Math Message Boards FAQ & Community Help | AoPS

Art of Problem Solving

AoPS Online

2022-23 IOQM India

3

1

A triangle $ABC$ with $AC=20$ is inscribed in a circle $\omega$ . A tangent $t$ to $\omega$ is drawn through $B$ . The distance $t$ from $A$ is $25$ and that from $C$ is $16$ .If $S$ denotes the area of the triangle $ABC$ , find the largest integer not exceeding $\frac{S}{20}$

lifeismathematics

2

In a paralleogram $ABCD$ , a point $P$ on the segment $AB$ is taken such that $\frac{AP}{AB}=\frac{61}{2022}$
and a point $Q$ on the segment $AD$ is taken such that $\frac{AQ}{AD}=\frac{61}{2065}$ .If $PQ$ intersects $AC$ at $T$ , find $\frac{AC}{AT}$ to the nearest integer

lifeismathematics

3

In a trapezoid $ABCD$ , 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 $AB$ at $M$ and side $BE$ at the point $P$ . Find the angle $DAE$ in degrees, if $AB:MP=2$ .

lifeismathematics

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 $M+N$

lifeismathematics

5

Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\cdots+(m+10)^2$ is the square of a positive integer $n$ . Find $m+n$

lifeismathematics

6

Let $a,b$ be positive integers satisfying $a^3-b^3-ab=25$ . Find the largest possible value of $a^2+b^3$ .

lifeismathematics

7

Find the number of ordered pairs $(a,b)$ such that $a,b \in \{10,11,\cdots,29,30\}$ and
$\hspace{1cm}$ $GCD(a,b)+LCM(a,b)=a+b$ .

lifeismathematics

8

Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$ .Write $\frac{p}{q}$ as $l+\frac{m}{n}$ , where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$ . Find the maximum value of $l+m+n$ .

lifeismathematics

9

Two sides of an integer sided triangle have lengths $18$ and $x$ . If there are exactly $35$ possible integer $y$ such that $18,x,y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have.

lifeismathematics

10

Consider the $10$ -digit number $M=9876543210$ . 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 $M$ .

lifeismathematics

11

Let $AB$ be diameter of a circle $\omega$ and let $C$ be a point on $\omega$ , different from $A$ and $B$ . The perpendicular from $C$ intersects $AB$ at $D$ and $\omega$ at $E(\neq C)$ . The circle with centre at $C$ and radius $CD$ intersects $\omega$ at $P$ and $Q$ . If the perimeter of the triangle $PEQ$ is $24$ , find the length of the side $PQ$

lifeismathematics

12

Given $\triangle{ABC}$ with $\angle{B}=60^{\circ}$ and $\angle{C}=30^{\circ}$ , let $P,Q,R$ be points on the sides $BA,AC,CB$ respectively such that $BPQR$ is an isosceles trapezium with $PQ \parallel BR$ and $BP=QR$ .
Find the maximum possible value of $\frac{2[ABC]}{[BPQR]}$ where $[S]$ denotes the area of any polygon $S$ .

lifeismathematics

13

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC$ such that $AD=BC$ .
Suppose $\angle{CAD}=x^{\circ}, \angle{ABC}=y^{\circ}$ and $\angle{ACB}=z^{\circ}$ and $x,y,z$ 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 $\angle{ABC}$ in degrees.

lifeismathematics

14

Let $x,y,z$ be complex numbers such that
$\hspace{ 2cm} \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}=9$
$\hspace{ 2cm} \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}=64$
$\hspace{ 2cm} \frac{x^3}{y+z}+\frac{y^3}{z+x}+\frac{z^3}{x+y}=488$

If $\frac{x}{yz}+\frac{y}{zx}+\frac{z}{xy}=\frac{m}{n}$ where $m,n$ are positive integers with $GCD(m,n)=1$ , find $m+n$ .

lifeismathematics

15

Let $x,y$ be real numbers such that $xy=1$ . Let $T$ and $t$ be the largest and smallest values of the expression
$\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$
.

If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $GCD(m,n)=1$ , find the value of $m+n$ .

lifeismathematics

16

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

Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s}\in \mathbb{Q}$ and $\frac{t}{u}\in \mathbb{Q}$ , in lowest form, find $r+s+t+u$ .

lifeismathematics

17

For a positive integer $n>1$ , let $g(n)$ denote the largest positive proper divisor of $n$ and $f(n)=n-g(n)$ . For example, $g(10)=5, f(10)=5$ and $g(13)=1,f(13)=12$ . Let $N$ be the smallest positive integer such that $f(f(f(N)))=97$ . Find the largest integer not exceeding $\sqrt{N}$

lifeismathematics

18

Let $m,n$ be natural numbers such that
$\hspace{2cm} m+3n-5=2LCM(m,n)-11GCD(m,n).$
Find the maximum possible value of $m+n$ .

lifeismathematics

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 $n=190$ , one may put $+$ signs so as to get $11$ ninety times and $1$ ten times , and get the sum $1000$ . If $a$ 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 $a$ .

lifeismathematics

20

For an integer $n\ge 3$ and a permutation $\sigma=(p_{1},p_{2},\cdots ,p_{n})$ of $\{1,2,\cdots , n\}$ , we say $p_{l}$ is a $landmark$ point if $2\le l\le n-1$ and $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$ . For example , for $n=7$ ,
the permutation $(2,7,6,4,5,1,3)$ has four landmark points: $p_{2}=7$ , $p_{4}=4$ , $p_{5}=5$ and $p_{6}=1$ . For a given $n\ge 3$ , let $L(n)$ denote the number of permutations of $\{1,2,\cdots ,n\}$ with exactly one landmark point. Find the maximum $n\ge 3$ for which $L(n)$ is a perfect square.

lifeismathematics

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 $N$ .

lifeismathematics

22

A binary sequence is a sequence in which each term is equal to $0$ or $1$ . 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 $0,1,1,0,0,1,1,1$ is $\text{friendly}$ . Let $F_{n}$ denote the number of $\text{friendly}$ binary sequences with $n$ terms. Find the smallest positive integer $n\ge 2$ such that $F_{n}>100$

lifeismathematics

23

In a triangle $ABC$ , the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$ . Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$ . Given that $BE=3,BA=4$ , find the integer nearest to $BC^2$ .

lifeismathematics

24

Let $N$ 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 $6$ If $N=100a+b$ , where $a,b$ are positive integers less than $100$ , find $a+b.$

lifeismathematics

© 2025 AoPS Incorporated

a