Olympiad problems

OMMC POTM, 2023 3

Tags: number theory

Three natural numbers are such that the product of any two of them is divisible by the sum of those two numbers. Prove that these numbers have a common divisor greater than $1$. [i]Proposed by Evan Chang (squareman), USA[/i]

2000 Switzerland Team Selection Test, 5

Tags: combinatorics , letters

Consider all words of length $n$ consisting of the letters $I,O,M$. How many such words are there, which contain no two consecutive $M$’s?

Croatia MO (HMO) - geometry, 2018.3

Tags: tangent , circumcenter , circumcircle , reflection , geometry

Let $k$ be a circle centered at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangent on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersect the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$, so that $|BC| = |BM|$. Prove that the circumcenter of the triangle $ADM$ is the reflection of $O$ across the line $AD$

2025 Spain Mathematical Olympiad, 5

Tags: combinatorics

Let $S$ be a finite set of cells in a square grid. On each cell of $S$ we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of $S$ still contains one grasshopper. [list] [*] Prove that, for every set $S$, the number of Asturian arrangements is a perfect square. [*] Compute the number of Asturian arrangements if $S$ is the following set:

1948 Moscow Mathematical Olympiad, 142

Tags: distance , combinatorics , combinatorial geometry

Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$. For what ratios of $a : b$ are such arrangements possible?

2008 Silk Road, 3

Tags: graph , graph coloring , coloring , combinatorics

Let $ G$ be a graph with $ 2n$ vertexes and $ 2n(n\minus{}1)$ edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red. Prove that it is possible to color some vertexes and edges in $ G$,such that all red vertexes has exactly $ n$ red edges.

2001 Korea - Final Round, 2

Tags: incenter , circumcircle , geometry

In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.

2002 District Olympiad, 3

Tags: number theory , inequalities , logarithm

Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $ Show that $ a<b. $

2019 Bulgaria National Olympiad, 2

Tags: circumcenter , geometry , orthocenter , circumcircle , perpendicular bisector

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$

2019 ELMO Shortlist, N3

Tags: number theory

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

2023 BMT, Tie 1

Tags: algebra

Wen finds $17$ consecutive positive integers that sum to $2023$. Compute the smallest of these integers.

2022-2023 OMMC, 1

Tags:

John has cut out these two polygons made out of unit squares. He joins them to each other to form a larger polygon (but they can't overlap). Find the smallest possible perimeter this larger polygon can have. He can rotate and reflect the cut out polygons.

1988 Federal Competition For Advanced Students, P2, 4

Tags: system of equations , algebra

Let $ a_{ij}$ be nonnegative integers such that $ a_{ij}\equal{}0$ if and only if $ i>j$ and that $ \displaystyle\sum_{j\equal{}1}^{1988}a_{ij}\equal{}1988$ holds for all $ i\equal{}1,...,1988$. Find all real solutions of the system of equations: $ \displaystyle\sum_{j\equal{}1}^{1988} (1\plus{}a_{ij})x_j\equal{}i\plus{}1, 1 \le i \le 1988$.

2017 BMT Spring, 3

Tags: algebra , calculus , integration

Compute $\int^9_{-9}17x^3 \cos (x^2) dx.$

2007 Princeton University Math Competition, 10

Tags: modular arithmetic

Find all primes $p$ such that there exist positive integers $q$ and $r$ such that $p \nmid q$, $3 \nmid q$, $p^3 = r^3 - q^2$.

2021 JBMO Shortlist, A1

Tags: algebra

Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

2001 Tuymaada Olympiad, 4

Tags: function , combinatorics

Is it possible to colour all positive real numbers by 10 colours so that every two numbers with decimal representations differing in one place only are of different colours? (We suppose that there is no place in a decimal representations such that all digits starting from that place are 9's.) [i]Proposed by A. Golovanov[/i]

1966 IMO Shortlist, 57

Tags: geometry , 3d geometry , symmetry , rotation , cube

Is it possible to choose a set of $100$ (or $200$) points on the boundary of a cube such that this set is fixed under each isometry of the cube into itself? Justify your answer.

1984 All Soviet Union Mathematical Olympiad, 390

Tags: chessboard , combinatorics

The white fields of $1983\times 1984 $1983x1984 are filled with either $+1$ or $-1$. For every black field, the product of neighbouring numbers is $+1$. Prove that all the numbers are $+1$.

2017 Baltic Way, 18

Tags: number theory , abstract algebra

Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?

2020/2021 Tournament of Towns, P2

Tags: number theory

Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) [i]Boris Frenkin[/i]

2004 AMC 10, 16

Tags: induction

The $ 5\times 5$ grid shown contains a collection of squares with sizes from $ 1\times 1$ to $ 5\times 5$. How many of these squares contain the black center square? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)); for(int i=0; i<=5; ++i) { draw((0,i)--(5,i)); draw((i,0)--(i,5)); } fill((2,2)--(2,3)--(3,3)--(3,2)--cycle);[/asy]$ \textbf{(A)}\ 12\qquad \textbf{(B)}\ 15\qquad \textbf{(C)}\ 17\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 20$

2020 Ukrainian Geometry Olympiad - December, 5

Tags: geometry , circumcircle , altitude , angle

In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.

2013 AMC 8, 8

Tags: probability , complementary counting

A fair coin is tossed 3 times. What is the probability of at least two consecutive heads? $\textbf{(A)}\ \frac18 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac34$

2011 Switzerland - Final Round, 8

Tags: parallelogram , ratio , similar triangles , power of a point , geometry

Let $ABCD$ be a parallelogram and $H$ the Orthocentre of $\triangle{ABC}$. The line parallel to $AB$ through $H$ intersects $BC$ at $P$ and $AD$ at $Q$ while the line parallel to $BC$ through $H$ intersects $AB$ at $R$ and $CD$ at $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic. [i](Swiss Mathematical Olympiad 2011, Final round, problem 8)[/i]