Olympiad problems

2005 IMO, 2

Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$. Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.

2022 Latvia Baltic Way TST, P10

Tags: circumcircle , geometry

Let $\triangle ABC$ be a triangle satisfying $AB<AC$. Let $D$ be a point on the segment $AC$ such that $AB=AD$. Let then $X$ be a point on the segment $BC$ satisfying $BD^2=BX\cdot BC$. Let the circumcircles of the triangles $\triangle XDC$ and $\triangle ABC$ intersect at $M \neq C$. Prove that the line $MD$ goes through the midpoint of the arc $\widehat{BAC}$ of the circumcircle of $\triangle ABC$.

2025 Korea - Final Round, P2

Tags: algebra , function , functional equation

Let $\mathbb{R}$ be the set of real numbers. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ that satisfy the following condition. Here, $f^{100}(x)$ is the function obtained by composing $f(x)$ $100$ times, that is, $(\underbrace{f \circ f \circ \cdots \circ f}_{100 \ \text{times}})(x).$ [b](Condition)[/b] For all $x, y \in \mathbb{R}$, $$f(x + f^{100}(y)) = x + y \ \ \ \text{or} \ \ \ f(f^{100}(x) + y) = x + y$$

2018 India PRMO, 8

Tags: chord , angle , geometry

Let $AB$ be a chord of a circle with centre $O$. Let $C$ be a point on the circle such that $\angle ABC =30^o$ and $O$ lies inside the triangle $ABC$. Let $D$ be a point on $AB$ such that $\angle DCO = \angle OCB = 20^o$. Find the measure of $\angle CDO$ in degrees.

2011 AMC 10, 21

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Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1,3,4,5,6,$ and $9$. What is the sum of the possible values for $w$? $ \textbf{(A)}\ 16 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 62 \qquad \textbf{(E)}\ 93 $

2005 All-Russian Olympiad Regional Round, 11.1

Tags: trigonometry , algebra

Find all pairs of numbers $x, y \in \left( 0, \frac{\pi}{2}\right)$ , satisfying the equality $$\sin x + \sin y = \sin (xy)$$

2018 Taiwan TST Round 1, 1

Tags: function , algebra

Find all functions $ f: \mathbb{R} \to \mathbb{R} $ such that $$ f\left(f\left(x\right)+y\right) = f\left(x^2-y\right)+4\left(y-2\right)\left(f\left(x\right)+2\right) $$ holds for all $ x, y \in \mathbb{R} $

2003 Gheorghe Vranceanu, 2

Tags: function , real analysis , limit point

Let be a real number $ a $ and a function $ f:[a,\infty )\longrightarrow\mathbb{R} $ that is continuous at $ a. $ Prove that $ f $ is primitivable on $ (a,\infty ) $ if and only if $ f $ is primitivable on $ [a,\infty ) . $

2002 Kazakhstan National Olympiad, 3

Tags: modular arithmetic , combinatorics , extremal combinatorics , maximization

Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.

1972 Bundeswettbewerb Mathematik, 4

Tags: combinatorics

$p>2$ persons participate at a chess tournament, two players play at most one game against each other. After $n$ games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that $n \leq \frac{p^{2}}4$.

2012 Purple Comet Problems, 16

Tags: number theory , relatively prime

Let $a$, $b$, and $c$ be non-zero real number such that $\tfrac{ab}{a+b}=3$, $\tfrac{bc}{b+c}=4$, and $\tfrac{ca}{c+a}=5$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{abc}{ab+bc+ca}=\tfrac{m}{n}$. Find $m+n$.

2019 Peru EGMO TST, 2

Tags: arc midpoint , midpoints arc , concyclic , circumcircle , orthocenter , geometry

Let $\Gamma$ be the circle of an acute triangle $ABC$ and let $H$ be its orthocenter. The circle $\omega$ with diameter $AH$ cuts $\Gamma$ at point $D$ ($D\ne A$). Let $M$ be the midpoint of the smaller arc $BC$ of $\Gamma$ . Let $N$ be the midpoint of the largest arc $BC$ of the circumcircle of the triangle $BHC$. Prove that there is a circle that passes through the points $D, H, M$ and $N$.

1968 German National Olympiad, 3

Tags: algebra , functional , functional equation

Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation $$a \cdot f(x^n) + f(-x^n) = bx$$ suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.

1966 IMO Shortlist, 62

Tags: algebra , system of equations

Solve the system of equations \[ |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 \] \[ |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 \] \[ |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 \] \[ |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 \] where $a_1, a_2, a_3, a_4$ are four different real numbers.

II Soros Olympiad 1995 - 96 (Russia), 11.5

Tags: geometry , 3d geometry , polyhedron , angle

$6$ points are taken on the surface of the sphere, forming three pairs of diametrically opposite points on the sphere. Consider a convex polyhedron with vertices at these points. Prove that if this polyhedron has one right dihedral angle, then it has exactly $6$ right dihedral angles.

2010 Sharygin Geometry Olympiad, 15

Tags: circumcircle , geometry

Let $AA_1, BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC.$ $AA_1$ meets $B_1C_1$ in a point $K.$ The circumcircles of triangles $A_1KC_1$ and $A_1KB_1$ intersect the lines $AB$ and $AC$ for the second time at points $N$ and $L$ respectively. Prove that [b]a)[/b] The sum of diameters of these two circles is equal to $BC,$ [b] b)[/b] $\frac{A_1N}{BB_1} + \frac{A_1L}{CC_1}=1.$

2019 CHKMO, 4

Tags: combinatorial geometry , geometry , circumcircle

Find all integers $n \geq 3$ with the following property: there exist $n$ distinct points on the plane such that each point is the circumcentre of a triangle formed by 3 of the points.

2024 MMATHS, 11

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Let $n$ be the least possible value of $$\sqrt{x^2+y^2-2x+6y+19}+\sqrt{x^2+y^2+8x-4y+21}.$$ Find $n^2.$

1986 IMO Shortlist, 13

Tags: probability , combinatorics , counting

A particle moves from $(0, 0)$ to $(n, n)$ directed by a fair coin. For each head it moves one step east and for each tail it moves one step north. At $(n, y), y < n$, it stays there if a head comes up and at $(x, n), x < n$, it stays there if a tail comes up. Let$ k$ be a fixed positive integer. Find the probability that the particle needs exactly $2n+k$ tosses to reach $(n, n).$

2013 BMT Spring, 8

Tags: number theory

The three-digit prime number $p$ is written in base $2$ as $p_2$ and in base $5$ as $p_5$, and the two representations share the same last $2$ digits. If the ratio of the number of digits in $p_2$ to the number of digits in $p_5$ is $5$ to $2$, find all possible values of $p$.

2023 AMC 12/AHSME, 22

Tags: function

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$ $$f(a + b) + f(a - b) = 2f(a) f(b).$$ Which one of the following cannot be the value of $f(1)?$ $ \textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } -1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } -2$

2009 Today's Calculation Of Integral, 506

Tags: calculus , integration , function , parabola , calculus computations , conic , geometry

Let $ a,\ b$ be the real numbers such that $ 0\leq a\leq b\leq 1$. Find the minimum value of $ \int_0^1 |(x\minus{}a)(x\minus{}b)|\ dx$.

1985 National High School Mathematics League, 1

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$a,b$ are real numbers (neither is $0$).Given two conditions: A: $a>0$. B: $a>b$ and $a^{-1}>b^{-1}$. Then, which one of the followings are true? $(\text{A})$A is sufficient but unnecessary condition of B. $(\text{B})$A is necessary but insufficient condition of B. $(\text{C})$A is sufficient and necessary condition of B. $(\text{D})$A is insufficient and unnecessary condition of B.

2002 Manhattan Mathematical Olympiad, 4

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Somebody placed digits $1,2,3, \ldots , 9$ around the circumference of a circle in an arbitrary order. Reading clockwise three consecutive digits you get a $3$-digit whole number. There are nine such $3$-digit numbers altogether. Find their sum.

2024 USAJMO, 2

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Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq 2m$ and $1\leq y\leq 2n$. A configuration of $mn$ rectangles is called [i]happy[/i] if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd. [i]Proposed by Serena An and Claire Zhang[/i]