Olympiad problems

2022 Azerbaijan EGMO/CMO TST, A2

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

2013 Dutch BxMO/EGMO TST, 4

Tags: functional equation , Determine all functions , TST , algebra

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying \[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]

2018 China Team Selection Test, 2

Tags: combinatorics , TST , graph theory

Let $G$ be a simple graph with 100 vertices such that for each vertice $u$, there exists a vertice $v \in N \left ( u \right )$ and $ N \left ( u \right ) \cap N \left ( v \right ) = \o $. Try to find the maximal possible number of edges in $G$. The $ N \left ( . \right )$ refers to the neighborhood.

2021 Israel TST, 1

Tags: sum of digits , TST , number theory

A pair of positive integers $(a,b)$ is called an [b]average couple[/b] if there exist positive integers $k$ and $c_1, \dots, c_k$ for which \[\frac{c_1+c_2+\cdots+c_k}{k}=a\qquad \text{and} \qquad \frac{s(c_1)+s(c_2)+\cdots+s(c_k)}{k}=b\] where $s(n)$ denotes the sum of digits of $n$ in decimal representation. Find the number of average couples $(a,b)$ for which $a,b<10^{10}$.

2017 Ukraine Team Selection Test, 10

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

2023 Chile TST IMO, 3

Tags: algebra , Chile , TST

Solve the system of equations in real numbers: \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} \] \[ x^2 + y^2 + z^2 = 294 \] \[ x + y + z = 0 \]

2018 China Team Selection Test, 3

Tags: geometry , TST

Circle $\omega$ is tangent to sides $AB$,$AC$ of triangle $ABC$ at $D$,$E$ respectively, such that $D\neq B$, $E\neq C$ and $BD+CE<BC$. $F$,$G$ lies on $BC$ such that $BF=BD$, $CG=CE$. Let $DG$ and $EF$ meet at $K$. $L$ lies on minor arc $DE$ of $\omega$, such that the tangent of $L$ to $\omega$ is parallel to $BC$. Prove that the incenter of $\triangle ABC$ lies on $KL$.

2024 Germany Team Selection Test, 2

Tags: number theory , IMO Shortlist , AZE IMO TST , TST

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square. [i]Proposed by Tahjib Hossain Khan, Bangladesh[/i]

2024 Chile TST Ibero., 2

Tags: TST , algebra , Chile

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

2017 China Team Selection Test, 1

Tags: TST , combinatorics , equation , China , generating functions

Prove that :$$\sum_{k=0}^{58}C_{2017+k}^{58-k}C_{2075-k}^{k}=\sum_{p=0}^{29}C_{4091-2p}^{58-2p}$$

2023 Israel TST, P1

Tags: TST , combinatorics , pentagon

A real number is written next to each vertex of a regular pentagon. All five numbers are different. A triple of vertices is called [b] successful[/b] if they form an isosceles triangle for which the number written on the top vertex is either larger than both numbers written on the base vertices, or smaller than both. Find the maximum possible number of successful triples.

2014 JBMO Shortlist, 5

Tags: number theory , AZE JBMO TST , TST

Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$

2019 IMO Shortlist, G2

Tags: geometry , IMO Shortlist , TST

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

2024 Azerbaijan IZhO TST, 1

Tags: functional equation , algebra , TST , AZE IzHO TST

Let $\alpha\neq0$ be a real number. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x;y\in\mathbb{R}$

2025 JBMO TST - Turkey, 6

Tags: algebra , Inequality , TST

Find the minimum value of \[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\] where $x,y,z>1$ are reals.

2024 Israel TST, P2

Tags: TST , combinatorics , graph theory , analytic geometry

Let $n>1$ be an integer. Given a simple graph $G$ on $n$ vertices $v_1, v_2, \dots, v_n$ we let $k(G)$ be the minimal value of $k$ for which there exist $n$ $k$-dimensional rectangular boxes $R_1, R_2, \dots, R_n$ in a $k$-dimensional coordinate system with edges parallel to the axes, so that for each $1\leq i<j\leq n$, $R_i$ and $R_j$ intersect if and only if there is an edge between $v_i$ and $v_j$ in $G$. Define $M$ to be the maximal value of $k(G)$ over all graphs on $n$ vertices. Calculate $M$ as a function of $n$.

2004 Germany Team Selection Test, 2

Tags: geometry proposed , geometry , Locus problems , Geometric Inequalities , Germany , TST , Team Selection Test

Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$. Find all points $B$ on the diameter $d$ in the interior of $k$ such that \[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\] (i. e. give an explicit description of these points without using the points $M$ and $N$).

2023 Germany Team Selection Test, 1

Tags: algebra , IMO Shortlist , AZE IMO TST , TST

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2023 Azerbaijan IMO TST, 4

Tags: number theory , IMO Shortlist , AZE IMO TST , TST

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2023 Chile TST Ibero., 4

Tags: Chile , geometry , Iberoamerican , TST

Let $ABC$ be a triangle with $AB < AC$ and let $\omega$ be its circumcircle. Let $M$ denote the midpoint of side $BC$ and $N$ the midpoint of arc $BC$ of $\omega$ that contains $A$. The circumcircle of triangle $AMN$ intersects sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Prove that $BP = CQ$.

2024 Azerbaijan BMO TST, 1

Tags: IMO Shortlist , number theory , AZE BMO TST , AZE EGMO TST , TST , BRA Cono Sur TST

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

2023 Azerbaijan BMO TST, 3

Tags: algebra , functional equation , AZE BMO TST , AZE EGMO TST , BMO Shortlist , TST

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2019 India IMO Training Camp, 3

Tags: indian olympiad , graph , combinatorics , TST

There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.

2017 Saudi Arabia JBMO TST, 5

Tags: algebra , Inequality , TST

Let $a,b,c>0$ and $a+b+c=6$ . Prove that $$ \frac{1}{a^2b+16}+\frac{1}{b^2c+16}+\frac{1}{c^2a+16} \ge \frac{1}{8}.$$

2019 Moldova Team Selection Test, 4

Tags: geometry , TST

Quadrilateral $ABCD$ is inscribed in circle $\Gamma$ with center $O$. Point $I$ is the incenter of triangle $ABC$, and point $J$ is the incenter of the triangle $ABD$. Line $IJ$ intersects segments $AD, AC, BD, BC$ at points $P, M, N$ and, respectively $Q$. The perpendicular from $M$ to line $AC$ intersects the perpendicular from $N$ to line $BD$ at point $X$. The perpendicular from $P$ to line $AD$ intersects the perpendicular from $Q$ to line $BC$ at point $Y$. Prove that $X, O, Y$ are colinear.