Olympiad problems

1975 Dutch Mathematical Olympiad, 4

Given is a rectangular plane coordinate system. (a) Prove that it is impossible to find an equilateral triangle whose vertices have integer coordinates. (b) In the plane the vertices $A, B$ and $C$ lie with integer coordinates in such a way that $AB = AC$. Prove that $\frac{d(A,BC)}{BC}$ is rational.

2008 Balkan MO, 4

Tags: inequalities , polynomial , modular arithmetic , induction , number theory , relatively prime , algebra

Let $ c$ be a positive integer. The sequence $ a_1,a_2,\ldots$ is defined as follows $ a_1\equal{}c$, $ a_{n\plus{}1}\equal{}a_n^2\plus{}a_n\plus{}c^3$ for all positive integers $ n$. Find all $ c$ so that there are integers $ k\ge1$ and $ m\ge2$ so that $ a_k^2\plus{}c^3$ is the $ m$th power of some integer.

2011 Morocco National Olympiad, 3

Tags: linear algebra , matrix , algebra

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

2000 Mexico National Olympiad, 2

Tags: pascal triangle , number theory

A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row? 1 2 3 4 5 3 5 7 9 8 12 16 20 28 4

2024 LMT Fall, 6

Tags: guts

Let $P$ be a point in rectangle $ABCD$ such that the area of $PAB$ is $20$ and the area of $PCD$ is $24$. Find the area of $ABCD$.

2020 LIMIT Category 2, 13

Tags: sum of digits , limit

For every $n \in N $, let $d(n)$ denote the sum of digits of $n$. It is easy to see that the sequence $d(n), d(d(n))$, $d(d(d(n))), ... $ will eventually become a constant integer between $1$ and $9$ (both inclusive). This number is called the digital root of $n$ . Denote it by $b(n)$. Then for how many natural numbers $k<1000 , \lim_{n \to \infty} b(k^n)$ exists.

2014 Turkey Junior National Olympiad, 4

Tags: circumcircle , trapezoid , geometry

$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

2001 All-Russian Olympiad Regional Round, 10.1

Tags: algebra , geometry , combinatorial geometry

The lengths of the sides of the polygon are $a_1$, $a_2$,. $..$ ,$a_n$. The square trinomial $f(x)$ is such that $f(a_1) = f(a_2 +...+ a_n)$. Prove that if $A$ is the sum of the lengths of several sides of a polygon, $B$ is the sum of the lengths of its remaining sides, then $f(A) = f(B)$.

2011 Lusophon Mathematical Olympiad, 2

Tags: geometry , rectangle , symmetry , geometric transformation , parallelogram , reflection , rotation

Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.

2020 New Zealand MO, 2

Tags: midpoint , square , geometry

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.

2006 Abels Math Contest (Norwegian MO), 3

Tags: integer , number theory , diophantine , rational

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

1991 Austrian-Polish Competition, 3

Tags: midpoint , equal segments , combinatorial geometry , combinatorics , geometry

Given two distinct points $A_1,A_2$ in the plane, determine all possible positions of a point $A_3$ with the following property: There exists an array of (not necessarily distinct) points $P_1,P_2,...,P_n$ for some $n \ge 3$ such that the segments $P_1P_2,P_2P_3,...,P_nP_1$ have equal lengths and their midpoints are $A_1, A_2, A_3, A_1, A_2, A_3, ...$ in this order.

Gheorghe Țițeica 2024, P3

Tags: olympiad number theory

We know there is some positive integer $k$ such that $\overline{3a\dots a20943}$ is prime (where $a$ appears $k$ times). Find the digit $a$. [i]Dorel Miheț[/i]

2024 CMIMC Team, 1

Tags: team

Solve for $x$ if $\sqrt{x + 1}+ \sqrt{x} = 5.$ [i]Proposed by Eric Oh[/i]

2023 Israel Olympic Revenge, P3

Tags: functional equation , algebra , function

Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which \[f(f(x)+y)=f(f(y)+x)\] holds for all $x, y\in \mathbb{R}$.

2003 Argentina National Olympiad, 4

Tags: geometry , trapezoid , right triangle , circumcenter , area of a triangle

The trapezoid $ABCD$ of bases $AB$ and $CD$, has $\angle A = 90^o, AB = 6, CD = 3$ and $AD = 4$. Let $E, G, H$ be the circumcenters of triangles $ABC, ACD, ABD$, respectively. Find the area of the triangle $EGH$.

1997 Denmark MO - Mohr Contest, 4

Tags: number theory , diophantine , diophantine equation

Find all pairs $x,y$ of natural numbers that satisfy the equation $$x^2-xy+2x-3y=1997$$

2010 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , algebra , polynomial , function

Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.

2000 Romania National Olympiad, 4

Tags: geometry , square , geometric inequality

In the square $ABCD$ we consider $ E \in (AB)$, $ F \in (AD)$ and $EF \cap AC = \{P\}$. Show that: a) $\frac{1}{AE} + \frac{1}{AF} = \frac{\sqrt2}{AP}$ b) $AP^2 \le \frac{AE \cdot AF}{2}$

2010 Junior Balkan Team Selection Tests - Romania, 4

Tags: circumcircle , parallelogram , radical axis , geometry

Let a triangle $ABC$ , $O$ it's circumcenter , $H$ ortocenter and $M$ the midpoint of $AH$. The perpendicular at $M$ to line $OM$ meets $AB$ and $AC$ at points $P$, respective $Q$. Prove that $MP=MQ$. Babis

2002 China Team Selection Test, 1

Tags: inequalities

Given $ n \geq 3$, $ n$ is a integer. Prove that: \[ (2^n \minus{} 2) \cdot \sqrt{2i\minus{}1} \geq \left( \sum_{j\equal{}0}^{i\minus{}1}C_n^j \plus{} C_{n\minus{}1}^{i\minus{}1} \right) \cdot \sqrt{n}\] where if $ n$ is even, then $ \displaystyle 1 \leq i \leq \frac{n}{2}$; if $ n$ is odd, then $ \displaystyle 1 \leq i \leq \frac{n\minus{}1}{2}$.

2022 Kyiv City MO Round 2, Problem 3

Tags: permutation , number theory , algebra

Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$. [i](Proposed by Oleksii Masalitin)[/i]

Russian TST 2018, P4

Tags: algebra , inequalities

Let $a_1,\ldots,a_{n+1}$ be positive real numbers satisfying $1/(a_1+1)+\cdots+1/(a_{n+1}+1)=n$. Prove that \[\sum_{i=1}^{n+1}\prod_{j\neq i}\sqrt[n]{a_j}\leqslant\frac{n+1}{n}.\]

2009 Argentina Iberoamerican TST, 1

Tags: number theory

Find all positive integers $ (x,y)$ such that $ \frac{y^2x}{x\plus{}y}$ is a prime number

2005 Singapore Senior Math Olympiad, 2

Tags: geometry , equal segments

Consider the nonconvex quadrilateral $ABCD$ with $\angle C>180$ degrees. Let the side $DC$ extended to meet $AB$ at $F$ and the side $BC$ extended to meet $AD$ at $E$. A line intersects the interiors of the sides $AB,AD,BC,CD$ at points $K,L,J,I$ respectively. Prove that if $DI=CF$ and $BJ=CE$, then $KJ=IL$