Olympiad problems

2003 Swedish Mathematical Competition, 5

Given two positive numbers $a, b$, how many non-congruent plane quadrilaterals are there such that $AB = a$, $BC = CD = DA = b$ and $\angle B = 90^o$ ?

2016 Thailand Mathematical Olympiad, 3

Tags: functional equation , algebra , functional

Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$.

2014 ASDAN Math Tournament, 2

Tags: geometry test

Let $ABC$ be a triangle with sides $AB=19$, $BC=21$, and $AC=20$. Let $\omega$ be the incircle of $ABC$ with center $I$. Extend $BI$ so that it intersects $AC$ at $E$. If $\omega$ is tangent to $AC$ at the point $D$, then compute the length of $DE$.

2010 Estonia Team Selection Test, 1

Tags: number theory , gcd , power of prime , coprime , prime

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

2007 Turkey Team Selection Test, 1

Tags: combinatorics

Find the number of the connected graphs with 6 vertices. (Vertices are considered to be different)

2009 National Olympiad First Round, 17

Tags:

$ ABC$ is an equilateral triangle. $ D$ is a point inside $ \triangle ABC$ such that $ AD \equal{} 8$, $ BD \equal{} 13$, and $ \angle ADC \equal{} 120^\circ$. What is the length of $ DC$? $\textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16$

2013 Greece Team Selection Test, 4

Tags: trapezoid , combinatorics , tiling

Let $n$ be a positive integer. An equilateral triangle with side $n$ will be denoted by $T_n$ and is divided in $n^2$ unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also $m$ be a positive integer with $m<n$ and suppose that $T_n$ and $T_m$ can be tiled with "trapezoids". Prove that, if from $T_n$ we remove a $T_m$ with the same orientation, then the rest can be tiled with "trapezoids".

2018 All-Russian Olympiad, 6

Tags: number theory

$a$ and $b$ are given positive integers. Prove that there are infinitely many positive integers $n$ such that $n^b+1$ doesn't divide $a^n+1$.

2014 PUMaC Number Theory A, 2

Tags:

What is the last digit of ${17^{17^{17^{17}}}}$?

2025 JBMO TST - Turkey, 4

Tags: number theory

Let $p$ and $q$ be prime numbers. Prove that if $pq(p+1)(q+1)+1$ is a perfect square, then $pq + 1$ is also a perfect square.

2015 Junior Balkan MO, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]

1998 All-Russian Olympiad Regional Round, 10.7

Tags: combinatorics

A cube of side length $n$ is divided into unit cubes by [i]partitions[/i] (each [i]partition[/i] separates a pair of adjacent unit cubes). What is the smallest number of [i]partitions[/i] that can be removed so that from each cube, one can reach the surface of the cube without passing through a partition ?

2005 MOP Homework, 2

Tags: geometric transformation , reflection , circumcircle , symmetry , perpendicular bisector , geometry

Let $ABC$ be a triangle, and let $D$ be a point on side $AB$. Circle $\omega_1$ passes through $A$ and $D$ and is tangent to line $AC$ at $A$. Circle $\omega_2$ passes through $B$ and $D$ and is tangent to line $BC$ at $B$. Circles $\omega_1$ and $\omega_2$ meet at $D$ and $E$. Point $F$ is the reflection of $C$ across the perpendicular bisector of $AB$. Prove that points $D$, $E$, and $F$ are collinear.

2007 Bulgaria Team Selection Test, 3

Tags: ratio , parallelogram , gauss , geometry

Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.

2010 Indonesia TST, 2

Tags: combinatorics , graph theory

Let $T$ be a tree with$ n$ vertices. Choose a positive integer $k$ where $1 \le k \le n$ such that $S_k$ is a subset with $k$ elements from the vertices in $T$. For all $S \in S_k$, define $c(S)$ to be the number of component of graph from $S$ if we erase all vertices and edges in $T$, except all vertices and edges in $S$. Determine $\sum_{S\in S_k} c(S)$, expressed in terms of $n$ and $k$.

2015 Costa Rica - Final Round, 1

Tags: geometry , circumcircle , angle

Let $\vartriangle ABC$ be such that $\angle BAC$ is acute. The line perpendicular on side $AB$ from $C$ and the line perpendicular on $AC$ from $B$ intersect the circumscribed circle of $\vartriangle ABC$ at $D$ and $E$ respectively. If $DE = BC$ , calculate $\angle BAC$.

2014 China Second Round Olympiad, 3

Tags: combinatorics

Let $S=\{1,2,3,\cdots,100\}$. Find the maximum value of integer $k$, such that there exist $k$ different nonempty subsets of $S$ satisfying the condition: for any two of the $k$ subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

2023 ELMO Shortlist, A2

Tags: algebra , functional equation

Let $\mathbb R_{>0}$ denote the set of positive real numbers. Find all functions $f:\mathbb R_{>0}\to\mathbb R_{>0}$ such that for all positive real numbers $x$ and $y$, \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] [i]Proposed by Luke Robitaille[/i]

2022 Stanford Mathematics Tournament, 6

Tags:

Let the incircle of $\triangle ABC$ be tangent to $AB,BC,AC$ at points $M,N,P$, respectively. If $\measuredangle BAC=30^\circ$, find $\tfrac{[BPC]}{[ABC]}\cdot\tfrac{[BMC]}{[ABC]}$, where $[ABC]$ denotes the area of $\triangle ABC$.

2009 IMO Shortlist, 3

Tags: geometry , symmetry , triangle , parallelogram

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

1995 Grosman Memorial Mathematical Olympiad, 7

Tags: algebra , function , lattice

For a given positive integer $n$, let $A_n$ be the set of all points $(x,y)$ in the coordinate plane with $x,y \in \{0,1,...,n\}$. A point $(i, j)$ is called internal if $0 < i, j < n$. A real function $f$ , defined on $A_n$, is called [i]good [/i] if it has the following property: For every internal point $x$, the value of $f(x)$ is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance $1$ from $x$). Prove that if $f$ and $g$ are good functions that coincide at the non-internal points of $A_n$, then $f \equiv g$.

2004 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , combinatorics

Given is a convex polygon with $n\geq 5$ sides. Prove that there exist at most $\displaystyle \frac{n(2n-5)}3$ triangles of area 1 with the vertices among the vertices of the polygon.

2020/2021 Tournament of Towns, P7

Tags: geometry

There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that: [list=a] [*]one of its angles is not greater than $60^\circ{}$; [*]one of its angles is at least $120^\circ$. [/list] [i]Maxim Didin[/i]

2021 Balkan MO Shortlist, G5

Tags: shortlist , geometry , reflection , parallel

Let $ABC$ be an acute triangle with $AC > AB$ and circumcircle $\Gamma$. The tangent from $A$ to $\Gamma$ intersects $BC$ at $T$. Let $M$ be the midpoint of $BC$ and let $R$ be the reflection of $A$ in $B$. Let $S$ be a point so that $SABT$ is a parallelogram and finally let $P$ be a point on line $SB$ such that $MP$ is parallel to $AB$. Given that $P$ lies on $\Gamma$, prove that the circumcircle of $\triangle STR$ is tangent to line $AC$. [i]Proposed by Sam Bealing, United Kingdom[/i]

2013 Math Prize For Girls Problems, 17

Tags: function , trigonometry , algebra , domain

Let $f$ be the function defined by $f(x) = -2 \sin(\pi x)$. How many values of $x$ such that $-2 \le x \le 2$ satisfy the equation $f(f(f(x))) = f(x)$?