Olympiad problems

2017 Taiwan TST Round 2, 3

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2001 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , calculus , integration

Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.

Indonesia MO Shortlist - geometry, g3

Tags: circumcircle , geometry , tangent , incircle

Given $ABC$ triangle with incircle $L_1$ and circumcircle $L_2$. If points $X, Y, Z$ lie on $L_2$, such that $XY, XZ$ are tangent to $L_1$, then prove that $YZ$ is also tangent to $L_1$.

2011 National Olympiad First Round, 4

Tags: inequalities

How many subsets, which does not contain two consecutive numbers, are there of the set $\{1,2,\dots ,20\}$ with $8$ elements? $\textbf{(A)}\ {{13}\choose{8}} \qquad\textbf{(B)}\ {{13}\choose{9}} \qquad\textbf{(C)}\ {{14}\choose{8}} \qquad\textbf{(D)}\ {{14}\choose{9}} \qquad\textbf{(E)}\ {{20}\choose{15}}$

1979 IMO Longlists, 46

Tags: set systems , intersection , combinatorics

Let $K$ denote the set $\{a, b, c, d, e\}$. $F$ is a collection of $16$ different subsets of $K$, and it is known that any three members of $F$ have at least one element in common. Show that all $16$ members of $F$ have exactly one element in common.

2018 Pan-African Shortlist, C6

Tags: combinatorics

A circle is divided into $n$ sectors ($n \geq 3$). Each sector can be filled in with either $1$ or $0$. Choose any sector $\mathcal{C}$ occupied by $0$, change it into a $1$ and simultaneously change the symbols $x, y$ in the two sectors adjacent to $\mathcal{C}$ to their complements $1-x$, $1-y$. We repeat this process as long as there exists a zero in some sector. In the initial configuration there is a $0$ in one sector and $1$s elsewhere. For which values of $n$ can we end this process?

2017 IMC, 1

Tags: linear algebra , matrix

Determine all complex numbers $\lambda$ for which there exists a positive integer $n$ and a real $n\times n$ matrix $A$ such that $A^2=A^T$ and $\lambda$ is an eigenvalue of $A$.

1987 Poland - Second Round, 6

Tags: geometry , similarity , similar

We assign to any quadrilateral $ ABCD $ the centers of the circles circumscribed in the triangles $ BCD $, $ CDA $, $ DAB $, $ ABC $. Prove that if the vertices of a convex quadrilateral $ Q $ do not lie on the circle, then a) the four points assigned to quadrilateral Q in the above manner are the vertices of the convex quadrilateral. Let us denote this quadrilateral by $ t(Q) $, b) the vertices of the quadrilateral $ t(Q) $ do not lie on the circle, c) quadrilaterals $ Q $ and $ t(t(Q) $ are similar.

2010 National Chemistry Olympiad, 16

Tags:

Moist air is less dense than dry air at the same temperature and barometric pressure. Which is the best explanation for this observation? $ \textbf{(A)}\hspace{.05in}\ce{H2O} \text{ is a polar molecule but } \ce{N2} \text{ and } \ce{O2} \text{ are not} \qquad$ $\textbf{(B)}\hspace{.05in} \ce{H2O} \text{has a higher boiling point than } \ce{N2} \text{or} \ce{O2}\qquad$ $\textbf{(C)}\hspace{.05in}\ce{H2O} \text{has a lower molar mass than} \ce{N2} \text{or} \ce{O2}\qquad$ $\textbf{(D)}\hspace{.05in}\ce{H2O} \text{has a higher heat capacity than} \ce{N2} \text{or} \ce{O2}\qquad$

2010 Danube Mathematical Olympiad, 2

Tags: parallelogram , geometric transformation , geometry , reflection , homothety , ratio

Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

2019 Korea - Final Round, 1

Tags: combinatorics

There are $n$ cards such that for each $i=1,2, \cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations: [list] [*] $A$ : One can take the top card of the pile and move it to the bottom; [*] $B$ : One can remove the top card from the pile. [/list] The operation $ABBABBABBABB \cdots $ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$.

2008 Pre-Preparation Course Examination, 3

Tags: 3d geometry , sphere , probability and stats , geometry

Prove that we can put $ \Omega(\frac1{\epsilon})$ points on surface of a sphere with radius 1 such that distance of each of these points and the plane passing through center and two of other points is at least $ \epsilon$.

1999 All-Russian Olympiad, 2

Tags: induction , number theory

Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]

2023 Yasinsky Geometry Olympiad, 1

Tags: geometry , perimeter

In triangle $ABC$, let points $M$ and $N$ be the midpoints of sides $AB$ and $BC$ respectively. It is known that the perimeter of the triangle $MBN$ is $12$ cm, and the perimeter of the quadrilateral $AMNC$ is $20$ cm. Find the length of the segment $MN$.

1979 Austrian-Polish Competition, 4

Tags: functional equation , algebra , function

Determine all functions $f : N_0 \to R$ satisfying $f (x+y)+ f (x-y)= f (3x)$ for all $x,y$.

2024 JBMO TST - Turkey, 1

Tags: geometry

In the acute-angled triangle $ABC$, $P$ is the midpoint of segment $BC$ and the point $K$ is the foot of the altitude from $A$. $D$ is a point on segment $AP$ such that $\angle BDC=90$. Let $(ADK) \cap BC=E$ and $(ABC) \cap AE=F$. Prove that $\angle AFD=90$.

2025 Belarusian National Olympiad, 11.1

Tags: algebra , polynomial

Numbers $1,\ldots,2025$ are written in a circle in increasing order. For every three consecutive numbers $i,j,k$ we consider the polynomial $(x-i)(x-j)(x-k)$. Let $s(x)$ be the sum of all $2025$ these polynomials. Prove that $s(x)$ has an integral root. [i]A. Voidelevich[/i]

2016 Peru IMO TST, 4

Tags: max , divide , min , positive integer , number theory

Let $N$ be the set of positive integers. Find all the functions $f: N\to N$ with $f (1) = 2$ and such that $max \{f(m)+f(n), m+n\}$ divides $min\{2m+2n,f (m+ n)+1\}$ for all $m, n$ positive integers

2009 Princeton University Math Competition, 4

Tags:

How many strings of ones and zeroes of length 10 are there such that there is an even number of ones, and no zero follows another zero?

1997 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 3

Tags: function

Let $ f_i (x), i \equal{} 1,2,3 \cdots$ be defined by $ f_1 (x) \equal{} \frac{1}{1 \minus{} x}$ and $ f_{i\plus{}1} (x) \equal{} f_i (f_1 (x))$. Then $ f_{1998} (1998)$ equals A. 0 B. 1998 C. -1/1997 D. 1997/1998 E. None of these

2015 Belarus Team Selection Test, 2

Tags: geometry , reflection , circumcircle , cyclic quadrilateral , geometric transformation

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

2024 Baltic Way, 20

Tags: diophantine equation , discriminant , number theory

Positive integers $a$, $b$ and $c$ satisfy the system of equations \begin{align*} (ab-1)^2&=c(a^2+b^2)+ab+1,\\ a^2+b^2&=c^2+ab. \end{align*} a) Prove that $c+1$ is a perfect square. b) Find all such triples $(a,b,c)$.

2013 China Team Selection Test, 1

Tags: real analysis , modular arithmetic , geometric transformation , geometry , number theory

Let $p$ be a prime number and $a, k$ be positive integers such that $p^a<k<2p^a$. Prove that there exists a positive integer $n$ such that \[n<p^{2a}, C_n^k\equiv n\equiv k\pmod {p^a}.\]

Ukrainian TYM Qualifying - geometry, II.1

Tags: geometry , 3d geometry , sphere , cylinder

Inside a right cylinder with a radius of the base $R$ are placed $k$ ($k\ge 3$) of equal balls, each of which touches the side surface and the lower base of the cylinder and, in addition, exactly two other balls. After that, another equal ball is placed inside the cylinder so that it touches the upper base of the cylinder and all other balls. Find the volume $V (R, k)$ of the cylinder.

2022 China Northern MO, 2

Tags: number theory , divide

(1) Find the smallest positive integer $a$ such that $221|3^a -2^a$, (2) Let $A=\{n\in N^*: 211|1+2^n+3^n+4^n\}$. Are there infinitely many numbers $n$ such that both $n$ and $n+1$ belong to set $A$?