2023-24 IOQM India

Part of India IOQM

Subcontests

(30)

1

IOQM 2023-24 P-30

d(m)

denote the number of positive integer divisors of a positive integer

m

r

is the number of integers

n \leqslant 2023

\sum_{i=1}^{n} d(i)

is odd. , find the sum of digits of

r.

1

IOQM 2023-24 P-27

(a,b,c,d)

of distinct integers is said to be

balanced

a+c=b+d

\mathcal{S}

be any set of quadruples

(a,b,c,d)

1 \leqslant a<b<d<c \leqslant 20

and where the cardinality of

\mathcal{S}

4411

. Find the least number of balanced quadruples in

\mathcal{S}.

1

IOQM 2023-24 P-25

Find the least positive integer

n

such that there are at least

1000

unordered pairs of diagonals in a regular polygon with

n

vertices that intersect at a right angle in the interior of the polygon.

1

IOQM 2023-24 P-24

A trapezium in the plane is a quadrilateral in which a pair of opposite sides are parallel. A trapezium is said to be non-degenerate if it has positive area. Find the number of mutually non-congruent, non-degenerate trapeziums whose sides are four distinct integers from the set

\{5,6,7,8,9,10\}

1

IOQM 2023-24 P-23

In the coordinate plane, a point is called a

\text{lattice point}

if both of its coordinates are integers. Let

A

(12,84)

. Find the number of right angled triangles

ABC

in the coordinate plane

B

C

are lattice points, having a right angle at vertex

A

and whose incenter is at the origin

(0,0)

1

IOQM 2023-24 P-21

n \in \mathbb{N}

, consider non-negative valued functions

f

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

f(i) \geqslant f(j)

i>j

\sum_{i=1}^{n} (i+ f(i))=2023.

n

\sum_{i=1}^{n} f(i)

is at least. How many such functions exist in that case?

1

IOQM 2023-24 P-18

\mathcal{P}

be a convex polygon with

50

vertices. A set

\mathcal{F}

of diagonals of

\mathcal{P}

minimally friendly

if any diagonal

d \in \mathcal{F}

intersects at most one other diagonal in

\mathcal{F}

at a point interior to

\mathcal{P}.

Find the largest possible number of elements in a

\text{minimally friendly}

\mathcal{F}

1

IOQM 2023-24 P-19

n \in \mathbb{N}

P(n)

denote the product of the digits in

n

S(n)

denote the sum of the digits in

n

. Consider the set

A=\{n \in \mathbb{N}: P(n)

is non-zero, square free and

S(n)

is a proper divisor of

P(n)\}

. Find the maximum possible number of digits of the numbers in

A

1

IOQM 2023-24 P-20

For any finite non empty set

X

of integers, let

\max (X)

denote the largest element of

X

|X|

denote the number of elements in

X

N

is the number of ordered pairs

(A, B)

of finite non-empty sets of positive integers, such that

\begin{aligned} & \max (A) \times|B|=12 ; \text { and } \\ & |A| \times \max (B)=11 \end{aligned}

N

can be written as

100 a+b

a, b

are positive integers less than 100 , find

a+b

1

IOQM 2023-24 P-29

A positive integer

n>1

is called beautiful if

n

can be written in one and only one way as

n=a_1+a_2+\cdots+a_k=a_1 \cdot a_2 \cdots a_k

for some positive integers

a_1, a_2, \ldots, a_k

k>1

a_1 \geq a_2 \geq \cdots \geq a_k

. (For example 6 is beautiful since

6=3 \cdot 2 \cdot 1=3+2+1

, and this is unique. But 8 is not beautiful since

8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1

8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1

, so uniqueness is lost.) Find the largest beautiful number less than 100.

1

IOQM 2023-24 P-28

On each side of an equilateral triangle with side length

n

n

1 \leq n \leq 100

n-1

points that divide the side into

n

equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of

n

for which it is possible to turn all coins tail up after a finite number of moves.

1

IOQM 2023-24 P-22

In an equilateral triangle of side length 6 , pegs are placed at the vertices and also evenly along each side at a distance of 1 from each other. Four distinct pegs are chosen from the 15 interior pegs on the sides (that is, the chosen ones are not vertices of the triangle) and each peg is joined to the respective opposite vertex by a line segment. If

N

denotes the number of ways we can choose the pegs such that the drawn line segments divide the interior of the triangle into exactly nine regions, find the sum of the squares of the digits of

N

1

IOQM 2023-24 P-17

Consider the set

\mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\}

a, b, c, d, e

are integers. If

D

is the average value of the fourth element of such a tuple in the set, taken over all the elements of

\mathcal{S}

, find the largest integer less than or equal to

D

1

IOQM 2023-24 P-16

The six sides of a convex hexagon

A_1 A_2 A_3 A_4 A_5 A_6

are colored red. Each of the diagonals of the hexagon is colored either red or blue. If

N

is the number of colorings such that every triangle

A_i A_j A_k

1 \leq i<j<k \leq 6

, has at least one red side, find the sum of the squares of the digits of

N

1

IOQM 2023-24 P-15

A B C D

be a unit square. Suppose

M

N

B C

C D

respectively such that the perimeter of triangle

M C N

O

be the circumcentre of triangle

M A N

P

be the circumcentre of triangle

M O N

\left(\frac{O P}{O A}\right)^2=\frac{m}{n}

for some relatively prime positive integers

m

n

, find the value of

m+n

1

IOQM 2023-24 P-14

A B C

be a triangle in the

x y

B

is at the origin

(0,0)

B C

D

B C: C D=1: 1, C A

E

C A: A E=1: 2

A B

F

A B: B F=1: 3

G(32,24)

be the centroid of the triangle

A B C

K

be the centroid of the triangle

D E F

. Find the length

G K

1

IOQM 2023-24 P-13

The ex-radii of a triangle are

10\frac{1}{2}, 12

14

. If the sides of the triangle are the roots of the cubic

x^3-px^2+qx-r=0

p, q,r

are integers , find the nearest integer to

\sqrt{p+q+r}.

1

IOQM 2023-24 P-12

P(x)=x^3+ax^2+bx+c

be a polynomial where

a,b,c

are integers and

c

p_{i}

be the value of

P(x)

x=i

p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}

, find the value of

p_{2}+2p_{1}-3p_{0}.

1

IOQM 2023-24 P-11

A positive integer

m

has the property that

m^2

is expressible in the form

4n^2-5n+16

n

is an integer (of any sign). Find the maximum value of

|m-n|.

1

IOQM 2023-24 P-10

\{a_{n}\}_{n \geqslant 0}

a_{0}=1, a_{1}=-4

a_{n+2}=-4a_{n+1}-7a_{n}

n \geqslant 0

. Find the number of positive integer divisors of

a^2_{50}-a_{49}a_{51}

1

IOQM 2023-24 P26

In the land of Binary , the unit of currency is called Ben and currency notes are available in denominations

1,2,2^2,2^3,..

Bens. The rules of the Government of Binary stipulate that one can not use more than two notes of any one denomination in any transaction. For example, one can give change for

2

Bens in two ways :

2

one Ben notes or

1

two Ben note. For

5

Ben one can given

1

1

four Ben note or

1

2

two Ben notes. Using

5

one Ben notes or

3

one Ben notes and

1

two Ben notes for a

5

Ben transaction is prohibited. Find the number of ways in which one can give a change

100

Bens following the rules of the Government.

1

IOQM 2023-24 P-9

Find the number of triples

(a, b, c)

of positive integers such that (a)

a b

is a prime; (b)

b c

is a product of two primes; (c)

a b c

is not divisible by square of any prime and (d)

a b c \leq 30

1

IOQM 2023-24 P-8

2 \times 2

tile and seven dominoes (

2 \times 1

tile), find the number of ways of tiling (that is, cover without leaving gaps and without overlapping of any two tiles) a

2 \times 7

rectangle using some of these tiles.

1

IOQM 2023-24 P-7

Unconventional dice are to be designed such that the six faces are marked with numbers from

1

6

1

2

appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.

1

IOQM 2023-24 P-6

X

be the set of all even positive integers

n

such that the measure of the angle of some regular polygon is

n

degrees. Find the number of elements in

X

1

IOQM 2023-24 P-5

A B C

E

be the midpoint of

A C

F

be the midpoint of

A B

B E

C F

G

Y

Z

be the midpoints of

B E

C F

respectively. If the area of triangle

A B C

is 480 , find the area of triangle

G Y Z

1

IOQM 2023-24 P-4

x, y

be positive integers such that

x^4=(x-1)\left(y^3-23\right)-1 .

Find the maximum possible value of

x+y

1

IOQM 2023-24 P-3

\alpha

\beta

be positive integers such that

\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16} .

Find the smallest possible value of

\beta

1

IOQM 2023-24 P-2

Find the number of elements in the set

\left\{(a, b) \in \mathbb{N}: 2 \leq a, b \leq 2023, \log _a(b)+6 \log _b(a)=5\right\}

1

IOQM 2023-24 P-1

n

be a positive integer such that

1 \leq n \leq 1000

M_n

be the number of integers in the set

X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots, \sqrt{4 n+1000}\}

a=\max \left\{M_n: 1 \leq n \leq 1000\right\} \text {, and } b=\min \left\{M_n: 1 \leq n \leq 1000\right\} \text {. }

a-b