2022-23 IOQM India

Part of India IOQM

Subcontests

(24)

1

IOQM 2022-23 P-24

N

be the number of ways of distributing

52

identical balls into

4

distinguishable boxes such that no box is empty and the difference between the number of balls in any two of the boxes is not a multiple of

6

N=100a+b

a,b

are positive integers less than

100

a+b.

1

IOQM 2022-23 P-23

ABC

AD

\angle{BAC}

1:2

AD

E

EB

is perpendicular

AB

BE=3,BA=4

, find the integer nearest to

BC^2

1

IOQM 2022-23 P-22

A binary sequence is a sequence in which each term is equal to

0

1

. A binary sequence is called

\text{friendly}

if each term is adjacent to at least on term that is equal to

1

. For example , the sequence

0,1,1,0,0,1,1,1

\text{friendly}

F_{n}

denote the number of

\text{friendly}

binary sequences with

n

terms. Find the smallest positive integer

n\ge 2

F_{n}>100

1

IOQM 2022-23 P-21

An ant is at vertex of a cube. Every

10

minutes it moves to an adjacent vertex along an edge. If

N

is the number of one hour journeys that end at the starting vertex, find the sum of the squares of the digits of

N

1

IOQM 2022-23 P-20

n\ge 3

and a permutation

\sigma=(p_{1},p_{2},\cdots ,p_{n})

\{1,2,\cdots , n\}

p_{l}

landmark

2\le l\le n-1

(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0

. For example , for

n=7

,\\ the permutation

(2,7,6,4,5,1,3)

has four landmark points:

p_{2}=7

p_{4}=4

p_{5}=5

p_{6}=1

n\ge 3

L(n)

denote the number of permutations of

\{1,2,\cdots ,n\}

with exactly one landmark point. Find the maximum

n\ge 3

L(n)

is a perfect square.

1

IOQM 2022-23 P-19

Consider a string of

n

1's

. We wish to place some

+

signs in between so that the sum is

1000

. For instance, if

n=190

+

signs so as to get

11

ninety times and

1

ten times , and get the sum

1000

a

is the number of positive integers

n

for which it is possible to place

+

signs so as to get the sum

1000

, then find the sum of digits of

a

1

IOQM 2022-23 P-18

m,n

be natural numbers such that \\

\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).

\\ Find the maximum possible value of

m+n

1

IOQM 2022-23 P-17

For a positive integer

n>1

g(n)

denote the largest positive proper divisor of

n

f(n)=n-g(n)

g(10)=5, f(10)=5

g(13)=1,f(13)=12

N

be the smallest positive integer such that

f(f(f(N)))=97

. Find the largest integer not exceeding

\sqrt{N}

1

IOQM 2022-23 P-16

a,b,c

be reals satisfying\\

\hspace{2cm} 3ab+2=6b, \hspace{0.5cm} 3bc+2=5c, \hspace{0.5cm} 3ca+2=4a.

\mathbb{Q}

denote the set of all rational numbers. Given that the product

abc

can take two values

\frac{r}{s}\in \mathbb{Q}

\frac{t}{u}\in \mathbb{Q}

, in lowest form, find

r+s+t+u

1

IOQM 2022-23 P-15

x,y

be real numbers such that

xy=1

T

t

be the largest and smallest values of the expression \\

\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}

T+t

can be expressed in the form

\frac{m}{n}

m,n

are nonzero integers with

GCD(m,n)=1

, find the value of

m+n

1

IOQM 2022-23 P-14

x,y,z

be complex numbers such that\\

\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9

\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64

\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488

\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}

m,n

are positive integers with

GCD(m,n)=1

m+n

1

IOQM 2022-23 P-13

ABC

be a triangle and let

D

be a point on the segment

BC

AD=BC

\angle{CAD}=x^{\circ}, \angle{ABC}=y^{\circ}

\angle{ACB}=z^{\circ}

x,y,z

are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of

\angle{ABC}

1

IOQM 2022-23 P-12

\triangle{ABC}

\angle{B}=60^{\circ}

\angle{C}=30^{\circ}

P,Q,R

be points on the sides

BA,AC,CB

respectively such that

BPQR

is an isosceles trapezium with

PQ \parallel BR

BP=QR

.\\ Find the maximum possible value of

\frac{2[ABC]}{[BPQR]}

[S]

denotes the area of any polygon

S

1

IOQM 2022-23 P-11

AB

be diameter of a circle

\omega

C

\omega

, different from

A

B

. The perpendicular from

C

AB

D

\omega

E(\neq C)

. The circle with centre at

C

CD

\omega

P

Q

. If the perimeter of the triangle

PEQ

24

, find the length of the side

PQ

1

IOQM 2022-23 P-10

10

M=9876543210

. We obtain a new

10

-digit number from

M

according to the following rule: we can choose one or more disjoint pairs of adjacent digits in

M

and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from

M=9\underline{87}6 \underline{54} 3210

by interchanging the

2

underlined pairs, and keeping the others in their places, we get

M_{1}=9786453210

. Note that any number of (disjoint) pairs can be interchanged. Find the number of new numbers that can be so obtained from

M

1

IOQM 2022-23 P-9

Two sides of an integer sided triangle have lengths

18

x

. If there are exactly

35

possible integer

y

18,x,y

are the sides of a non-degenerate triangle, find the number of possible integer values

x

1

IOQM 2022-23 P-8

Suppose the prime numbers

p

q

q^2+3p=197p^2+q

\frac{p}{q}

l+\frac{m}{n}

l,m,n

are positive integers ,

m<n

GCD(m,n)=1

. Find the maximum value of

l+m+n

1

IOQM 2022-23 P-7

Find the number of ordered pairs

(a,b)

a,b \in \{10,11,\cdots,29,30\}

\hspace{1cm}

GCD(a,b)+LCM(a,b)=a+b

1

IOQM 2022-23 P-6

a,b

be positive integers satisfying

a^3-b^3-ab=25

. Find the largest possible value of

a^2+b^3

1

IOQM 2022-23 P-5

m

be the smallest positive integer such that

m^2+(m+1)^2+\cdots+(m+10)^2

is the square of a positive integer

n

m+n

1

IOQM 2022-23 P-4

Starting with a positive integer

M

written on the board , Alice plays the following game: in each move, if

x

is the number on the board, she replaces it with

3x+2

.Similarly, starting with a positive integer

N

written on the board, Bob plays the following game: in each move, if

x

is the number on the board, he replaces it with

2x+27

.Given that Alice and Bob reach the same number after playing

4

moves each, find the smallest value of

M+N

1

IOQM 2022-23 P-3

ABCD

, the internal bisector of angle

A

intersects the base

BC

(or its extension) at the point

E

. Inscribed in the triangle

ABE

is a circle touching the side

AB

M

BE

P

. Find the angle

DAE

AB:MP=2

1

IOQM 2022-23 P-2

In a paralleogram

ABCD

P

AB

is taken such that

\frac{AP}{AB}=\frac{61}{2022}

Q

AD

is taken such that

\frac{AQ}{AD}=\frac{61}{2065}

PQ

AC

T

\frac{AC}{AT}

to the nearest integer

1

alternate segments IOQM 2022-23 P-1

ABC

AC=20

is inscribed in a circle

\omega

t

\omega

is drawn through

B

t

A

25

C

16

S

denotes the area of the triangle

ABC

, find the largest integer not exceeding

\frac{S}{20}