Olympiad problems

2015 Switzerland Team Selection Test, 11

In Thailand there are $n$ cities. Each pair of cities is connected by a one-way street which can be borrowed, depending on its type, only by bike or by car. Show that there is a city from which you can reach any other city, either by bike or by car. [i]Remark : It is not necessary to use the same means of transport for each city[/i]

2018 Polish MO Finals, 2

Tags: combinatorics , number theory , fixed point

A subset $S$ of size $n$ of a plane consisting of points with both coordinates integer is given, where $n$ is an odd number. The injective function $f\colon S\rightarrow S$ satisfies the following: for each pair of points $A, B\in S$, the distance between points $f(A)$ and $f(B)$ is not smaller than the distance between points $A$ and $B$. Prove there exists a point $X$ such that $f(X)=X$.

2013 Balkan MO Shortlist, C4

Tags: coloring , chessboard , combinatorics

A closed, non-self-intersecting broken line $L$ is drawn over a $(2n+1) \times (2n+1)$ chessboard in such a way that the set of L's vertices coincides with the set of the vertices of the board’s squares and every edge in $L$ is a side of some board square. All board squares lying in the interior of $L$ are coloured in red. Prove that the number of neighbouring pairs of red squares in every row of the board is even.

2018 MIG, 5

Tags:

Some of the values produced by two functions, $f(x)$ and $g(x)$, are shown below. Find $f(g(3))$ \begin{tabular}{c||c|c|c|c|c} $x$ & 1 & 3 & 5 & 7 & 9 \\ \hline\hline $f(x)$ & 3 & 7 & 9 & 13 & 17 \\ \hline $g(x)$ & 54 & 9 & 25 & 19 & 44 \end{tabular} $\textbf{(A) }3\qquad\textbf{(B) }7\qquad\textbf{(C) }8\qquad\textbf{(D) }13\qquad\textbf{(E) }17$

2018 ELMO Shortlist, 1

Tags: geometry

Let $ABC$ be an acute triangle with orthocenter $H$, and let $P$ be a point on the nine-point circle of $ABC$. Lines $BH, CH$ meet the opposite sides $AC, AB$ at $E, F$, respectively. Suppose that the circumcircles $(EHP), (FHP)$ intersect lines $CH, BH$ a second time at $Q,R$, respectively. Show that as $P$ varies along the nine-point circle of $ABC$, the line $QR$ passes through a fixed point. [i]Proposed by Brandon Wang[/i]

2025 Taiwan TST Round 1, G

Tags: geometry

Suppose $I$ and $I_A$ are the incenter and the $A$-excenter of triangle $ABC$, respectively. Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the perpendicular from $I_A$ to $BC$. The line $MI$ intersects the circumcircle again at $T$ . For any point $X$ on the circumcircle of triangle $ABC$, let $XT$ intersect $BC$ at $Y$ . Prove that $A, D, X, Y$ are concyclic.

2008 All-Russian Olympiad, 4

Tags: limit , algebra

The sequences $ (a_n),(b_n)$ are defined by $ a_1\equal{}1,b_1\equal{}2$ and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\] Show that $ a_{2008} < 5$.

1995 Turkey Team Selection Test, 2

Tags: combinatorics

Let $n$ be a positive integer. Find the number of permutations $\sigma$ of the set $\{1, 2, ..., n\}$ such that $\sigma(j) \geq j$ holds for exactly two values of $j$.

1953 Czech and Slovak Olympiad III A, 1

Tags: complex number , plane

Find the locus of all numbers $z\in\mathbb C$ in complex plane satisfying $$z+\bar z=a\cdot|z|,$$ where $a\in\mathbb R$ is given.

2012 IFYM, Sozopol, 2

Tags: algebra , sequence

The sequence $\{x_n\}_{n=0}^\infty$ is defined by the following equations: $x_n=\sqrt{x_{n-1} x_{n-2}+\frac{n}{2}}$ ,$\forall$ $n\geq 2$, $x_0=x_1=1$. Prove that there exist a real number $a$, such that $an<x_n<an+1$ for each natural number $n$.

2025 ISI Entrance UGB, 8

Tags: number theory

Let $n \geq 2$ and let $a_1 \leq a_2 \leq \cdots \leq a_n$ be positive integers such that $\sum_{i=1}^{n} a_i = \prod_{i=1}^{n} a_i$. Prove that $\sum_{i=1}^{n} a_i \leq 2n$ and determine when equality holds.

1992 Bundeswettbewerb Mathematik, 3

Tags: trigonometry , geometry

Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.

2010 Balkan MO Shortlist, N3

Tags: function , greatest common divisor , relatively prime , totient function , number theory

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

1981 Yugoslav Team Selection Test, Problem 2

Tags: geometry

Suppose that there is a point $S$ inside a quadrilateral $ABCD$ such that segments $SA,SB,SC,SD$ divide the quadrilateral into four triangles of equal areas. Prove that one of the diagonals of the quadrilateral bisects the other one.

2021 JBMO Shortlist, G1

Tags: geometry

Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$. Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.

2016 CHMMC (Fall), 9

Tags: geometry

In quadrilateral $ABCD$, $AB = DB$ and $AD = BC$. If $\angle ABD = 36^{\circ}$ and $\angle BCD = 54^{\circ}$, find $\angle ADC$ in degrees.

2000 Austrian-Polish Competition, 10

Tags: combinatorial geometry , combinatorics

The plan of the castle in Baranow Sandomierski can be presented as the graph with $16$ vertices on the picture. A night guard plans a closed round along the edges of this graph. (a) How many rounds passing through each vertex exactly once are there? The directions are irrelevant. (b) How many non-selfintersecting rounds (taking directions into account) containing each edge of the graph exactly once are there? [img]https://cdn.artofproblemsolving.com/attachments/1/f/27ca05fc689fd8d873130db9d8cc52acf49bb4.png[/img]

2013 Romania National Olympiad, 2

Tags: superior algebra , polynomial , algebra

Given a ring $\left( A,+,\cdot \right)$ that meets both of the following conditions: (1) $A$ is not a field, and (2) For every non-invertible element $x$ of $ A$, there is an integer $m>1$ (depending on $x$) such that $x=x^2+x^3+\ldots+x^{2^m}$. Show that (a) $x+x=0$ for every $x \in A$, and (b) $x^2=x$ for every non-invertible $x\in A$.

1990 Federal Competition For Advanced Students, P2, 2

Tags: algebra , inequalities

Show that for all integers $ n \ge 2$, $ \sqrt { 2\sqrt[3]{3 \sqrt[4]{4...\sqrt[n]{n}}}}<2$

2017 CIIM, Problem 2

Tags: college math , real analysis

Let $f :\mathbb{R} \to \mathbb{R}$ a derivable function such that $f(0) = 0$ and $|f'(x)| \leq |f(x)\cdot log |f(x)||$ for every $x \in \mathbb{R}$ such that $0 < |f(x)| < 1/2.$ Prove that $f(x) = 0$ for every $x \in \mathbb{R}$.

2023 Brazil EGMO TST -wrong source, 3

Tags: combinatorics

There are $n$ cards. Max and Lewis play, alternately, the following game Max starts the game, he removes exactly $1$ card, in each round the current player can remove any quantity of cards, from $1$ card to $t+1$ cards, which $t$ is the number of removed cards by the previous player, and the winner is the player who remove the last card. Determine all the possible values of $n$ such that Max has the winning strategy.

2022 Stanford Mathematics Tournament, 7

Tags:

Let \[A_j=\left\{(x,y):0\le x\sin\left(\frac{j\pi}{3}\right)+y\cos\left(\frac{j\pi}{3}\right)\le6-\left(x\cos\left(\frac{j\pi}{3}\right)-y\sin\left(\frac{j\pi}{3}\right)\right)^2\right\}\] The area of $\cup_{j=0}^5A_j$ can be expressed as $m\sqrt{n}$. What is the area?

2000 Turkey Team Selection Test, 3

Tags: function , limit , inequalities , algebra

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that \[|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.\] Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$

2003 China Second Round Olympiad, 2

Tags: geometry , perimeter , floor function , calculus , integration , modular arithmetic , algebra

Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell>m>n$ and $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$, where $\{x\}=x-\lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle.

2018 Thailand TST, 3

Tags: geometry , incenter

A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_a, I_b, I_c$ and $I_d$ be the incenters of the triangles $DAB, ABC, BCD$ and $CDA$, respectively. Suppose that the common external tangents of the circles $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.