Olympiad problems

2022 Kosovo & Albania Mathematical Olympiad, 1

Find all pairs of integers $(m, n)$ such that $$m+n = 3(mn+10).$$

2022 Thailand Online MO, 5

Tags: similar triangles , angle chasing , reflection , geometry

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. Let $M_B$ and $M_C$ be the midpoints of $AC$ and $AB$, respectively. Place points $X$ and $Y$ on line $BC$ such that $\angle HM_BX = \angle HM_CY = 90^{\circ}$. Prove that triangles $OXY$ and $HBC$ are similar.

2021 Iranian Combinatorics Olympiad, P1

Tags: combinatorics

In the lake, there are $23$ stones arranged along a circle. There are $22$ frogs numbered $1, 2, \cdots, 22$ (each number appears once). Initially, each frog randomly sits on a stone (several frogs might sit on the same stone). Every minute, all frogs jump at the same time as follows: the frog number $i$ jumps $i$ stones forward in the clockwise direction. (In particular, the frog number $22$ jumps $1$ stone in the counter-clockwise direction.) Prove that at some point, at least $6$ stones will be empty.

2022 Saudi Arabia BMO + EGMO TST, p2

Tags: algebra

Determine if there exist functions $f, g : R \to R$ satisfying for every $x \in R$ the following equations $f(g(x)) = x^3$ and $g(f(x)) = x^2$.

1996 Austrian-Polish Competition, 7

Tags: number theory

Prove there are no such integers $ k, m $ which satisfy $ k \ge 0, m \ge 0 $ and $ k!+48=48(k+1)^m $.

2023 Malaysian Squad Selection Test, 6

Tags: geometry

Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

2012 Korea Junior Math Olympiad, 3

Tags: number theory

Find all $l,m,n \in\mathbb{N}$ that satisfies the equation $5^l43^m+1=n^3$

2013 Hitotsubashi University Entrance Examination, 2

Tags: geometry

Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$.

2000 IMO, 4

Tags: function , arithmetic sequence , combinatorics , set partition

A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician locates the box from which no card has been drawn. How many ways are there to put the cards in the three boxes so that the trick works?

1979 IMO Longlists, 12

Tags: 3d geometry , prism , combinatorics , pentagon , geometry

We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots,5$ is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

ICMC 7, 1

Tags: number theory

Prove that there exist distinct positive integers $a_1, a_2,\ldots , a_{2024}$ such that for each $i\in\{1,2,\ldots,2024\}$\[a_i\mid a_1a_2\cdots a_{i-1}a_{i+1}\cdots a_{2024}+k,\]where a) $k=1$ and b) $k=2024.$ [i]Proposed by Ishan Nath[/i]

2014 NIMO Problems, 3

Tags:

Call an integer $k$ [i]debatable[/i] if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.) [i]Proposed by Lewis Chen[/i]

2021 JBMO Shortlist, C5

Tags: combinatorics

Let $M$ be a subset of the set of $2021$ integers $\{1, 2, 3, ..., 2021\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a + b - c | > 10$. Determine the largest possible number of elements of $M$.

2002 National Olympiad First Round, 2

Tags: modular arithmetic

What is $3^{2002}$ in $\bmod 11$? $ \textbf{a)}\ 1 \qquad\textbf{b)}\ 3 \qquad\textbf{c)}\ 4 \qquad\textbf{d)}\ 5 \qquad\textbf{e)}\ \text{None of above} $

2004 China Team Selection Test, 2

Tags: matrix , combinatorics , ramsey theory , extremal combinatorics

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

2024 Bulgaria MO Regional Round, 11.3

Tags: number theory , divisor , extremal

A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

2023 Brazil Cono Sur TST, 3

Tags: invariant

The numbers $1, 2, \dots , 50$ are written on a board. Letícia performs the following actions: she erases two numbers $a$ and $b$ on the board, writes the number $a+b$ on it and notes the number $ab(a+b)$ in her notebook. After performing these operations $49$ times, when there is only one number written on the board, Letícia calculates the sum $S$ of the $49$ numbers in the notebook. a) Prove that $S$ doesn't depend on the order Letícia chooses the numbers to perform the operations. b) Find the value of $S$.

2015 All-Russian Olympiad, 2

Tags: circumcircle , parallelogram , geometry

Given is a parallelogram $ABCD$, with $AB <AC <BC$. Points $E$ and $F$ are selected on the circumcircle $\omega$ of $ABC$ so that the tangenst to $\omega$ at these points pass through point $D$ and the segments $AD$ and $CE$ intersect. It turned out that $\angle ABF = \angle DCE$. Find the angle $\angle{ABC}$. A. Yakubov, S. Berlov

2014 Contests, 2

Tags: modular arithmetic , number theory

Find all integers $n$, $n>1$, with the following property: for all $k$, $0\le k < n$, there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$.

2006 China Team Selection Test, 1

Tags: inequalities , combinatorics

Let $A$ be a non-empty subset of the set of all positive integers $N^*$. If any sufficient big positive integer can be expressed as the sum of $2$ elements in $A$(The two integers do not have to be different), then we call that $A$ is a divalent radical. For $x \geq 1$, let $A(x)$ be the set of all elements in $A$ that do not exceed $x$, prove that there exist a divalent radical $A$ and a constant number $C$ so that for every $x \geq 1$, there is always $\left| A(x) \right| \leq C \sqrt{x}$.

2005 Georgia Team Selection Test, 10

Tags: 3-variable inequality , algebra , inequalities

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

2018 CMIMC Geometry, 4

Tags: geometric transformation , rotation , geometry

Suppose $\overline{AB}$ is a segment of unit length in the plane. Let $f(X)$ and $g(X)$ be functions of the plane such that $f$ corresponds to rotation about $A$ $60^\circ$ counterclockwise and $g$ corresponds to rotation about $B$ $90^\circ$ clockwise. Let $P$ be a point with $g(f(P))=P$; what is the sum of all possible distances from $P$ to line $AB$?

2023 Durer Math Competition Finals, 5

Tags: geometry

We are given a triangle $ABC$ and two circles ($k_1$ and $k_2$) so the diameter of $k_1$ is $AB$ and the diameter of $k_2$ is $AC$. Let the intersection of $BC$ line segment and $k_1$ (that isn’t $B$) be $P,$ and the intersection of $BC$ line segment and $k_2$ (that isn’t $B$) be $Q$. We know, that $AB = 3003$ and $AC = 4004$ and $BC = 5005$. What is the distance between $P$ and $Q$?

2002 Moldova Team Selection Test, 3

Tags: locus , minimum , geometry

A triangle $ABC$ is inscribed in a circle $G$. For any point $M$ inside the triangle, $A_1$ denotes the intersection of the ray $AM$ with $G$. Find the locus of point $M$ for which $\frac{BM\cdot CM}{MA_1}$ is minimal, and find this minimum value.

2024 Thailand TSTST, 4

Tags: number theory , prime number

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences