Olympiad problems

1994 Portugal MO, 5

Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?

1989 Turkey Team Selection Test, 5

Tags: inequalities

There are $n\geq2$ weights such that each weighs a positive integer less than $n$ and their total weights is less than $2n$. Prove that there is a subset of these weights such that their total weights is equal to $n$.

1998 National Olympiad First Round, 11

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If two faces of a dice have a common edge, the two faces are called adjacent faces. In how many ways can we construct a dice with six faces such that any two consecutive numbers lie on two adjacent faces? $\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 14 \qquad\textbf{(C)}\ 18 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ \text{None}$

1975 All Soviet Union Mathematical Olympiad, 218

Tags: tournament , maximum , combinatorics

The world and the european champion are determined in the same tournament carried in one round. There are $20$ teams and $k$ of them are european. The european champion is determined according to the results of the games only between those $k$ teams. What is the greatest $k$ such that the situation, when the single european champion is the single world outsider, is possible if: a) it is hockey (draws allowed)? b) it is volleyball (no draws)?

2010 Paenza, 5

Tags: geometry , 3d geometry , combinatorics

In $4$-dimensional space, a set of $1 \times 2 \times 3 \times 4$ bricks is given. Decide whether it is possible to build boxes of the following sizes using these bricks: [list]i) $2 \times 5 \times 7 \times 12$ ii) $5 \times 5 \times 10 \times 12$ iii) $6 \times 6 \times 6 \times 6$.[/list]

2019 MIG, 15

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Alice, Bob, and Catherine decide to have a race. Alice runs at a speed of $3$ feet per second, and Bob runs at a speed of $5$ feet per second. In the end, Bob finishes the same amount of time before Catherine as Catherine finishes before Alice. What was Catherine's speed, in feet per second? $\textbf{(A) }\dfrac{15}4\qquad\textbf{(B) }4\qquad\textbf{(C) }\dfrac{17}4\qquad\textbf{(D) }\dfrac92\qquad\textbf{(E) }\text{impossible to determine}$

2015 Latvia Baltic Way TST, 2

Tags: functional equation , functional , algebra

It is known about the function $f : R \to R$ that $\bullet$ $f(x) > f(y)$ for all real $x > y$ $\bullet$ $f(x) > x$ for all real $x$ $\bullet$ $f(2x - f (x)) = x$ for all real $x$. Prove that $f(x) = x + f(0)$ for all real numbers $x$.

2024 CCA Math Bonanza, I5

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Triangle $ABC$ has points $D$,$E$,$F$ on segment $BC$ in that order, where $D$ is between $B$ and $E$, and $AD$ and $AE$ trisect angle $BAF$. If $\angle BAF = 60^{\circ}$, $\frac{EF}{EC}=\frac{2}{3}$, and $\frac{AE}{AC} = 2$, find $\angle BAC$. [i]Individual #5[/i]

2008 Romania Team Selection Test, 1

Tags: induction , graph theory , algebra

Let $ n$ be a nonzero positive integer. Find $ n$ such that there exists a permutation $ \sigma \in S_{n}$ such that \[ \left| \{ |\sigma(k) \minus{} k| \ : \ k \in \overline{1, n} \}\right | = n.\]

2014-2015 SDML (High School), 2

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A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?

2019 May Olympiad, 3

Tags: geometry , equal angles

On the sides $AB, BC$ and $CA$ of a triangle $ABC$ are located the points $P, Q$ and $R$ respectively, such that $BQ = 2QC, CR = 2RA$ and $\angle PRQ = 90^o$. Show that $\angle APR =\angle RPQ$.

2010 Singapore Junior Math Olympiad, 5

Tags: algebra

The numbers $\frac{1}{1}, \frac{1}{2}, ... , \frac{1}{2010}$ are written on a blackboard. A student chooses any two of the numbers, say $x$, $y$, erases them and then writes down $x + y + xy$. He continues to do this until only one number is left on the blackboard. What is this number?

2014 Baltic Way, 19

Tags: greatest common divisor , number theory

Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]

2007 South africa National Olympiad, 3

Tags: incenter , geometry

In acute-angled triangle $ ABC$, the points $ D,E,F$ are on sides $ BC,CA,AB$, respectively such that $ \angle AFE \equal{} \angle BFD, \angle FDB \equal{} \angle EDC, \angle DEC \equal{} \angle FEA$. Prove that $ AD$ is perpendicular to $ BC$.

1946 Moscow Mathematical Olympiad, 120

Tags: combinatorics , combinatorial geometry

a) A bus network is organized so that: 1) one can reach any stop from any other stop without changing buses; 2) every pair of routes has a single stop at which one can change buses; 3) each route has exactly three stops? How many bus routes are there? It is assumed that there are at least two routes. b) A town has $57$ bus routes. How many stops does each route have if it is known that 1) one can reach any stop from any other stop without changing buses; 2) for every pair of routes there is a single stop where one can change buses; 3) each route has three or more stops?

1988 Irish Math Olympiad, 3

Tags: geometry

$ABC$ is a triangle inscribed in a circle, and $E$ is the mid-point of the arc subtended by $BC$ on the side remote from $A$. If through $E$ a diameter $ED$ is drawn, show that the measure of the angle $DEA$ is half the magnitude of the difference of the measures of the angles at $B$ and $C$.

2007 Pre-Preparation Course Examination, 2

Tags: algorithm , combinatorics

There is a WORD game with the following rules. There are finite number of relations $U_{i}\longrightarrow V_{i}$($U_{i},V_{i}$ are words). There is are two words $A,B$. We start from $A$, and we want to reach to $B$. At each step we can change one subword $U_{i}$ to $V_{i}$. Prove that there does not exist an algorithm that picks up $A,B$ and $U_{i}$'s,$V_{i}$'s and decides whether we can reach from $A$ to $B$ or not.

2009 Baltic Way, 3

Tags: algebra , polynomial , number theory

Let $ n$ be a given positive integer. Show that we can choose numbers $ c_k\in\{\minus{}1,1\}$ ($ i\le k\le n$) such that \[ 0\le\sum_{k\equal{}1}^nc_k\cdot k^2\le4.\]

2022 JBMO Shortlist, A3

Tags: inequality , algebra

Let $a, b,$ and $c$ be positive real numbers such that $a + b + c = 1$. Prove the following inequality $$a \sqrt[3]{\frac{b}{a}} + b \sqrt[3]{\frac{c}{b}} + c \sqrt[3]{\frac{a}{c}} \le ab + bc + ca + \frac{2}{3}.$$ Proposed by [i]Anastasija Trajanova, Macedonia[/i]

1926 Eotvos Mathematical Competition, 1

Tags: diophantine , diophantine equation , system of equations , system , number theory

Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

1996 APMO, 1

Tags: perimeter , ratio , rhombus , trigonometry , invariant , symmetry , geometry

Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two segments perpendicular to the diagonal $BD$ and such that the distance between them is $d > \frac{BD}{2}$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$. Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of $MN$ and $PQ$ so long as the distance between them remains constant.

2025 Harvard-MIT Mathematics Tournament, 5

Tags: team

Let $\triangle{ABC}$ be an acute triangle with orthocenter $H.$ Points $E$ and $F$ are on segments $\overline{AC}$ and $\overline{AB},$ respectively, such that $\angle{EHF}=90^\circ.$ Let $X$ be the foot of the perpendicular from $H$ to $\overline{EF}.$ Prove that $\angle{BXC}=90^\circ.$

2024 Kazakhstan National Olympiad, 4

Tags: analytic geometry , combinatorics

Players $A$ and $B$ play the following game on the coordinate plane. Player $A$ hides a nut at one of the points with integer coordinates, and player $B$ tries to find this hidden nut. In one move $B$ can choose three different points with integer coordinates, then $A$ tells whether these three points together with the nut's point lie on the same circle or not. Can $B$ be guaranteed to find the nut in a finite number of moves?

Kvant 2019, M2566

Tags: number theory , lcm , least common multiple

Determine if there exist five consecutive positive integers such that their LCM is a perfect square.

2010 Oral Moscow Geometry Olympiad, 3

Tags: geometry , fixed , fixed point , area of a triangle , area

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.