Olympiad problems

1996 IMC, 2

Evaluate the definite integral $$\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,$$ where $n$ is a natural number.

2015 NIMO Summer Contest, 7

Tags: knight , chess , puzzle , combinatorics

The NIMO problem writers have invented a new chess piece called the [i]Oriented Knight[/i]. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16\times 16$ chess board? [i] Proposed by Tony Kim and David Altizio [/i]

2023 EGMO, 6

Tags: mixtilinear , config geo , radax

Let $ABC$ be a triangle with circumcircle $\Omega$. Let $S_b$ and $S_c$ respectively denote the midpoints of the arcs $AC$ and $AB$ that do not contain the third vertex. Let $N_a$ denote the midpoint of arc $BAC$ (the arc $BC$ including $A$). Let $I$ be the incenter of $ABC$. Let $\omega_b$ be the circle that is tangent to $AB$ and internally tangent to $\Omega$ at $S_b$, and let $\omega_c$ be the circle that is tangent to $AC$ and internally tangent to $\Omega$ at $S_c$. Show that the line $IN_a$, and the lines through the intersections of $\omega_b$ and $\omega_c$, meet on $\Omega$.

2007 Ukraine Team Selection Test, 5

Tags: geometry , circumcircle , parallelogram , geometric transformation , reflection , euler , symmetry

$ AA_{3}$ and $ BB_{3}$ are altitudes of acute-angled $ \triangle ABC$. Points $ A_{1}$ and $ B_{1}$ are second points of intersection lines $ AA_{3}$ and $ BB_{3}$ with circumcircle of $ \triangle ABC$ respectively. $ A_{2}$ and $ B_{2}$ are points on $ BC$ and $ AC$ respectively. $ A_{1}A_{2}\parallel AC$, $ B_{1}B_{2}\parallel BC$. Point $ M$ is midpoint of $ A_{2}B_{2}$. $ \angle BCA \equal{} x$. Find $ \angle A_{3}MB_{3}$.

2018 Iran MO (3rd Round), 4

Tags: algebra , polynomial

Let $P(x)$ be a non-zero polynomial with real coefficient so that $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.

2002 Hong kong National Olympiad, 1

Tags: trigonometry , geometry

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

2025 Olympic Revenge, 2

Tags: geometry , circumcircle

Let $ABC$ be a scalene triangle with $\Omega_A, \Omega_B,\Omega_C$ its excircles. $T_A$ is the intersection point of the external tangent (different of $AB$) of $\Omega_A,\Omega_B$ with the external tangent (different of $AC$) of $\Omega_A, \Omega_C$. Define $T_B, T_C$ in a similar way. If $I_A, I_B, I_C$ are the excenters of $ABC$, prove that the circumcircles of $AI_AT_A, BI_BT_B, CI_CT_C$ concur in exactly two points.

2017 IMO Shortlist, N8

Tags: number theory , euclidean algorithm

Let $p$ be an odd prime number and $\mathbb{Z}_{>0}$ be the set of positive integers. Suppose that a function $f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}$ satisfies the following properties: [list] [*] $f(1,1)=0$. [*] $f(a,b)+f(b,a)=1$ for any pair of relatively prime positive integers $(a,b)$ not both equal to 1; [*] $f(a+b,b)=f(a,b)$ for any pair of relatively prime positive integers $(a,b)$. [/list] Prove that $$\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.$$

2002 All-Russian Olympiad Regional Round, 9.2

Tags: quadratic , polynomial , function , algebra

A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.

2014 MMATHS, 2

Tags: algebra

Let $(a_n)^{\infty}_{n =1}$ be a sequence of positive integers with $a_1 < a_2 < a_3 < ...$ , and for n = 1, 2, 3,..., $$a_{2n} = a_n + n.$$ Furthermore, whenever $n$ is prime, so is $a_n$. Prove that $a_n = n$.