2023 IOQM Problems

Problem 1

Let $n$ be a positive integer such that $1 \leq n \leq 1000$ . Let $M_n$ be the number of integers in the set

$X_n = \left\{\sqrt{4n + 1}, \sqrt{4n + 2}, \ldots, \sqrt{4n + 1000}\right\}$ . Let $a = \max\{M_n : 1 \leq n \leq 1000\}$ , and $b = \min\{M_n : 1 \leq n \leq 1000\}$ .

Find $a - b$ .

Solution

Video link containing all the solutions for this paper https://youtu.be/NXzyDJKbM1k?si=uUByYIteDqHY9-_L

Problem 2

Find the number of elements in the set

$\lbrace(a.b)\in N: 2 \leq a,b \leq2023,\:\: \log_{a}{b}+6\log_{b}{a}=5\rbrace$

Problem 3

Let $\alpha$ and $\beta$ be positive integers such that $\frac{16}{37}<\frac{\alpha}{\beta}<\frac{7}{16}$ Find the smallest possible value of $\beta$ .

Problem 4

Let $x, y$ be positive integers such that $x^{4}=(x-1)(y^{3}-23)-1$ Find the maximum possible value of $x + y$ .

Problem 5

In a triangle $ABC$ , let $E$ be the midpoint of $AC$ and $F$ be the midpoint of $AB$ . The medians $BE$ and $CF$ intersect at $G$ . Let $Y$ and $Z$ be the midpoints of $BE$ and $CF$ respectively. If the area of triangle $ABC$ is $480$ , find the area of triangle $GYZ$ .

Problem 6

Let $X$ 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 $X$ .

Problem 7

Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ 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.

Problem 8

Given a 2 x 2 tile and seven dominoes (2 x 1 tile), find the number of ways of tilling a 2 x 7 rectangle using some of these tiles.

Problem 9

Find the number of triples $(a, b, c)$ of positive integers such that (a) $ab$ is a prime;

(b) $bc$ is a product of two primes;

(c) $abc$ is not divisible by square of any prime and

(d) $abc\leq30$

Problem 10

The sequence $\{a_n\}_{n\geq0}$ is defined by $a_0 = 1$ , $a_1 = -4$ , and $a_{n+2} = -4a_{n+1} - 7a_n, \text{for } n \geq 0.$ Find the number of positive integer divisors of $a_{50}^2 - a_{49}a_{51}$ .

Problem 11

A positive integer $m$ haas the property that $m^2$ is expressible in the form $4n^2-5n+16$ , where n is an integer. Find the maximum value of $|m-n|$ .

Problem 13

The ex-radii of a triangle are 10 1/ 2, 12 and 14. If the sides of the triangle are the roots of the cubic $x^3-px^2+qx-r=0$ , where $p, q, r$ are integers, find the integer nearest to $\sqrt{p+q+r}$

Problem 15

Let $ABCD$ be a unit square. Suppose $M$ and $N$ are points on $BC$ and $CD$ respectively such that the perimeter of triangle $MCN$ is $2$ . Let $O$ be the circumcenter of triangle $MAN$ , and $P$ be the circumcenter of triangle $MON$ . If $\left(\dfrac{OP}{OA}\right)^2=\dfrac{m}{n}$ for some relatively prime positive integers $m$ and $n$ , find the value of $m+n$ .

Problem 16

The six sides of a convex hexagon $A_1A_2A_3A_4A_5A_6$ 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 $A_iA_jA_k$ , where $1 \leq i < j < k \leq 6$ , has at least one red side, find the sum of the squares of the digits of $N$ .

Problem 17

Consider the set $S = \{ (a, b, c, d, e) : 0 < a < b < c < d < e < 100 \}$ where $a, b, c, d, e$ 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 S, find the largest integer less than or equal to D.

Problem 18

Let $P$ be a convex polygon with $50$ vertices. A set $F$ of diagonals of $P$ is said to be minimally friendly if any diagonal $d$ ∈ $F$ intersects at most one other diagonal in $F$ at a point interior to $P$ . Find the largest possible number of elements in a minimally friendly set $F$ .

Problem 19

For $n$ ∈ $N$ , let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$ . Consider the set $A=\{n \in N : P(n) \text{is non-zero, square free and } S(n) \text{ is a proper divisor of } P(n) \}$ . Find the maximum possible number of digits of the numbers in $A$ .

Problem 20

For any finite non empty set $X$ of integers, let $\text{max} (X)$ denote the largest element of $X$ and $|X|$ denote the number of elements in $X$ . If $N$ is the number of ordered pairs $(A, B)$ of finite non-empty sets of positive integers, such that $\text{max} (A)$ × $|B|=12$ ; and $|A|$ × $\text{max} (B)=11$ and $N$ can be written as $100a + b$ where $a, b$ are positive integers less than $100$ , find $a + b$ .

Problem 21

For n ∈ N, consider non-negative integer valued function f on {1,2,...,n} satisfying $f(i)\ge f(j)$ for $i>j$ and $\sum_{i=1}^{n} i+f(i) = 2023$ . Choose n such that $\sum_{i=1}^{n}f(i)$ is the least. How many such functions exist in that case?

Problem 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 $N$ .

Problem 23

In the coordinate plane, a point is called a $\textit{lattice point}$ if both of its coordinates are integers. Let $A$ be the point $(12, 84)$ . Find the number of right angled triangles $ABC$ in the coordinate plane where $B$ and $C$ are lattice points, having a right angle at the vertex $A$ and whose incenter is at the origin $(0, 0)$ .

Problem 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 $\{5, 6, 7, 8, 9, 10\}$ .

Problem 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.

Problem 26

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 a change for 2 Bens in two ways: 2 one Ben notes or 1 two Ben note. For 5 Ben one can give 1 one Ben note and 1 four Ben note or 1 one Ben note and 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 change for 100 Bens, following the rules of the Government.

Problem 27

A quadruple $(a, b, c, d)$ of distinct integers is said to be $\textit{balanced}$ if $a + c = b + d$ . Let $\mathcal{S}$ be any set of quadruples $(a, b, c, d)$ where $1 \leq a < b < d < c \leq 20$ and where the cardinality of $\mathcal{S}$ is $4411$ . Find the least number of balanced quadruples in $\mathcal{S}.$

Problem 28

On each side of an equilateral triangle with side length n units, where n is an integer, $1 \le n \le 100$ , consider $n-1$ 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.

Problem 29

A positive integer $n > 1$ is called $beautiful$ if $n$ can be written in one and only one way as $n = a_1 + a_2 +... + a_k = a_1 a_2 ... a_k$ for some positive integers $a_1, a_2, . . . , a_k$ , where $k > 1$ and $a_1 \geq a_2 \geq ... \geq a_k$ . (For example 6 is beautiful since 6 = 3 · 2 · 1 = 3 + 2 + 1 , and this is unique. But 8 is not beautiful since 8 = 4 + 2 + 1 + 1 = 4 · 2 · 1 · 1 as well as 8 = 2 + 2 + 2 + 1 + 1 = 2 · 2 · 2 · 1 · 1 , so uniqueness is lost.) Find the largest beautiful number less than 100.

Problem 30

Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$ . If $r$ is the no of integers $n \leq 2023$ for which $\sum_{i = 1}^{n} d(i)$ is odd, find the sum of the digits of $r$ .

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