Olympiad problems

2022-23 IOQM India, 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 $AB:MP=2$.

2020-21 IOQM India, 19

Tags: IOQM , geometry , parallelogram

Let $ABCD$ be a parallelogram. Let $E$ and $F$ be the midpoints of sides $AB$ and $BC$ respectively. The lines $EC$ and $FD$ intersect at $P$ and form four triangles $APB, BPC, CPD, DPA$. If the area of the parallelogram is $100$, what is the maximum area of a triangles among these four triangles?

2022-23 IOQM India, 8

Tags: number theory , prime numbers , IOQM

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 $l+m+n$.

2022-23 IOQM India, 9

Tags: geometry , IOQM

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.

2020-21 IOQM India, 10

Tags: statistics , median , mean , algebra , IOQM

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? [i](The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)[/i]

2022-23 IOQM India, 17

Tags: nt , IOQM

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 $\sqrt{N}$

2022-23 IOQM India, 19

Tags: combinatorics , counting , IOQM

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 $a$.

2022-23 IOQM India, 7

Tags: nt , IOQM

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

2022 IOQM India, 11

Tags: combinatorics , IOQM

In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife?

2022 IOQM India, 1

Tags: geometry , IOQM

Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$ and $C$ lies on $L_2$. Find the area of the square.

2022 IOQM India, 2

Tags: IOQM

Ria writes down the numbers $1,2,\cdots, 101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen?

2020-21 IOQM India, 13

Tags: number theory , IOQM , Olympiad , Olympiads

Find the sum of all positive integers $n$ for which $\mid 2^n + 5^n - 65 \mid$ is a perfect square.

2022 IOQM India, 6

Tags: IOQM , Problem solving strategies , folklore

Let $x,y,z$ be positive real numbers such that $x^2 + y^2 = 49, y^2 + yz + z^2 = 36$ and $x^2 + \sqrt{3}xz + z^2 = 25$. If the value of $2xy + \sqrt{3}yz + zx$ can be written as $p \sqrt{q}$ where $p,q \in \mathbb{Z}$ and $q$ is squarefree, find $p+q$.

2022 IOQM India, 12

Tags: combinatorics , IOQM

A $12 \times 12$ board is divided into $144$ unit squares by drawing lines parallel to the sides. Two rooks placed on two unit squares are said to be non-attacking if they are not in the same column or same row. Find the least number $N$ such that if $N$ rooks are placed on the unit squares, one rook per square, we can always find $7$ rooks such that no two are attacking each other.

2023-24 IOQM India, 30

Tags: IOQM

Let $d(m)$ denote the number of positive integer divisors of a positive integer $m$. If $r$ is the number of integers $n \leqslant 2023$ for which $\sum_{i=1}^{n} d(i)$ is odd. , find the sum of digits of $r.$

2022-23 IOQM India, 4

Tags: IOQM , algebra , diophantine

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 $M+N$

2022-23 IOQM India, 11

Tags: geometry , perimeter , IOQM

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 $PQ$

2022-23 IOQM India, 13

Tags: geometry , trig bash , arithmetic sequence , IOQM

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.

2020-21 KVS IOQM India, 5

Tags: IOQM , KV

Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$.

2023-24 IOQM India, 29

Tags: IOQM , IOQM 2023 , number theory

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 $a_1, a_2, \ldots, a_k$, where $k>1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. (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 $8=4+2+1+1=4 \cdot 2 \cdot 1 \cdot 1$ as well as $8=2+2+2+1+1=2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost.) Find the largest beautiful number less than 100.

2023-24 IOQM India, 5

Tags: geometry , IOQM , areas , area , area of a triangle , barycentric coordinates , barycentric

In a triangle $A B C$, let $E$ be the midpoint of $A C$ and $F$ be the midpoint of $A B$. The medians $B E$ and $C F$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $B E$ and $C F$ respectively. If the area of triangle $A B C$ is 480 , find the area of triangle $G Y Z$.

2022-23 IOQM India, 5

Tags: nt , IOQM

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 $m+n$

2022-23 IOQM India, 18

Tags: number theory , IOQM

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 $m+n$.

2022-23 IOQM India, 16

Tags: algebra , IOQM , system of equations

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 $r+s+t+u$.

2022-23 IOQM India, 10

Tags: combinatorics , Permutations and Combination , Enumerative Combinatorics , india , IOQM

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 $M$.