smallest element of

B

. Find the sum of digits of

a + b

.

23

1

KVS IOQM P23 max N such divisors by 3 of {1,2,3,...,N} equal to those by 5 or 7

Find the largest positive integer

N

such that the number of integers In the set

{1,2,3,...,N}

which are divisible by

3

is equal to the number of integers which are divisible by

5

or

7

(or both),

24

1

KVS IOQM P24 area of ABCD, tangent unequal circles

Two circles

S_1

and

S_2

, of radii

6

units and

3

units respectively, are tangent to each other, externally. Let

AC

and

BD

be their direct common tangents with

A

and

B

on

S_1

, and

C

and

D

on

S_2

. Find the area of quadrilateral

ABDC

to the nearest Integer.

25

1

KVS IOQM P25 remainder of abcdc by 100

A five digit number

n= \overline{abcdc}

. Is such that when divided respectively by

2,3,4,5,6

the remainders are

a,b,c,d,c

. What is the remainder when

n

is divided by

100

?

26

1

KVS IOQM P26 min a+b+c when sum of any 2 is pefrect square

Let

a,b,c

be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum

a + b + c

. Find this sum.

27

1

KVS IOQM P27 max angle of triangle by recflections of acute ABC

Let

ABC

be an acute-angled triangle and

P

be a point in its interior. Let

P_A,P_B

and

P_c

be the images of

P

under reflection in the sides

BC,CA

, and

AB

, respectively. If

P

is the orthocentre of the triangle

P_AP_BP_C

and if the largest angle of the triangle that can be formed by the line segments

PA, PB

. and

PC

is

x^o

, determine the value of

x

.

16

1

KV IOQM P16

If

x

and

y

are positive integers such that

(x-4)(x-10)=2^y

, then Find maximum value of

x+y

17

1

KVS IOQM P17

Two sides of a regular polygon of

n

sides when extended meet at

28

degrees. What is smallest possible value of

n

28

1

KVS IOQM P28 n' denote no obtained by deleting zero digits

For a natural number

n

, let

n'

denote the number obtained by deleting zero digits, if any. (For example, if

n = 260

,

n' = 26

, if

n = 2020

,

n' = 22

.),Find the number of

3

-digit numbers

n

for which

n'

is a divisor of

n

, different from

n

.

13

1

KVS IOQM 13

The sides of triangle are

x

,

2x+1

and

x+2

for some positive rational

x

. Angle of triangle is

60

degree. Find perimeter

29

1

KVS IOQM P29 flawless permutations of {1,2,3,4,5}

Consider a permutation

(a_1,a_2,a_3,a_4,a_5)

of

\{1,2,3,4,5\}

. We say the

5

-tuple

(a_1,a_2,a_3,a_4,a_5)

is dlawless if for all

1 \le i<j<k \le 5

, the sequence

(a_i,a_j,a_k)

is not an arithmetic progression (in that order). Find the number of flawless

5

-tuples.

30

1

KVS IOQM P30 Ari chooses 7 from n balls

Ari chooses

7

balls at random from

n

balls numbered

1

to

n

. If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find

n

.

11

1

KVS IOQM P11

The prime numbers

a,b

and

c

are such that

a+b^2=4c^2

. Determine the sum of all possible values of

a+b+c

.

10

1

KVS IOQM P10

Let

A

and

B

be two finite sets such that there are exactly

144

sets which are subsets of

A

or subsets of

B

. Find the number of elements in

A \cup B

.

9

1

KVS IOQM P9

find the number of ordered triples

(x,y,z)

of real numbers that satisfy the system of equations:

x+y+z=7; x^2+y^2+z^2=27; xyz=5

.

8

1

KVS IOQM P8

Find the largest

2

-digit number

N

which is divisible by

4

, such that all integral powers of

N

end with

N

.

7

1

KVS IOQM P7

a,b,c

are positive real numbers such that

a^2+b^2=c^2

and

ab=c

. Determine the value of

\left\lvert{\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}}\right\rvert

6

1

KVS IOQM P6

Let

ABCD

be a square with side length

100

. A circle with centre

C

and radius

CD

is drawn. Another circle of radius

r

, lying inside

ABCD

, is drawn to touch this circle externally and such that the circle also touches

AB

and

AD

. If

r=m+n\sqrt{k}

, where

m,n

are integers and

k

is a prime number, find the value of

\frac{m+n}k

.

5

1

KVS IOQM P5

Find the number of positive integers

n

such that the highest power of

7

dividing

n!

is

8

.

4

1

KVS IOQM P4

Let

ABC

be an isosceles triangle with

AB=AC

and incentre

I

. If

AI=3

and the distance from

I

to

BC

is

2

, what is the square of length on

BC

?

3

1

KVS IOQM P3

Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?

2

1

KVS IOQM P2

If

ABCD

is a rectangle and

P

is a point inside it such that

AP=33, BP=16, DP=63

. Find

CP

.

1

KVS IOQM P1

If

a,b,c

are real numbers and

(a+b-5)^2+(b+2c+3)^2+(c+3a-10)^2=0

find the integer nearest to

a^3+b^3+c^3

.

2020-21 KVS IOQM India

Subcontests

KVS IOQM 21 x in {1,2..,8}, у in {9,10,.,.,16}, z in {17,18,...,24} , x+y+z=36

KVS IOQM 20 two people travel around a circular track

KVS IOQM 22 sum r+r_1+ r_2, inradii in right triangle

KVS IOQM 18 ratios and cevians

KVS IOQM 15 4 green and 8 red marbles around a circle

KVS IOQM 14 square inscribed in equilateral

KVS IOQM 19 folding a semicircle along a chord

KVS IOQM 12 x^2 + 2020x + n = 0, x^2 + mx + 2020 = 0

KVS IOQM P23 max N such divisors by 3 of {1,2,3,...,N} equal to those by 5 or 7

KVS IOQM P24 area of ABCD, tangent unequal circles

KVS IOQM P25 remainder of abcdc by 100

KVS IOQM P26 min a+b+c when sum of any 2 is pefrect square

KVS IOQM P27 max angle of triangle by recflections of acute ABC

KV IOQM P16

KVS IOQM P17

KVS IOQM P28 n' denote no obtained by deleting zero digits

KVS IOQM 13

KVS IOQM P29 flawless permutations of {1,2,3,4,5}

KVS IOQM P30 Ari chooses 7 from n balls

KVS IOQM P11

KVS IOQM P10

KVS IOQM P9

KVS IOQM P8

KVS IOQM P7

KVS IOQM P6

KVS IOQM P5

KVS IOQM P4

KVS IOQM P3

KVS IOQM P2

KVS IOQM P1