Olympiad problems

2009 Bulgaria National Olympiad, 1

The natural numbers $a$ and $b$ satisfy the inequalities $a > b > 1$ . It is also known that the equation $\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$. Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).

2012 USAJMO, 3

Tags: inequalities , polynomial , ratio , algebra

Let $a,b,c$ be positive real numbers. Prove that $\frac{a^3+3b^3}{5a+b}+\frac{b^3+3c^3}{5b+c}+\frac{c^3+3a^3}{5c+a} \geq \frac{2}{3}(a^2+b^2+c^2)$.

1973 IMO, 1

Tags: geometry , triangle inequality , minimization , triangle , optimization

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

2003 Swedish Mathematical Competition, 5

Tags: geometry , equal segments

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.