Olympiad problems

2001 Croatia Team Selection Test, 1

Consider $A = \{1, 2, ..., 16\}$. A partition of $A$ into nonempty sets $A_1, A_2,..., A_n$ is said to be good if none of the Ai contains elements $a, b, c$ (not necessarily distinct) such that $a = b + c$. (a) Find a good partition $\{A_1, A_2, A_3, A_4\}$ of $A$. (b) Prove that no partition $\{A_1, A_2, A_3\}$ of $A$ is good

2015 Canadian Mathematical Olympiad Qualification, 6

Tags: geometry , angle bisector

Let $\triangle ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$, and $AB < AC$. Let points $D, E, F$ be located on side $BC$ such that $AD$ is the altitude, $AE$ is the internal angle bisector, and $AF$ is the median. Prove that $3AD + AF > 4AE$.

2011 Indonesia TST, 2

Tags: induction , abstract algebra , vector , group theory

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).

1983 IMO Longlists, 7

Tags: inequalities , number theory

Find all numbers $x \in \mathbb Z$ for which the number \[x^4 + x^3 + x^2 + x + 1\] is a perfect square.

2017 AMC 12/AHSME, 9

Tags: set

Let $S$ be the set of points $(x,y)$ in the coordinate plane such that two of the three quantities $3$, $x+2$, and $y-4$ are equal and the third of the three quantities is no greater than this common value. Which of the following is a correct description of $S$? $\textbf{(A) } \text{a single point} \qquad \textbf{(B) } \text{two intersecting lines} \\ \\ \textbf{(C) } \text{three lines whose pairwise intersections are three distinct points} \\ \\ \textbf{(D) } \text{a triangle} \qquad \textbf{(E) } \text{three rays with a common endpoint}$

2022-23 IOQM India, 9

Tags: geometry

Two sides of an integer sided triangle have lengths $18$ and $x$. If there are exactly $35$ possible integer $y$ such that $18,x,y$ are the sides of a non-degenerate triangle, find the number of possible integer values $x$ can have.

2013 Costa Rica - Final Round, G3

Tags: geometry , rectangle

Let $ABCD$ be a rectangle with center $O$ such that $\angle DAC = 60^o$. Bisector of $\angle DAC$ cuts a $DC$ at $S$, $OS$ and $AD$ intersect at $L$, $BL$ and $AC$ intersect at $M$. Prove that $SM \parallel CL$.

2019 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , geometry

Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.

2024 Lusophon Mathematical Olympiad, 6

Tags: number theory

A positive integer $n$ is called $oeirense$ if there exist two positive integers $a$ and $b$, not necessarily distinct, such that $n=a^2+b^2$. Determine the greatest integer $k$ such that there exist infinitely many positive integers $n$ such that $n$, $n+1$, $\dots$, $n+k$ are oeirenses.

2004 Thailand Mathematical Olympiad, 16

Tags: digit , number theory

What are last three digits of $2^{2^{2004}}$ ?

2014 German National Olympiad, 6

Tags: geometry , angle , circumscribed , quadrilateral

Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$

1984 AIME Problems, 7

Tags: function , algebra , domain , functional equation

The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.

1998 Gauss, 20

Tags: gauss

Each of the 12 edges of a cube is coloured either red or green. Every face of the cube has at least one red edge. What is the smallest number of red edges? $\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

MOAA Team Rounds, 2022.2

Tags: geometry

While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.

2007 Chile National Olympiad, 3

Tags: number theory , combinatorics , game , game strategy

Two players, Aurelio and Bernardo, play the following game. Aurelio begins by writing the number $1$. Next it is Bernardo's turn, who writes number $2$. From then on, each player chooses whether to add $1$ to the number just written by the previous player, or whether multiply that number by $2$. Then write the result and it's the other player's turn. The first player to write a number greater than $ 2007$ loses the game. Determine if one of the players can ensure victory no matter what the other does.

2018 Estonia Team Selection Test, 4

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

1993 All-Russian Olympiad, 1

Tags: modular arithmetic , parameterization , quadratic , number theory

For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.

2002 JBMO ShortLists, 11

Tags: geometry

Let $ ABC$ be an isosceles triangle with $ AB\equal{}AC$ and $ \angle A\equal{}20^\circ$. On the side $ AC$ consider point $ D$ such that $ AD\equal{}BC$. Find $ \angle BDC$.

III Soros Olympiad 1996 - 97 (Russia), 9.1

Tags: radical , algebra

Without using a calculator, find out which number is greater: $$|\sqrt[3]{5}-\sqrt3|-\sqrt3| \,\,\,\, \text{or} \,\,\,\, 0.01$$

1999 Harvard-MIT Mathematics Tournament, 6

Tags:

You want to sort the numbers 5 4 3 2 1 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. For example, [i]6 5 3[/i] 4 2 1 -> 4 2 [i]6 5 3[/i] 1 is a valid block move for 6 numbers. What is the minimum number of block moves necessary to get 1 2 3 4 5?

1968 All Soviet Union Mathematical Olympiad, 106

Tags: geometry , incircle , median

Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.

2020 Balkan MO Shortlist, A2

Tags: algebra , inequality

Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that $\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$. [i]Albania[/i]

2021 Nigerian Senior MO Round 2, 2

Tags: combinatorics

$N$ boxes are arranged in a circle and are numbered $1,2,3,.....N$ In a clockwise direction. A ball is assigned a number from${1,2,3,....N}$ and is placed in one of the boxes.A round consist of the following; if the current number on the ball is $n$, the ball is moved $n$ boxes in the clockwise direction and the number on the ball is changed to $n+1$ if $n<N$ and to $1$ if $n=N$. Is it possible to choose $N$, the initial number on the ball, and the first position of the ball in such a way that the ball gets back to the same box with the same number on it for the first time after exactly $2020$ rounds

2024 Yasinsky Geometry Olympiad, 3

Tags: geometry , orthocenter , diameter

Let $ H $ be the orthocenter of an acute triangle $ ABC $, and let $ AT $ be the diameter of the circumcircle of this triangle. Points $ X $ and $ Y $ are chosen on sides $ AC $ and $ AB $, respectively, such that $ TX = TY $ and $ \angle XTY + \angle XAY = 90^\circ $. Prove that $ \angle XHY = 90^\circ $. [i] Proposed by Matthew Kurskyi[/i]

1989 IMO Shortlist, 19

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.