Olympiad problems

2004 Kazakhstan National Olympiad, 2

A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?

2006 Junior Tuymaada Olympiad, 6

Tags: number theory , combinatorics , sum

[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $. For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$. Find the number of all [i]palindromic partitions[/i] of the number $2006$.

2013 BMT Spring, 4

Tags: number theory , algebra

Given $f_1(x)=2x-2$ and, for $k\ge2$, defined $f_k(x)=f(f_{k-1}(x))$ to be a real-valued function of $x$. Find the remainder when $f_{2013}(2012)$ is divided by the prime $2011$.

1988 Mexico National Olympiad, 4

Tags: integer , combinatorics

In how many ways can one select eight integers $a_1,a_2, ... ,a_8$, not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$?

1964 Spain Mathematical Olympiad, 4

Tags: geometry

We are given an equilateral triangle $ABC$, of side $a$, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?

2024 CCA Math Bonanza, L4.3

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Byan Rai is currently standing on the origin of a $2$D plane. In each second: [list] [*] he jumps one unit up with probability $\frac{6}{11}$, [*] he jumps three units down with probability $\frac{2}{11}$, [*] he jumps four units right with probability $\frac{3}{22}$, [*] he jumps four units left with probability $\frac{3}{22}$. [/list] Suppose Byan ends up at $(x, y)$ after $2024$ seconds. The expected value of $x^2 + y^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Lightning 4.3[/i]

2003 AMC 10, 10

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Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased? $ \textbf{(A)}\ \frac{26}{10} \qquad \textbf{(B)}\ \frac{26^2}{10^2} \qquad \textbf{(C)}\ \frac{26^2}{10} \qquad \textbf{(D)}\ \frac{26^3}{10^3} \qquad \textbf{(E)}\ \frac{26^3}{10^2}$

Estonia Open Senior - geometry, 2011.2.3

Tags: ratio , geometry , rational , area

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

2018 Stanford Mathematics Tournament, 3

Tags: geometry

Let $ABC$ be a triangle and $D$ be a point such that $A$ and $D$ are on opposite sides of $BC$. Give that $\angle ACD = 75^o$, $AC = 2$, $BD =\sqrt6$, and $AD$ is an angle bisector of both $\vartriangle ABC$ and $\vartriangle BCD$, find the area of quadrilateral $ABDC$.

2025 Alborz Mathematical Olympiad, P1

Tags: positive integer , functional equation , number theory , algebra

Let $ \mathbb{Z^{+}} $ denote the set of all positive integers. Find all functions $ f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} $ such that for every pair of positive integers $ a $ and $ b $, there exists a positive integer $ c $ satisfying: $$ f(a)f(b) - ab = 2^{c-1} - 1. $$ Proposed by Matin Yousefi

2000 IMC, 5

Tags: function , algebra , domain , symmetry , real analysis

Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

2000 Baltic Way, 5

Tags: ratio , trigonometry , geometry

Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$.

2021 Thailand TST, 1

Tags: geometry

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

MathLinks Contest 6th, 2.1

Tags: algebra , equation

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

2016 BMT Spring, 6

Tags: combinatorics , number theory

Bob plays a game on the whiteboard. Initially, the numbers $\{1, 2, ...,n\}$ are shown. On each turn, Bob takes two numbers from the board $x$, $y$, erases them both, and writes down $2x + y$ onto the board. In terms of n, what is the maximum possible value that Bob can end up with?

Durer Math Competition CD 1st Round - geometry, 2012.D3

Tags: geometry , 3d geometry , cube , parallel

Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter? [img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]

2020 USOMO, 6

Tags: inequalities

Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$ [i]Proposed by David Speyer and Kiran Kedlaya[/i]

2014 IFYM, Sozopol, 7

Tags: combinatorics , graph theory

It is known that each two of the 12 competitors, that participated in the finals of the competition “Mathematical duels”, have a common friend among the other 10. Prove that there is one of them that has at least 5 friends among the group.

2024 Malaysian IMO Team Selection Test, 6

Tags: geometry

Let $\omega_1$, $\omega_2$, $\omega_3$ are three externally tangent circles, with $\omega_1$ and $\omega_2$ tangent at $A$. Choose points $B$ and $C$ on $\omega_1$ so that lines $AB$ and $AC$ are tangent to $\omega_3$. Suppose the line $BC$ intersect $\omega_3$ at two distinct points, and $X$ is the intersection further away to $B$ and $C$ than the other one. Prove that one of the tangent lines of $\omega_2$ passing through $X$, is also tangent to an excircle of triangle $ABC$. [i]Proposed by Ivan Chan Kai Chin[/i]

2014 Iran Team Selection Test, 1

Tags: circumcircle , geometry

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.

2014 Online Math Open Problems, 24

Tags: floor function , vector , function , inequalities

Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$? [i]Proposed by Sammy Luo[/i]

2021 Canadian Junior Mathematical Olympiad, 4

Tags: algebra , inequalities , n-variable inequality

Let $n\geq 2$ be some fixed positive integer and suppose that $a_1, a_2,\dots,a_n$ are positive real numbers satisfying $a_1+a_2+\cdots+a_n=2^n-1$. Find the minimum possible value of $$\frac{a_1}{1}+\frac{a_2}{1+a_1}+\frac{a_3}{1+a_1+a_2}+\cdots+\frac{a_n}{1+a_1+a_2+\cdots+a_{n-1}}$$

1979 Yugoslav Team Selection Test, Problem 1

Tags: sum , inequalities , number theory

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

2009 Swedish Mathematical Competition, 6

Tags: combinatorics , game

On a table lie $289$ coins that form a square array $17 \times 17$. All coins are facing with the crown up. In one move, it is possible to reverse any five coins lying in a row: vertical, horizontal or diagonal. Is it possible that after a number of such moves, all the coins to be arranged with tails up?

2004 India IMO Training Camp, 1

Tags: ratio , function , geometry

Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively. (a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$ (b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]