Olympiad problems

2020 Lusophon Mathematical Olympiad, 2

a) Find a pair(s) of integers $(x,y)$ such that: $y^2=x^3+2017$ b) Prove that there isn't integers $x$ and $y$, with $y$ not divisible by $3$, such that: $y^2=x^3-2017$

2012 Balkan MO Shortlist, A6

Tags: algebra , function , domain , inequalities , inequalities proposed , Balkan

Let $k$ be a positive integer. Find the maximum value of \[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\] where $a$, $b$, $c$ are non-negative reals such that $a+b+c=3k$.

2022 Saudi Arabia BMO + EGMO TST, 2.1

Tags: recurrence relation , Recurrence , number theory

Define $a_0 = 2$ and $a_{n+1} = a^2_n + a_n -1$ for $n \ge 0$. Prove that $a_n$ is coprime to $2n + 1$ for all $n \in N$.

2024 Azerbaijan JBMO TST, 1

Tags: JBMO Shortlist , number theory , AZE JBMO TST

Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

2023 VN Math Olympiad For High School Students, Problem 2

Tags: algebra

a) Given a prime number $p$ and $2$ polynomials$$P(x)=a_nx^n+...+a_1x+a_0; Q(x)=b_mx^m+...+b_1x+b_0.$$ We know that the product $P(x)Q(x)$ is a polynomial whose coefficents are all divisible by $p.$ Prove that: at least $1$ in $2$ polynomials $P(x),Q(x)$ has all coefficents are all divisible by $p.$ b) Prove that the product of $2$ original polynomials is a original polynomial.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , Divisibility , FBH , Perfect Square

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

2014 Contests, 3

Tags: number theory , Divisibility

Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 | (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$.

2023 Estonia Team Selection Test, 4

Tags: geometry

A convex quadrilateral $ABCD$ has $\angle BAC = \angle ADC$. Let $M{}$ be the midpoint of the diagonal $AC$. The side $AD$ contains a point $E$ such that $ABME$ is a parallelogram. Let $N{}$ be the midpoint of the line segment $AE{}$. Prove that the line $AC$ touches the circumcircle of the triangle $DMN$ at point $M{}$.

1995 China Team Selection Test, 1

Tags: combinatorics unsolved , combinatorics

Let $S = \lbrace A = (a_1, \ldots, a_s) \mid a_i = 0$ or $1, i = 1, \ldots, 8 \rbrace$. For any 2 elements of $S$, $A = \lbrace a_1, \ldots, a_8\rbrace$ and $B = \lbrace b_1, \ldots, b_8\rbrace$. Let $d(A,B) = \sum_{i=1}{8} |a_i - b_i|$. Call $d(A,B)$ the distance between $A$ and $B$. At most how many elements can $S$ have such that the distance between any 2 sets is at least 5?

2006 IMO Shortlist, 8

Tags: inequalities , geometry , circumcircle , triangle inequality , IMO Shortlist

Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that \[\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ.\] Prove that $AB+CD \geq BC+AD$. [i]Proposed by Waldemar Pompe, Poland[/i]

1980 IMO, 5

Tags: geometry , concurrency , IMO Shortlist

Let $A_1A_2A_3$ be a triangle and, for $1 \leq i \leq 3$, let $B_i$ be an interior point of edge opposite $A_i$. Prove that the perpendicular bisectors of $A_iB_i$ for $1 \leq i \leq 3$ are not concurrent.

2022 Moldova EGMO TST, 4

Tags: algebra , polynomial , abstract algebra , modular arithmetic

Prove that there exists an integer polynomial $P(X)$ such that $P(n)+4^n \equiv 0 \pmod {27}$. for all $n \geq 0$.

2023 Bangladesh Mathematical Olympiad, P6

Tags: geometry , cyclic quadrilateral , Angle Chasing

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.

MOAA Team Rounds, 2019.7

Tags: number theory , team , 2019

Suppose $ABC$ is a triangle inscribed in circle $\omega$ . Let $A'$ be the point on $\omega$ so that $AA'$ is a diameter, and let $G$ be the centroid of $ABC$. Given that $AB = 13$, $BC = 14$, and $CA = 15$, let $x$ be the area of triangle $AGA'$ . If $x$ can be expressed in the form $m/n$ , where m and n are relatively prime positive integers, compute $100n + m$.

2013 IMC, 1

Tags: inequalities , trigonometry , quadratics , function , complex analysis , IMC , college contests

Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$. [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

2019 IMO Shortlist, A2

Tags: algebra , IMO Shortlist , IMO Shortlist 2019 , inequalities

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

2002 USAMO, 4

Tags: functional equation , USAMO

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 - y^2) = x f(x) - y f(y) \] for all pairs of real numbers $x$ and $y$.

1949 Moscow Mathematical Olympiad, 163

Tags: hexagon , geometry , parallel

Prove that if opposite sides of a hexagon are parallel and the diagonals connecting opposite vertices have equal lengths, a circle can be circumscribed around the hexagon.

1988 All Soviet Union Mathematical Olympiad, 486

Tags: sphere , tetrahedron , radical , geometric inequality , 3D geometry , geometry

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

2023 Princeton University Math Competition, A8

Tags: geometry

Let $\vartriangle ABC$ be a triangle with $AB = 4$ and $AC = \frac72$ . Let $\omega$ denote the $A$-excircle of $\vartriangle ABC$. Let $\omega$ touch lines $AB$, $AC$ at the points $D$, $E$, respectively. Let $\Omega$ denote the circumcircle of $\vartriangle ADE$. Consider the line $\ell$ parallel to $BC$ such that $\ell$ is tangent to $\omega$ at a point $F$ and such that $\ell$ does not intersect $\Omega$. Let $\ell$ intersect lines $AB$, $AC$ at the points $X$, $Y$ , respectively, with $XY = 18$ and $AX = 16$. Let the perpendicular bisector of $XY$ meet the circumcircle of $\vartriangle AXY$ at $P$, $Q$, where the distance from $P$ to $F$ is smaller than the distance from $Q$ to$ F$. Let ray $\overrightarrow {PF}$ meet $\Omega$ for the first time at the point $Z$. If $PZ^2 = \frac{m}{n}$ for relatively prime positive integers $m$, $n$, find $m + n$.

2008 Czech and Slovak Olympiad III A, 3

Tags: number theory , greatest common divisor , number theory proposed

Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.

2016 Turkey EGMO TST, 1

Tags: inequalities

Prove that \[ x^4y+y^4z+z^4x+xyz(x^3+y^3+z^3) \geq (x+y+z)(3xyz-1) \] for all positive real numbers $x, y, z$.

2016 AMC 8, 1

Tags: 2016 AMC 8 , AMC 8

The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this? $\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$

1993 Baltic Way, 13

Tags: combinatorics proposed , combinatorics

An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of small triangles that can be chosen so that no two of them lie on a line that is parallel to any of the sides of the triangle $ABC$?

1998 Yugoslav Team Selection Test, Problem 3

Tags: algebra , Sequence

Prove that there are no positive integers $n$ and $k\le n$ such that the numbers $$\binom nk,\binom n{k+1},\binom n{k+2},\binom n{k+3}$$in this order form an arithmetic progression.