2024-25 IOQM India

Part of India IOQM

Subcontests

(30)

1

IOQM P30 2024

ABC

be a right-angled triangle with

\angle B = 90^{\circ}

. Let the length of the altitude

BD

12

. What is the minimum possible length of

AC

AC

and the perimeter of triangle

ABC

1

IOQM P29 2024

n = 2^{19}3^{12}

M

denote the number of positive divisors of

n^2

which are less than

n

but would not divide

n

.What is the number formed by taking the last two digits of

M

(in the same order)?

1

IOQM P28 2024

Find the largest positive integer

n <30

\frac{1}{2}(n^8 + 3n^4 -4)

is not divisible by the square of any prime number.

1

IOQM P27 2024

ABC

P

in the interior of

ABC

\angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB.

\angle BAC = 30^{\circ}

AP = 12

D,E,F

be the feet of perpendiculars from

P

BC,CA,AB

respectively. If

m \sqrt{n}

is the area of the triangle DEF where

m,n

are integers with

n

prime, then what is the value of the product

mn

1

IOQM P26 2024

\lfloor x \rfloor

for all real numbers

x

satisfying the equation

16 + 15x + 15x^2 = \lfloor x \rfloor ^3

1

IOQM P25 2024

M

of positive integers consists of distinct perfect squares and the number

92

. The average of the numbers in

M

85

92

M

, the average drops to

84

N^2

is the largest possible square in

M

, what is the value of

N

1

IOQM P24 2024

Consider the set

F

of all polynomials whose coefficients are in the set of

\{0,1\}

q(x) = x^3 + x +1

. The number of polynomials

p(x)

F

14

such that the product

p(x)q(x)

F

1

IOQM P23 2024

Consider the fourteen numbers,

1^4,2^4,...,14^4

. The smallest natural numebr

n

such that they leave distinct remainders when divided by

n

1

IOQM P22 2024

ABC

\angle BAC = 90^{\circ}

D

be the point on

BC

AB + BD = AC + CD

BD : DC = 2:1

\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}

m,n

are relatively prime positive integers and

p

is a prime number, determine the value of

m+n+p

1

IOQM P21 2024

n

\Bigl\lfloor \frac{n}{9} \Bigr\rfloor

is a three digit number with equal digits, and

\Bigl\lfloor \frac{n-172}{4} \Bigr\rfloor

4

digit number with the digits

2,0,2,4

in some order. What is the remainder when

n

100

1

IOQM P20 2024

On a natural number

n

you are allowed to operations :

(1)

n

2

(2)

3

n

. For example starting with

8

13

8 \longrightarrow 16 \longrightarrow 13

. You need two steps and you cannot do in less than two steps. Starting from

11

, what is the least number of steps required to reach

121

1

IOQM P19 2024

Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, so taht no three of the points form a triangle with lines of the same colour.

1

IOQM P18 2024

p,q

be two-digit number neither of which are divisible by

10

r

be the four-digit number by putting the digits of

p

followed by the digits of

q

p,q

very, a computer prints

r

on the screen if

\gcd(p,q) = 1

p+q

r

. Suppose that the largest number that is printed by the computer is

N

. Determine the number formed by the last two digits of

N

(in the same order).

1

IOQM P17 2024

Consider an isosceles triangle

ABC

BC = 30, CA = AB = 20

D

be the foot of the perpendicular from

A

BC

M

be the midpoint of

AD

PQ

be a chord of the circumcircle of triangle

ABC

M

PQ

PQ

BC

. The length of

PQ

1

IOQM P16 2024

f: \mathbb{R} \longrightarrow \mathbb{R}

be a function satisfying the relation

4f(3-x) + 3f(x) = x^2

x

. Find the value of

f(27) - f(25)

to the nearest integer. (Here

\mathbb{R}

denotes the set of real numbers.)

1

IOQM P15 2024

X

be the set of consisting of twenty positive integers

n,n+2,...,n+38

. The smallest value of

n

for which any three numbers

a,b,c \in X

, not necessarily distinct, form the sides of an acute-angled triangle is:

1

IOQM P14 2024

Initially, there are

3^{80}

particles at the origin

(0,0)

. At each step the particles are moved to points above the

x

-axis as follows: if there are

n

particles at any point

(x,y)

\Bigl \lfloor \frac{n}{3} \Bigr\rfloor

of them are moves to

(x+1,y+1)

\Bigl \lfloor \frac{n}{3} \Bigr\rfloor

(x,y+1)

and the remaining to

(x-1,y+1)

, For example, after the first step, there are

3^{79}

particles each at

(1,1),(0,1)

(-1,1)

. After the second step, there are

3^{78}

particles each at

(-2,2)

(2,2)

2 \times 3^{78}

particles each at

(-1,2)

(1,2)

3^{79}

(0,2)

80

steps, the number of particles at

(79,80)

1

IOQM P13 2024

Three positive integers

a,b,c

a>c

satisfy the following equations :

ac + b+c = bc + a + 66, \; \; \; \; a+b+c=32

Find the value of

a

1

IOQM P12 2024

Consider a square

ABCD

16

E,F

CD

CE = EF = FD

BF

AE

M

\bigtriangleup MAB

1

IOQM P11 2024

The positive real numbers

a,b,c,

\frac{a}{2b+1} + \frac{2b}{3c+1} + \frac{3c}{a+1} = 1

\frac{1}{a+1} + \frac{1}{2b+1} + \frac{1}{3c+1} = 2

What is the value of

\frac{1}{a} + \frac{1}{b} + \frac{1}{c}

1

IOQM P10 2024

Determine the number of positive integral values of

p

for which there exists a triangle with sides

a,b,

c

a^2 + (p^2 + 9)b^2 + 9c^2 - 6ab - 6pbc = 0.

1

IOQM P9 2024

Consider the grid of points

X = \{(m,n) | 0 \leq m,n \leq 4 \}

. We say a pair of points

\{(a,b),(c,d)\}

X

is a knight-move pair if

( c = a \pm 2

d = b \pm 1)

( c = a \pm 1

d = b \pm 2)

. The number of knight-move pairs in

X

1

IOQM P8 2024

n

be the smallest integer such that the sum of digits of

n

is divisible by

5

as well as the sum of digits of

(n+1)

is divisible by

5

. What are the first two digits of

n

in the same order.

1

IOQM P7 2024

Determine the sum of all possible surface area of a cube two of whose vertices are

(1,2,0)

(3,3,2)

1

IOQM P6 2024

Find the number of triples of real numbers

(a,b,c)

a^{20} + b^{20} + c^{20} = a^{24} + b^{24} + c^{24} =1

1

IOQM P5 2024

a = \frac{x}{y} +\frac{y}{z} +\frac{z}{x}

b = \frac{x}{z} +\frac{y}{x} +\frac{z}{y}

c = \left(\frac{x}{y} +\frac{y}{z} \right)\left(\frac{y}{z} +\frac{z}{x} \right)\left(\frac{z}{x} +\frac{x}{y} \right)

|ab-c|

1

IOQM P4 2024

ABCD

be a quadrilateral with

\angle ADC = 70^{\circ}

\angle ACD = 70^{\circ}

\angle ACB = 10^{\circ}

\angle BAD = 110^{\circ}

. The measure of

\angle CAB

(in degrees) is:

1

IOQM P3 2024

The number obtained by taking the last two digits of

5^{2024}

in the same order is:

1

IOQM P2 2024

The number of four-digit odd numbers having digits

1,2,3,4

, each occuring exactly once, is:

1

IOQM P1 2024

The smallest positive integer that does not divide

1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9