Olympiad problems

2001 All-Russian Olympiad Regional Round, 10.1

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 $KJ=IL$

1988 AMC 12/AHSME, 28

Tags: probability

An unfair coin has probability $p$ of coming up heads on a single toss. Let $w$ be the probability that, in $5$ independent toss of this coin, heads come up exactly $3$ times. If $w = 144 / 625$, then $ \textbf{(A)}\ p\text{ must be }2/5$ $ \textbf{(B)}\ p\text{ must be }3/5$ $ \textbf{(C)}\ p\text{ must be greater than }3/5$ $ \textbf{(D)}\ p\text{ is not uniquely determined}$ $ \textbf{(E)}\ \text{there is no value of }p\text{ for which }w = 144/625$

2009 Turkey Junior National Olympiad, 2

Tags:

In the beginnig, each square of a strip formed by $n$ adjacent squares contains $0$ or $1$. At each step, we are writing $1$ to the squares containing $0$ and to the squares having exactly one neighbour containing $1$, and we are writing $0$s into the other squares. Determine all possible values of $n$ such that whatever the initial arrangement of $0$ and $1$ is, after finite number of steps, all squares can turn into $0$.

2018 Romanian Master of Mathematics, 4

Tags: number theory

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

2012 Indonesia TST, 3

Tags: angle chasing , complex number , cyclic quadrilateral , geometry

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

2025 Bulgarian Winter Tournament, 12.2

Tags: geometry , tangency , fixed point

In the plane are fixed two internally tangent circles $\omega$ and $\Omega$, so that $\omega$ is inside $\Omega$. Denote their common point by $T$. The point $A \neq T$ moves on $\Omega$ and point $B$ on $\Omega$ is such that $AB$ is tangent to $\omega$. The line through $B$, perpendicular to $AB$, meets the external angle bisector of $\angle ATB$ at $P$. Prove that, as $A$ varies on $\Omega$, the line $AP$ passes through a fixed point.

2017 India IMO Training Camp, 1

Tags: inequalities

Let $a,b,c$ be distinct positive real numbers with $abc=1$. Prove that $$\sum_{\text{cyc}} \frac{a^6}{(a-b)(a-c)}>15.$$

2017 NIMO Summer Contest, 11

Tags: geometry

Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If \[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\] then compute $\lfloor a^r b^p c^q \rfloor$. [i]Proposed by Michael Tang[/i]