Olympiad problems

2014 Iran MO (3rd Round), 5

Tags: circumcircle , geometry

$X$ and $Y$ are two points lying on or on the extensions of side $BC$ of $\triangle{ABC}$ such that $\widehat{XAY} = 90$. Let $H$ be the orthocenter of $\triangle{ABC}$. Take $X'$ and $Y'$ as the intersection points of $(BH,AX)$ and $(CH,AY)$ respectively. Prove that circumcircle of $\triangle{CYY'}$,circumcircle of $\triangle{BXX'}$ and $X'Y'$ are concurrent.

2011 Bogdan Stan, 3

Tags: inequalities , cauchy schwarz

Prove that $$ a+b+c>\left( \sqrt\alpha +\sqrt\beta +\sqrt\gamma \right)^2, $$ for all positive real numbers $ a,b,c,\alpha ,\beta ,\gamma $ that are under the condition $$ abc>\alpha bc+\beta ac+\gamma ab. $$ [i]Țuțescu Lucian[/i] and [i]Chiriță Aurel[/i]

1994 Hong Kong TST, 3

Tags: number theory

Find all non-negative integers $x, y$ and $z$ satisfying the equation: \[7^{x}+1=3^{y}+5^z\]

2023 CUBRMC, 2

Tags: geometry

The concave decagon shown below is embedded in the Cartesian coordinate plane such that all of its vertices have integer coordinates. Two opposite edges have length $5$, whereas the remaining eight edges have length $\sqrt{10}$. Every pair of opposite edges is parallel. The sides of the decagon do not intersect each other, and the decagon has vertical and horizontal axes of symmetry. Find the area of the decagon. [img]https://cdn.artofproblemsolving.com/attachments/1/5/daa4ab3d71af4b3274cd222f9a091eea3be705.png[/img]

1953 AMC 12/AHSME, 39

Tags: logarithm

The product, $ \log_a b \cdot \log_b a$ is equal to: $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ a \qquad\textbf{(C)}\ b \qquad\textbf{(D)}\ ab \qquad\textbf{(E)}\ \text{none of these}$

2003 Singapore Senior Math Olympiad, 2

Tags: algebra , polynomial , binomial , combination

For each positive integer $k$, we define the polynomial $S_k(x)=1+x+x^2+x^3+...+x^{k-1}$ Show that $n \choose 1$ $S_1(x) +$ $n \choose 2$ $S_2(x) +$ $n \choose 3$ $S_3(x)+...+$ $n \choose n$ $S_n(x) = 2^{n-1}S_n\left(\frac{1+x}{2}\right)$ for every positive integer $n$ and every real number $x$.

2012 India IMO Training Camp, 3

Tags: arithmetic sequence , combinatorics , additive combinatorics

Determine the greatest positive integer $k$ that satisfies the following property: The set of positive integers can be partitioned into $k$ subsets $A_1, A_2, \ldots, A_k$ such that for all integers $n \geq 15$ and all $i \in \{1, 2, \ldots, k\}$ there exist two distinct elements of $A_i$ whose sum is $n.$ [i]Proposed by Igor Voronovich, Belarus[/i]

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: combinatorics , combinatorial geometry , equilateral

A given equilateral triangle of side $10$ is divided into $100$ equilateral triangles of side $1$ by drawing parallel lines to the sides of the original triangle. Find the number of equilateral triangles, having vertices in the intersection points of parallel lines whose sides lie on the parallel lines.

2025 Israel TST, P1

Tags: function , algebra

Let $\mathcal{F}$ be a family of functions from $\mathbb{R}^+ \to \mathbb{R}^+$. It is known that for all $ f, g \in \mathcal{F} $, there exists $ h \in \mathcal{F} $ such that for all $ x, y \in \mathbb{R}^+ $, the following equation holds: \[ y^2 \cdot f\left(\frac{g(x)}{y}\right) = h(xy) \] Prove that for all $ f \in \mathcal{F} $ and all $ x \in \mathbb{R}^+ $, the following identity is satisfied: \[ f\left(\frac{x}{f(x)}\right) = 1. \]

2016 HMNT, 4

Tags: hmmt

A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

2009 Indonesia TST, 3

Tags: function , algebra

Find all function $ f: \mathbb{R} \rightarrow \mathbb{R}$ such that \[ f(x \plus{} y)(f(x) \minus{} y) \equal{} xf(x) \minus{} yf(y) \] for all $ x,y \in \mathbb{R}$.

2010 Contests, 1

Tags: geometry

A square with side length $2$ cm is placed next to a square with side length $6$ cm, as shown in the diagram. Find the shaded area, in cm$^2$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/ceb4912a6e73ca751113b2b5c92cbfdbb6e0d1.png[/img]

2025 AIME, 4

Tags: algebra

Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$ inclusive, such that $12x^2-xy-6y^2=0$.

2021 Princeton University Math Competition, A7

Tags: geometry

Let $ABC$ be a triangle with side lengths $AB = 13$, $AC = 17$, and $BC = 20$. Let $E, F$ be the feet of the altitudes from $B$ onto $AC$ and $C$ onto $AB$, respectively. Let $P$ be the second intersection of the circumcircles of $ABC$ and $AEF$. Suppose that $AP$ can be written as $\frac{a \sqrt{b}}{c}$ where $a, c$ are relatively prime and $b$ is square-free. Compute $a$.

1999 Romania Team Selection Test, 2

Tags: incenter , geometry

Let $ABC$ be an acute triangle. The interior angle bisectors of $\angle ABC$ and $\angle ACB$ meet the opposite sides in $L$ and $M$ respectively. Prove that there is a point $K$ in the interior of the side $BC$ such that the triangle $KLM$ is equilateral if and only if $\angle BAC = 60^\circ$.

2011 Tournament of Towns, 1

Tags: divisor , number theory

An integer $N > 1$ is written on the board. Alex writes a sequence of positive integers, obtaining new integers in the following manner: he takes any divisor greater than $1$ of the last number and either adds it to, or subtracts it from the number itself. Is it always (for all $N > 1$) possible for Alex to write the number $2011$ at some point?

2014 ELMO Shortlist, 2

Tags: inequalities

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

2021 Regional Olympiad of Mexico West, 3

Tags: number theory , sequence , recurrence relation

The sequence of real numbers $a_1, a_2, a_3, ...$ is defined as follows: $a_1 = 2019$, $a_2 = 2020$, $a_3 = 2021$ and for all $n \ge 1$ $$a_{n+3} = 5a^6_{n+2} + 3a^3_{n+1} + a^2_n.$$ Show that this sequence does not contain numbers of the form $m^6$ where $m$ is a positive integer.

2019 Math Prize for Girls Problems, 13

Tags:

Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all four triangles.

2003 Chile National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Find all primes $p, q$ such that $p + q = (p-q)^3$.

2012 Iran MO (3rd Round), 2

Tags: floor function , number theory

Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition: \[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\] [i]Proposed by Mohammad Gharakhani[/i]

2012 National Olympiad First Round, 2

Tags:

Find the sum of distinct residues of the number $2012^n+m^2$ on $\mod 11$ where $m$ and $n$ are positive integers. $ \textbf{(A)}\ 55 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 43 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 37$

Kvant 2020, M2631

Tags: geometry

There is a convex quadrangle $ABCD$ such that no three of its sides can form a triangle. Prove that: [list=a] [*]one of its angles is not greater than $60^\circ{}$; [*]one of its angles is at least $120^\circ$. [/list] [i]Maxim Didin[/i]

1996 Korea National Olympiad, 7

Tags: algebra

Let $A_n$ be the set of real numbers such that each element of $A_n$ can be expressed as $1+\frac{a_1}{\sqrt{2}}+\frac{a_2}{(\sqrt{2})^2}+\cdots +\frac{a_n}{(\sqrt{n})^n}$ for given $n.$ Find both $|A_n|$ and sum of the products of two distinct elements of $A_n$ where each $a_i$ is either $1$ or $-1.$

2021 Final Mathematical Cup, 2

Tags: geometry , tangent

The altitudes $BB_1$ and $CC_1$, are drawn in an acute triangle $ABC$. Let $X$ and $Y$ be the points, which are symmetrical to the points $B_1$ and $C_1$, with respect to the midpoints of the sides$ AB$ and $AC$ of the triangle $ABC$ respectively. Let's denote with $Z$ the point of intersection of the lines $BC$ and $XY$. Prove that the line $ZA$ is tangent to the circumscribed circle of the triangle $AXY$ .