Olympiad problems

1988 IMO Longlists, 89

We match sets $ M$ of points in the coordinate plane to sets $ M*$ according to the rule that $ (x*,y*) \in M*$ if and only if $ x \cdot x* \plus{} y \cdot y* \leq 1$ whenever $ (x,y) \in M.$ Find all triangles $ Q$ such that $ Q*$ is the reflection of $ Q$ in the origin.

2023 Sharygin Geometry Olympiad, 21

Tags: geometry , humpty points , radical axis , projections , othorcenter , Sharygin Geometry Olympiad , Sharygin 2023

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$. Prove that $P_a,P_b,P_c,P_d$ are concyclic.

May Olympiad L1 - geometry, 2015.3

Tags: geometry

In the quadrilateral $ABCD$, we have $\angle C$ is triple of $\angle A$, let $P$ be a point in the side $AB$ such that $\angle DPA = 90º$ and let $Q$ be a point in the segment $DA$ where $\angle BQA = 90º$ the segments $DP$ and $CQ$ intersects in $O$ such that $BO = CO = DO$, find $\angle A$ and $\angle C$.

2002 Croatia National Olympiad, Problem 4

Tags: Sequence , algebra , number theory

Let $(a_n)_{n\in\mathbb N}$ be an increasing sequence of positive integers. A term $a_k$ in the sequence is said to be good if it a sum of some other terms (not necessarily distinct). Prove that all terms of the sequence, apart from finitely many of them, are good.

2011 Morocco National Olympiad, 2

Tags: inequalities , geometry , perimeter , trigonometry , inequalities proposed

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

1964 Vietnam National Olympiad, 4

Tags: recurrence relation , algebra , Integer sequence

Define the sequence of positive integers $f_n$ by $f_0 = 1, f_1 = 1, f_{n+2} = f_{n+1} + f_n$. Show that $f_n =\frac{ (a^{n+1} - b^{n+1})}{\sqrt5}$, where $a, b$ are real numbers such that $a + b = 1, ab = -1$ and $a > b$.

2025 Caucasus Mathematical Olympiad, 3

Tags: combinatorics

A circle is drawn on the board, and $2n$ points are marked on it, dividing it into $2n$ equal arcs. Petya and Vasya are playing the following game. Petya chooses a positive integer $d \leqslant n$ and announces this number to Vasya. To win the game, Vasya needs to color all marked points using $n$ colors, such that each color is assigned to exactly two points, and for each pair of same-colored points, one of the arcs between them contains exactly $(d - 1)$ marked points. Find all $n$ for which Petya will be able to prevent Vasya from winning.

2011 Ukraine Team Selection Test, 7

Tags: modular arithmetic , number theory , equation , IMO Shortlist , LTE Lemma

Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\] [i]Proposed by Angelo Di Pasquale, Australia[/i]

2006 Romania Team Selection Test, 2

Tags: algebra , polynomial , algebra proposed

Let $p$ a prime number, $p\geq 5$. Find the number of polynomials of the form \[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$. [i]Valentin Vornicu[/i]

1996 Iran MO (2nd round), 1

Tags: inequalities , inequalities proposed

Let $a, b, c$ be real numbers. Prove that there exists a triangle with side lengths $a, b, c$ if and only if \[2(a^4 + b^4 + c^4) < (a^2 + b^2 + c^2)^2.\]

2022 Dutch IMO TST, 4

Tags: geometry , RMM , RMM 2020

Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively. Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.

2016 Saudi Arabia IMO TST, 2

Tags: algebra , Integer Polynomial , polynomial

Find all pairs of polynomials $P(x),Q(x)$ with integer coefficients such that $P(Q(x)) = (x - 1)(x - 2)...(x - 9)$ for all real numbers $x$

1961 Miklós Schweitzer, 7

Tags: college contests

[b]7.[/b] For the differential equation $ \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}= 2\frac{\partial^2 u}{\partial x \partial y} $ find all solutions of the form $u(x,y)=f(x)g(y)$. [b](R. 14)[/b]

2009 AMC 10, 5

Tags: FTW , symmetry , AMC

What is the sum of the digits of the square of $ 111,111,111$? $ \textbf{(A)}\ 18 \qquad \textbf{(B)}\ 27 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 63 \qquad \textbf{(E)}\ 81$

2006 Lithuania Team Selection Test, 4

Tags: geometry , combinatorics unsolved , combinatorics

Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.

2016 AMC 10, 2

Tags: AMC10 , function , AMC , AMC 10 , AMC 10 B

If $n\heartsuit m=n^3m^2$, what is $\frac{2\heartsuit 4}{4\heartsuit 2}$? $\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ 4$

1995 Denmark MO - Mohr Contest, 2

Tags: algebra , Sum of Squares

Find all sets of five consecutive integers with that property that the sum of the squares of the first three numbers is equal to the sum of the squares on the last two.

2018 Malaysia National Olympiad, B3

Tags: number theory , factorial

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

2010 Contests, 1

Tags: modular arithmetic , number theory proposed , number theory

Suppose that $m$ and $k$ are non-negative integers, and $p = 2^{2^m}+1$ is a prime number. Prove that [b](a)[/b] $2^{2^{m+1}p^k} \equiv 1$ $(\text{mod } p^{k+1})$; [b](b)[/b] $2^{m+1}p^k$ is the smallest positive integer $n$ satisfying the congruence equation $2^n \equiv 1$ $(\text{mod } p^{k+1})$.

2014 Contests, 1

Tags: geometry , perpendicular , isosceles

As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]

2021 Science ON all problems, 2

Tags: combinatorics

There is a football championship with $6$ teams involved, such that for any $2$ teams $A$ and $B$, $A$ plays with $B$ and $B$ plays with $A$ ($2$ such games are distinct). After every match, the winning teams gains $3$ points, the loosing team gains $0$ points and if there is a draw, both teams gain $1$ point each.\\ \\ In the end, the team standing on the last place has $12$ points and there are no $2$ teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations. $\textit{(Andrei Bâra)}$

2018 India PRMO, 9

Tags: trinomial , Integers , Root , maximum

Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?

2012 China Team Selection Test, 3

Tags: pigeonhole principle , combinatorics proposed , combinatorics

Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have \[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]

2016 APMO, 3

Tags: geometry

Let $AB$ and $AC$ be two distinct rays not lying on the same line, and let $\omega$ be a circle with center $O$ that is tangent to ray $AC$ at $E$ and ray $AB$ at $F$. Let $R$ be a point on segment $EF$. The line through $O$ parallel to $EF$ intersects line $AB$ at $P$. Let $N$ be the intersection of lines $PR$ and $AC$, and let $M$ be the intersection of line $AB$ and the line through $R$ parallel to $AC$. Prove that line $MN$ is tangent to $\omega$. [i]Warut Suksompong, Thailand[/i]

2004 China Team Selection Test, 3

Tags: number theory , relatively prime , number theory unsolved

Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.