Olympiad problems

2024 Pan-American Girls’ Mathematical Olympiad, 6

Let $ABC$ be a triangle, and let $a$, $b$, and $c$ be the lengths of the sides opposite vertices $A$, $B$, and $C$, respectively. Let $R$ be its circumradius and $r$ its inradius. Suppose that $b + c = 2a$ and $R = 3r$. The excircle relative to vertex $A$ intersects the circumcircle of $ABC$ at points $P$ and $Q$. Let $U$ be the midpoint of side $BC$, and let $I$ be the incenter of $ABC$. Prove that $U$ is the centroid of triangle $QIP$.

2011 Middle European Mathematical Olympiad, 3

Tags: geometry , circumcircle , geometric transformation , reflection

In a plane the circles $\mathcal K_1$ and $\mathcal K_2$ with centers $I_1$ and $I_2$, respectively, intersect in two points $A$ and $B$. Assume that $\angle I_1AI_2$ is obtuse. The tangent to $\mathcal K_1$ in $A$ intersects $\mathcal K_2$ again in $C$ and the tangent to $\mathcal K_2$ in $A$ intersects $\mathcal K_1$ again in $D$. Let $\mathcal K_3$ be the circumcircle of the triangle $BCD$. Let $E$ be the midpoint of that arc $CD$ of $\mathcal K_3$ that contains $B$. The lines $AC$ and $AD$ intersect $\mathcal K_3$ again in $K$ and $L$, respectively. Prove that the line $AE$ is perpendicular to $KL$.

2009 Argentina Team Selection Test, 4

Tags: number theory

Find all positive integers $ n$ such that $ 20^n \minus{} 13^n \minus{} 7^n$ is divisible by $ 309$.

1981 All Soviet Union Mathematical Olympiad, 309

Tags: equilateral , geometry , vector

Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral

MBMT Guts Rounds, 2015.8

Tags:

The school store is running out of supplies, but it still has five items: one pencil (costing $\$1$), one pen (costing $\$1$), one folder (costing $\$2$), one pack of paper (costing $\$3$), and one binder (costing $\$4$). If you have $\$10$, in how many ways can you spend your money? (You don't have to spend all of your money, or any of it.)

2014 Contests, 4

Tags: number theory

Let $m\ge 3$ and $n$ be positive integers such that $n>m(m-2)$. Find the largest positive integer $d$ such that $d\mid n!$ and $k\nmid d$ for all $k\in\{m,m+1,\ldots,n\}$.

2006 Romania National Olympiad, 3

Tags: inequalities , limit , complex number , real analysis

We have in the plane the system of points $A_1,A_2,\ldots,A_n$ and $B_1,B_2,\ldots,B_n$, which have different centers of mass. Prove that there is a point $P$ such that \[ PA_1 + PA_2 + \ldots+ PA_n = PB_1 + PB_2 + \ldots + PB_n . \]

2021 New Zealand MO, 4

Tags: diophantine equation , diophantine , number theory

Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.

1998 Brazil Team Selection Test, Problem 4

Tags: geometry

Let $L$ be a circle with center $O$ and tangent to sides $AB$ and $AC$ of a triangle $ABC$ in points $E$ and $F$, respectively. Let the perpendicular from $O$ to $BC$ meet $EF$ at $D$. Prove that $A,D$ and $M$ are collinear, where $M$ is the midpoint of $BC$.

2012 District Olympiad, 2

Tags: number theory , irrational number , algebra

Let $a, b$ and $c$ be positive real numbers such that $$a^2+ab+ac-bc = 0.$$ a) Show that if two of the numbers $a, b$ and $c$ are equal, then at least one of the numbers $a, b$ and $c$ is irrational. b) Show that there exist infinitely many triples $(m, n, p)$ of positive integers such that $$m^2 + mn + mp -np = 0.$$

2017 IOM, 5

Tags: number theory

Let $x $ and $y $ be positive integers such that $[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]$.Prove that one of the two numbers $x $ and $y $ divide the other. (Here $[a,b] $ denote the least common multiple of $a $ and $b $). Proposed by Dusan Djukic.

2005 Flanders Junior Olympiad, 4

Tags:

(a) Be M an internal point of the convex quadrilateral ABCD. Prove that $|MA|+|MB| < |AD|+|DC|+|CB|$. (b) Be M an internal point of the triangle ABC. Note $k=\min(|MA|,|MB|,|MC|)$. Prove $k+|MA|+|MB|+|MC|<|AB|+|BC|+|CA|$.

2012 Kosovo Team Selection Test, 1

Tags: modular arithmetic , combinatorics

A student had $18$ papers. He seleced some of these papers, then he cut each of them in $18$ pieces.He took these pieces and selected some of them, which he again cut in $18$ pieces each.The student took this procedure untill he got tired .After a time he counted the pieces and got $2012$ pieces .Prove that the student was wrong during the counting.

2010 AMC 10, 25

Tags: polynomial , number theory , least common multiple , floor function , binomial theorem , algebra

Let $ a>0$, and let $ P(x)$ be a polynomial with integer coefficients such that \[ P(1)\equal{}P(3)\equal{}P(5)\equal{}P(7)\equal{}a\text{, and}\] \[ P(2)\equal{}P(4)\equal{}P(6)\equal{}P(8)\equal{}\minus{}a\text{.}\] What is the smallest possible value of $ a$? $ \textbf{(A)}\ 105 \qquad \textbf{(B)}\ 315 \qquad \textbf{(C)}\ 945 \qquad \textbf{(D)}\ 7! \qquad \textbf{(E)}\ 8!$

2007 Kazakhstan National Olympiad, 4

Tags: geometry , rectangle , combinatorics

Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.

1995 IMO Shortlist, 1

Tags: geometry , circumcircle , concurrency

Let $ A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $ AC$ and $ BD$ intersect at $ X$ and $ Y$. The line $ XY$ meets $ BC$ at $ Z$. Let $ P$ be a point on the line $ XY$ other than $ Z$. The line $ CP$ intersects the circle with diameter $ AC$ at $ C$ and $ M$, and the line $ BP$ intersects the circle with diameter $ BD$ at $ B$ and $ N$. Prove that the lines $ AM,DN,XY$ are concurrent.

2015 Ukraine Team Selection Test, 9

Tags: point , heptagon , pentagon , convex , combinatorics , combinatorial geometry

The set $M$ consists of $n$ points on the plane and satisfies the conditions: $\bullet$ there are $7$ points in the set $M$, which are vertices of a convex heptagon, $\bullet$ for arbitrary five points with $M$, which are vertices of a convex pentagon, there is a point that also belongs to $M$ and lies inside this pentagon. Find the smallest possible value that $n$ can take .

2025 JBMO TST - Turkey, 7

Tags: geometry , pentagon , ratio

$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?

2024 District Olympiad, P1

Tags: linear algebra , matrix

Consider the matrix $X\in\mathcal{M}_2(\mathbb{C})$ which satisfies $X^{2022}=X^{2023}.$ Prove that $X^2=X^3.$

2016 Saudi Arabia Pre-TST, 2.2

Tags: combinatorics , combinatorial geometry , coloring

Ten vertices of a regular $20$-gon $A_1A_2....A_{20}$ are painted black and the other ten vertices are painted blue. Consider the set consisting of diagonal $A_1A_4$ and all other diagonals of the same length. 1. Prove that in this set, the number of diagonals with two black endpoints is equal to the number of diagonals with two blue endpoints. 2. Find all possible numbers of the diagonals with two black endpoints.

1987 AMC 12/AHSME, 29

Tags:

Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\frac{1}{t_{(n-1)}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, the sum of the digits of $n$ is $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 17 \qquad\textbf{(C)}\ 19 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 23$

2021 BMT, 3

Tags: floor function , algebra

Let $x$ be a solution to the equation $\lfloor x \lfloor x + 2\rfloor + 2\rfloor = 10$. Compute the smallest $C$ such that for any solution $x$, $x < C$. Here, $\lfloor m \rfloor$ is defined as the greatest integer less than or equal to $m$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor -4.25\rfloor = -5$.

2017 Baltic Way, 14

Tags: geometry , perpendicular lines , angle chase

Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.

2021 IMC, 1

Tags: linear algebra

Let $A$ be a real $n\times n$ matrix such that $A^3=0$ a) prove that there is unique real $n\times n$ matrix $X$ that satisfied the equation $X+AX+XA^2=A$ b) Express $X$ in terms of $A$

2020 Malaysia IMONST 1, 9

Tags: number theory , digit

What is the smallest positive multiple of $225$ that can be written using digits $0$ and $1$ only?