Olympiad problems

2003 Junior Tuymaada Olympiad, 3

In the acute triangle $ ABC $, the point $ I $ is the center of the inscribed the circle, the point $ O $ is the center of the circumscribed circle and the point $ I_a $ is the center the excircle tangent to the side $ BC $ and the extensions of the sides $ AB $ and $ AC $. Point $ A'$ is symmetric to vertex $ A $ with respect to the line $ BC $. Prove that $ \angle IOI_a = \angle IA'I_a $.

1961 AMC 12/AHSME, 27

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Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: ${{ \textbf{(A)}\ \text{infinite} \qquad\textbf{(B)}\ \text{finite, but greater than 2} \qquad\textbf{(C)}\ \text{Two} \qquad\textbf{(D)}\ \text{One} }\qquad\textbf{(E)}\ \text{Zero} } $

2016 USA Team Selection Test, 2

Tags: functional equation , algebra

Let $n \ge 4$ be an integer. Find all functions $W : \{1, \dots, n\}^2 \to \mathbb R$ such that for every partition $[n] = A \cup B \cup C$ into disjoint sets, \[ \sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a,b) W(b,c) = |A| |B| |C|. \]

2008 Bundeswettbewerb Mathematik, 3

Tags: geometry , angle bisector

Prove: In an acute triangle $ ABC$ angle bisector $ w_{\alpha},$ median $ s_b$ and the altitude $ h_c$ intersect in one point if $ w_{\alpha},$ side $ BC$ and the circle around foot of the altitude $ h_c$ have vertex $ A$ as a common point.

2014 Costa Rica - Final Round, 2

Tags: number theory

Find all positive integers $n$ such that $n!+2$ divides $(2n)!$.

1958 Kurschak Competition, 2

Tags: number theory , divide , divisible

Show that if $m$ and $n$ are integers such that $m^2 + mn + n^2$ is divisible by $9$, then they must both be divisible by $3$.

2009 Indonesia TST, 2

Tags: combinatorics

Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.

2013 SEEMOUS, Problem 1

Tags: calculus , function , integration

Find all continuous functions $f:[1,8]\to\mathbb R$, such that $$\int^2_1f(t^3)^2dt+2\int^2_1f(t^3)dt=\frac23\int^8_1f(t)dt-\int^2_1(t^2-1)^2dt.$$

2023 Costa Rica - Final Round, 3.4

Tags: combinatorics

A teacher wants her $N$ students to know each other, so she creates various clubs of three people, so that each student can participate in several clubs. The clubs are formed in such a way that if $A$ and $B$ are two people, then there is a single club such that $A$ and $B$ are two of its three members. [b](1)[/b] Show that there is no way for the teacher to form the clubs if $N = 11$. [b](2)[/b] Show that the teacher can do it if $N = 9$.

2006 All-Russian Olympiad Regional Round, 10.8

Tags: polyhedron , geometry , 3d geometry , combinatorial geometry , combinatorics

A convex polyhedron has $2n$ faces ($n\ge 3$), and all faces are triangles. What is the largest number of vertices at which converges exactly $3$ edges at a such a polyhedron ?

2007 South East Mathematical Olympiad, 2

Tags: geometry , circumcircle , cyclic quadrilateral

In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.

2016 Balkan MO Shortlist, C2

Tags: maximal , combinatorics

There are $2016$ costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.) Find the maximal $k$ such that the following holds: There are $k$ customers such that either all of them were in the shop at a specic time instance or no two of them were both in the shop at any time instance.

2019 China Team Selection Test, 2

Tags: graph theory , combinatorics

A graph $G(V,E)$ is triangle-free, but adding any edges to the graph will form a triangle. It's given that $|V|=2019$, $|E|>2018$, find the minimum of $|E|$ .

2017 Iran MO (3rd round), 3

Tags: circumcircle , geometry

In triangle $ABC$ points $P$ and $Q$ lies on the external bisector of $\angle A$ such that $B$ and $P$ lies on the same side of $AC$. Perpendicular from $P$ to $AB$ and $Q$ to $AC$ intersect at $X$. Points $P'$ and $Q'$ lies on $PB$ and $QC$ such that $PX=P'X$ and $QX=Q'X$. Point $T$ is the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$. Prove that $P',Q'$ and $T$ are collinear if and only if $\angle PBA+\angle QCA=90^{\circ}$.

2022 Federal Competition For Advanced Students, P2, 6

Tags: tiles , tiling , combinatorics

(a) Prove that a square with sides $1000$ divided into $31$ squares tiles, at least one of which has a side length less than $1$. (b) Show that a corresponding decomposition into $30$ squares is also possible. [i](Walther Janous)[/i]

2022 Ecuador NMO (OMEC), 6

Tags: prime , number theory

Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$

2022 Iran Team Selection Test, 5

Tags: algebra , sequence

Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic. Proposed by Navid Safaei

IV Soros Olympiad 1997 - 98 (Russia), 10.4

Tags: radical , algebra

Solve the equation $$ \sqrt{\sqrt{2x^2+x-3}+2x^2-3}=x.$$

2018 ASDAN Math Tournament, 1

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Each vertex on a cube is colored black or white independently at random with equal probability. What is the expected number of edges on the cube that connect vertices of different colors?

2019 Jozsef Wildt International Math Competition, W. 64

Tags: number theory

Prove that exist different natural numbers $x$, $y$, $z$, $t$ for which $$256\times 2019^{180n+1}=2x^9-2y^6+z^5-t^4$$for all $n\in \mathbb{N}^*$

2014 IMS, 11

Tags: number theory

Let the equation $a^2 + b^2 + 1=abc$ have answer in $\mathbb{N}$.Prove that $c=3$.

2008 JBMO Shortlist, 1

Tags: number theory

Find all the positive integers $x$ and $y$ that satisfy the equation $x(x - y) = 8y - 7$

2021 USAMO, 5

Tags: algebra

Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations: \begin{align*} a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\ a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\ a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\ &\vdots & &\vdots \\ a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1} \end{align*}

2016 District Olympiad, 2

Tags: function , integration , calculus

Let $ f:\mathbb{R}\longrightarrow (0,\infty ) $ be a continuous and periodic function having a period of $ 2, $ and such that the integral $ \int_0^2 \frac{f(1+x)}{f(x)} dx $ exists. Show that $$ \int_0^2 \frac{f(1+x)}{f(x)} dx\ge 2, $$ with equality if and only if $ 1 $ is also a period of $ f. $

2020 Puerto Rico Team Selection Test, 1

Tags: combinatorics , equilateral , hexagon , combinatorial geometry

We have $10,000$ identical equilateral triangles. Consider the largest regular hexagon that can be formed with these triangles without overlapping. How many triangles will not be used?