Olympiad problems

2014 Tuymaada Olympiad, 7

Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

2008 APMO, 5

Tags: modular arithmetic , inequalities , geometric series , number theory , residue

Let $ a, b, c$ be integers satisfying $ 0 < a < c \minus{} 1$ and $ 1 < b < c$. For each $ k$, $ 0\leq k \leq a$, Let $ r_k,0 \leq r_k < c$ be the remainder of $ kb$ when divided by $ c$. Prove that the two sets $ \{r_0, r_1, r_2, \cdots , r_a\}$ and $ \{0, 1, 2, \cdots , a\}$ are different.

2004 Romania National Olympiad, 1

Tags: function , inequalities , algebra

Find the strictly increasing functions $f : \{1,2,\ldots,10\} \to \{ 1,2,\ldots,100 \}$ such that $x+y$ divides $x f(x) + y f(y)$ for all $x,y \in \{ 1,2,\ldots,10 \}$. [i]Cristinel Mortici[/i]

2020 BMT Fall, 1

Tags: combinatorics

Justin throws a standard six-sided die three times in a row and notes the number of dots on the top face after each roll. How many different sequences of outcomes could he get?

2022 SG Originals, Q5

Tags: number theory , polynomial , modulo , algebra , prime number

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

2004 Baltic Way, 5

Tags: function , algebra

Determine the range of the following function defined for integer $k$, \[f(k)=(k)_3+(2k)_5+(3k)_7-6k\] where $(k)_{2n+1}$ denotes the multiple of $2n+1$ closest to $k$

2012 AMC 10, 23

Tags: geometric transformation , rotation , graph theory

Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? $ \textbf{(A)}\ 60 \qquad\textbf{(B)}\ 170 \qquad\textbf{(C)}\ 290 \qquad\textbf{(D)}\ 320 \qquad\textbf{(E)}\ 660 $

2009 Cono Sur Olympiad, 3

Tags: geometry

Let $A$, $B$, and $C$ be three points such that $B$ is the midpoint of segment $AC$ and let $P$ be a point such that $<PBC=60$. Equilateral triangle $PCQ$ is constructed such that $B$ and $Q$ are on different half=planes with respect to $PC$, and the equilateral triangle $APR$ is constructed in such a way that $B$ and $R$ are in the same half-plane with respect to $AP$. Let $X$ be the point of intersection of the lines $BQ$ and $PC$, and let $Y$ be the point of intersection of the lines $BR$ and $AP$. Prove that $XY$ and $AC$ are parallel.

1989 IMO Longlists, 64

Tags: combinatorics , invariant , chessboard , algorithm

A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.

2019 Taiwan TST Round 3, 2

Tags: function , algebra

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

2024 Argentina National Olympiad Level 2, 4

Tags: algebra

Find all pairs $(a, b)$ of positive rational numbers such that $$\sqrt{a}+\sqrt{b} = \sqrt{2+\sqrt{3}}.$$

2016 Sharygin Geometry Olympiad, P3

Tags: geometry

Let $AH_1$, $BH_2$ be two altitudes of an acute-angled triangle $ABC$ , $D$ be the projection of $H_1$ to $AC$, $E$ be the projection of $D$ to $AB$, $F$ be the common point of $ED$ and $AH_1$. Prove that $H_2F \parallel BC$. [i](Proposed by E.Diomidov)[/i]

2025 Kosovo National Mathematical Olympiad`, P4

Tags: concurrency , lemma , points in sides , cevian , pop , geometry , circles

Let $ABC$ be a given triangle. Let $A_1$ and $A_2$ be points on the side $BC$. Let $B_1$ and $B_2$ be points on the side $CA$. Let $C_1$ and $C_2$ be points on the side $AB$. Suppose that the points $A_1,A_2,B_1,B_2,C_1$ and $C_2$ lie on a circle. Prove that the lines $AA_1, BB_1$ and $CC_1$ are concurrent if and only if $AA_2, BB_2$ and $CC_2$ are concurrent.

2012 Balkan MO Shortlist, N2

Tags: recurrence relation , number theory , sequence , number theory with sequences

Let the sequences $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ satisfy $a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c^2_r = c_kc_m$.

2008 Canada National Olympiad, 4

Tags: function , modular arithmetic , prime number , number theory

Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n\quad {\rm mod}\; f(p) \] for all $ n \in {\bf N}$ and all prime numbers $ p$.

2022 Taiwan TST Round 2, 5

Tags: geometry , circumcircle , miquel point , complete quadrilateral , tangent

Let $ABCDE$ be a pentagon inscribed in a circle $\Omega$. A line parallel to the segment $BC$ intersects $AB$ and $AC$ at points $S$ and $T$, respectively. Let $X$ be the intersection of the line $BE$ and $DS$, and $Y$ be the intersection of the line $CE$ and $DT$. Prove that, if the line $AD$ is tangent to the circle $\odot(DXY)$, then the line $AE$ is tangent to the circle $\odot(EXY)$. [i]Proposed by ltf0501.[/i]

2002 Polish MO Finals, 2

Tags: 3d geometry , sphere , geometry

There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$. There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$. Show that the sphere touches the circumsphere of $ABCP$.

2020 CMIMC Team, 15

Tags: team

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose $M$ is the midpoint of segment $\overline{BC}$, $N$ is the midpoint of $\overline{AM}$, and $E$ and $F$ are the feet of the altitudes of $M$ onto $\overline{AB}$ and $\overline{AC}$, respectively. Further suppose $BC$ intersects $NE$ at $S$ and $NF$ at $T$, and let $X$ and $Y$ be the circumcenters of $\triangle MES$ and $\triangle MFT$, respectively. If $XY$ is tangent to the circumcircle of $\triangle ABC$, what is the area of $\triangle ABC$?

2019 Jozsef Wildt International Math Competition, W. 30

Tags: zeta function , limit , summation , function

[list=1] [*] Prove that $$\lim \limits_{n \to \infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)=0$$ [*] Calculate $$\sum \limits_{n=1}^{\infty} \left(n+\frac{1}{4}-\zeta(3)-\zeta(5)-\cdots -\zeta(2n+1)\right)$$ [/list]

2006 National Olympiad First Round, 23

Tags:

What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 10 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 36 $