Olympiad problems

2022-23 IOQM India, 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 $S$.

2020-21 KVS IOQM India, 9

Tags: IOQM , KV , algebra , system of equations

find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equations: $x+y+z=7; x^2+y^2+z^2=27; xyz=5$.

2020-21 KVS IOQM India, 11

Tags: IOQM , KV , number theory , prime numbers

The prime numbers $a,b$ and $c$ are such that $a+b^2=4c^2$. Determine the sum of all possible values of $a+b+c$.

2009 Greece National Olympiad, 1

Tags: number theory unsolved , number theory , IOQM

Find all positive integers $n$ such that the number \[A=\sqrt{\frac{9n-1}{n+7}}\] is rational.

2022 IOQM India, 4

Tags: IOQM

Consider the set of all 6-digit numbers consisting of only three digits, $a,b,c$ where $a,b,c$ are distinct. Suppose the sum of all these numbers is $593999406$. What is the largest remainder when the three digit number $abc$ is divided by $100$?

2023-24 IOQM India, 11

Tags: IOQM

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

2020-21 KVS IOQM India, 7

Tags: IOQM , KV

$a,b,c$ are positive real numbers such that $a^2+b^2=c^2$ and $ab=c$. Determine the value of $\left\lvert{\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}}\right\rvert$

2020-21 KVS IOQM India, 2

Tags: IOQM , KV , geometry , rectangle

If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$. Find $CP$.

2022-23 IOQM India, 21

Tags: recursion , geometry , 3D geometry , IOQM

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 $N$.

2022 IOQM India, 5

Tags: geometry , parallelogram , IOQM

In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$

2022-23 IOQM India, 24

Tags: combinatorics , counting , distinguishability , IOQM

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 $a+b.$

2023-24 IOQM India, 10

Tags: IOQM , algebra

The Sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by $a_{0}=1, a_{1}=-4$ and $a_{n+2}=-4a_{n+1}-7a_{n}$ , for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50}-a_{49}a_{51}$.

2022-23 IOQM India, 15

Tags: algebra , inequalities , IOQM

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 $m+n$.

2022 IOQM India, 3

Tags: number theory , prime numbers , arithmetic sequence , geometry , perimeter , IOQM , trivial

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

2022-23 IOQM India, 23

Tags: geometry , ratio , IOQM , trigonometry , Ratio Lemma

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 $BC^2$.

2020-21 IOQM India, 15

Tags: IOQM , combinatorics , Enumerative Combinatorics

Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?

2020-21 IOQM India, 17

Tags: IOQM

How many two digit numbers have exactly $4$ positive factors? $($Here $1$ and the number $n$ are also considered as factors of $n. )$

2020-21 IOQM India, 28

Tags: number theory , IOQM , Olympiads , Olympiad

A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$.

2020-21 KVS IOQM India, 4

Tags: IOQM , KV , geometry , incenter

Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?

2020-21 KVS IOQM India, 10

Tags: IOQM , KV

Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.

2020-21 KVS IOQM India, 6

Tags: IOQM , KV

Let $ABCD$ be a square with side length $100$. A circle with centre $C$ and radius $CD$ is drawn. Another circle of radius $r$, lying inside $ABCD$, is drawn to touch this circle externally and such that the circle also touches $AB$ and $AD$. If $r=m+n\sqrt{k}$, where $m,n$ are integers and $k$ is a prime number, find the value of $\frac{m+n}k$.

2020-21 IOQM India, 25

Tags: IOQM , number theory , Olympiad , Olympiads

For a positive integer $n$, let $\langle n \rangle$ denote the perfect square integer closest to $n$. For example, $\langle 74 \rangle = 81$, $\langle 18 \rangle = 16$. If $N$ is the smallest positive integer such that $$ \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N $$ find the sum of the squares of the digits of $N$.

2020-21 KVS IOQM India, 8

Tags: IOQM , KV

Find the largest $2$-digit number $N$ which is divisible by $4$, such that all integral powers of $N$ end with $N$.

2023-24 IOQM India, 7

Tags: IOQM , Dice , combinatorics , counting , permutations

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.

2020-21 KVS IOQM India, 3

Tags: IOQM , KV

Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?