Olympiad problems

2022 Francophone Mathematical Olympiad, 1

find all functions $f:\mathbb{Z} \to \mathbb{Z} $ such that $f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)$ holds for all $m,n \in \mathbb{Z}$

2024 CMIMC Geometry, 6

Tags: geometry

Andrew Mellon found a piece of melon that is shaped like a octagonal prism where the bases are regular. Upon slicing it in half once, he found that he created a cross-section that is an equilateral hexagon. What is the minimum possible ratio of the height of the melon piece to the side length of the base? [i]Proposed by Lohith Tummala[/i]

2023 Junior Balkan Team Selection Tests - Romania, P3

Tags: combinatorics

Consider a grid with $n{}$ lines and $m{}$ columns $(n,m\in\mathbb{N},m,n\ge2)$ made of $n\cdot m \; 1\times1$ squares called ${cells}$. A ${snake}$ is a sequence of cells with the following properties: the first cell is on the first line of the grid and the last cell is on the last line of the grid, starting with the second cell each has a common side with the previous cell and is not above the previous cell. Define the ${length}$ of a snake as the number of cells it's made of. Find the arithmetic mean of the lengths of all the snakes from the grid.

1998 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , analytic geometry , graphing lines , slope , derivative

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

2019 Durer Math Competition Finals, 6

Tags: combinatorics , game , game strategy

(Game) At the beginning of the game, the organisers place paper disks on the table, grouped into piles which may contain various numbers of disks. The two players take turns. On a player’s turn, their opponent selects two piles (one if there is only one pile left), and the player must remove some number of disks from one of the piles selected. This means that at least one disk has to be removed, and removing all disks in the pile is also permitted. The player removing the last disk from the table wins. [i]Defeat the organisers in this game twice in a row! A starting position will be given and then you can decide whether you want to go first or second.[/i]

2019 Kyiv Mathematical Festival, 5

Tags: Kyiv mathematical festival , combinatorics , number theory

Is it possible to fill the cells of a table of size $2019\times2019$ with pairwise distinct positive integers in such a way that in each rectangle of size $1\times2$ or $2\times1$ the larger number is divisible by the smaller one, and the ratio of the largest number in the table to the smallest one is at most $2019^4?$

2019 Azerbaijan IMO TST, 2

Tags: geometry , circumcircle , equal segments

Let $ABC$ ($BC > AC$) be an acute triangle with circumcircle $k$ centered at $O$. The tangent to $k$ at $C$ intersects the line $AB$ at the point $D$. The circumcircles of triangles $BCD, OCD$ and $AOB$ intersect the ray $CA$ (beyond $A$) at the points $Q, P$ and $K$, respectively, such that $P \in (AK)$ and $K \in (PQ)$. The line $PD$ intersects the circumcircle of triangle $BKQ$ at the point $T$, so that $P$ and $T$ are in different halfplanes with respect to $BQ$. Prove that $TB = TQ$.

1996 Italy TST, 3

Tags: geometry

3.Let ABCD be a parallelogram with side AB longer than AD and acute angle $\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.

2024 Sharygin Geometry Olympiad, 8.1

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

A circle $\omega$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $XP$ and the circle $XOP$ meet $\omega$ for a second time at points $X_1$, $X_2$ respectively. Prove that all lines $X_1X_2$ are parallel.

2014 AMC 12/AHSME, 14

Tags: geometry , AMC

A rectangular box has a total surface area of $94$ square inches. The sum of the lengths of all its edges is $48$ inches. What is the sum of the lengths in inches of all of its interior diagonals? ${ \textbf{(A)}\ 8\sqrt{3}\qquad\textbf{(B)}\ 10\sqrt{2}\qquad\textbf{(C)}\ 16\sqrt{3}\qquad\textbf{(D)}}\ 20\sqrt{2}\qquad\textbf{(E)}\ 40\sqrt{2} $

2025 District Olympiad, P4

Tags: geometry , complex numbers

Let $ABCDEF$ be a convex hexagon with $\angle A = \angle C=\angle E$ and $\angle B = \angle D=\angle F$. [list=a] [*] Prove that there is a unique point $P$ which is equidistant from sides $AB,CD$ and $EF$. [*] If $G_1$ and $G_2$ are the centers of mass of $\triangle ACE$ and $\triangle BDF$, show that $\angle G_1PG_2=60^{\circ}$.

Kvant 2023, M2776

Tags: combinatorics , algebra

There are $n{}$ currencies in a country, numbered from 1 to $n{}.$ In each currency, only non-negative integers are possible amounts of money. A person can have only one currency at any time. A person can exchange all the money he has from currency $i{}$ to currency $j{}$ at the rate of $\alpha_{ij}$ which is a positive real number. If he had $d{}$ units of currency $i{}$ he instead receives $\alpha_{ij}d$ units of currency $j{}$ while this number is rounded to the nearest integer; a number of the form $t-1/2$ is rounded to $t{}$ for any integer $t{}.$ It is known that $\alpha_{ij}\alpha_{jk}=\alpha_{ik}$ and $\alpha_{ii}=1$ for every $i,j,k.$ Can there be a person who can get rich indefinitely? [i]Proposed by I. Bogdanov[/i]

2019 Philippine MO, 2

Tags: combinatorics , double counting

Twelve students participated in a theater festival consisting of $n$ different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of $n$.

2014 Saudi Arabia IMO TST, 3

Tags: analytic geometry , vector , induction , modular arithmetic , combinatorics unsolved , combinatorics

We are given a lattice and two pebbles $A$ and $B$ that are placed at two lattice points. At each step we are allowed to relocate one of the pebbles to another lattice point with the condition that the distance between pebbles is preserved. Is it possible after finite number of steps to switch positions of the pebbles?

2011 NZMOC Camp Selection Problems, 3

Tags: combinatorics , Equilateral , grid , game strategy , game , winning strategy

Chris and Michael play a game on a board which is a rhombus of side length $n$ (a positive integer) consisting of two equilateral triangles, each of which has been divided into equilateral triangles of side length $ 1$. Each has a single token, initially on the leftmost and rightmost squares of the board, called the “home” squares (the illustration shows the case $n = 4$). [img]https://cdn.artofproblemsolving.com/attachments/e/b/8135203c22ce77c03c144850099ad1c575edb8.png[/img] A move consists of moving your token to an adjacent triangle (two triangles are adjacent only if they share a side). To win the game, you must either capture your opponent’s token (by moving to the triangle it occupies), or move on to your opponent’s home square. Supposing that Chris moves first, which, if any, player has a winning strategy?

1966 Vietnam National Olympiad, 2

Tags: geometry open , geometry

$a, b$ are two fixed lines through $O$. Variable lines $x, y$ are parallel. $x$ intersects a at $A$ and $b$ at $C$, $y$ intersects $a$ at $B$ and $b$at $D$. The lines $AD$ and $BC$ meet at $M$. The line through $M$ parallel to $x$ meets $a$ at $L$ and $b$ at $N$. What can you say about $L, M, N$? Find the locus $M$.

2019 Thailand TST, 1

Tags: IMO Shortlist , number theory , Hi

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

2024 APMO, 5

Tags: geometry , APMO 2024

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

2009 Indonesia TST, 3

Tags: modular arithmetic , floor function , ceiling function , inequalities , number theory proposed , number theory

Let $ n \ge 2009$ be an integer and define the set: \[ S \equal{} \{2^x|7 \le x \le n, x \in \mathbb{N}\}. \] Let $ A$ be a subset of $ S$ and the sum of last three digits of each element of $ A$ is $ 8$. Let $ n(X)$ be the number of elements of $ X$. Prove that \[ \frac {28}{2009} < \frac {n(A)}{n(S)} < \frac {82}{2009}. \]

2002 India IMO Training Camp, 3

Tags: quadratics , algebra proposed , algebra

Let $X=\{2^m3^n|0 \le m, \ n \le 9 \}$. How many quadratics are there of the form $ax^2+2bx+c$, with equal roots, and such that $a,b,c$ are distinct elements of $X$?

2006 Romania National Olympiad, 3

Tags: group theory , abstract algebra , superior algebra , superior algebra unsolved

Let $\displaystyle G$ be a finite group of $\displaystyle n$ elements $\displaystyle ( n \geq 2 )$ and $\displaystyle p$ be the smallest prime factor of $\displaystyle n$. If $\displaystyle G$ has only a subgroup $\displaystyle H$ with $\displaystyle p$ elements, then prove that $\displaystyle H$ is in the center of $\displaystyle G$. [i]Note.[/i] The center of $\displaystyle G$ is the set $\displaystyle Z(G) = \left\{ a \in G \left| ax=xa, \, \forall x \in G \right. \right\}$.

2010 ELMO Problems, 2

Tags: modular arithmetic , algorithm , combinatorics proposed , combinatorics , Elmo

2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations: [list] [*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip. [*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list] Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it. [i]Brian Hamrick.[/i]

2012 India Regional Mathematical Olympiad, 5

Tags: geometry , circumcircle , perpendicular bisector , angle bisector , geometry proposed

Let $AL$ and $BK$ be the angle bisectors in a non-isosceles triangle $ABC,$ where $L$ lies on $BC$ and $K$ lies on $AC.$ The perpendicular bisector of $BK$ intersects the line $AL$ at $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK.$ Prove that $LN=NA.$

2001 Czech-Polish-Slovak Match, 1

Tags: inequalities , n-variable inequality , Holder , Convexity

Let $n\ge2$ be a natural number, and $a_i$ be positive numbers, where $i=1,2,\cdots,n.$ Show that \[\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)\]

2015 IMO, 5

Tags: functional equation , algebra , IMO Shortlist , Dorlir Ahmeti , IMO Proposal

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$. [i]Proposed by Dorlir Ahmeti, Albania[/i]