Olympiad problems

2023 Switzerland Team Selection Test, 4

Tags: geometry

Let $ABC$ and $AMN$ be two similar, non-overlapping triangles with the same orientation, such that $AB=AC$ and $AM=AN$. Let $O$ be the circumcentre of the triangle $MAB$. Prove that the points $O, C, N$ and $A$ lie on a circle if and only if the triangle $ABC$ is equilateral.

2014 India IMO Training Camp, 2

Tags: inequalities

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

2010 Brazil National Olympiad, 3

Tags: modular arithmetic , greatest common divisor , search , number theory

Find all pairs $(a, b)$ of positive integers such that \[ 3^a = 2b^2 + 1. \]

2012 China Western Mathematical Olympiad, 4

Tags: inequalities , circumcircle , geometry

$P$ is a point in the $\Delta ABC$, $\omega $ is the circumcircle of $\Delta ABC $. $BP \cap \omega = \left\{ {B,{B_1}} \right\}$,$CP \cap \omega = \left\{ {C,{C_1}} \right\}$, $PE \bot AC$，$PF \bot AB$. The radius of the inscribed circle and circumcircle of $\Delta ABC $ is $r,R$. Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$.

1975 IMO, 1

Tags: rearrangement inequality , inequality , permutation , algebra

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

2000 All-Russian Olympiad, 1

Tags: quadratic , algebra

Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root. Compute the sum $a+b+c$.

2008 Princeton University Math Competition, B1

Tags: algebra

Solve for $x$: $x = 2 + \frac{4(2^6)}{11-3}$

2024 Moldova EGMO TST, 5

Tags: geometry , angle bisector , circumcircle , geometric inequality , inequalities

$AD$ Is the angle bisector Of $\angle BAC$ Where $D$ lies on the The circumcircle of $\triangle ABC$. Show that $2AD>AB+AC$

1969 IMO Longlists, 65

Tags: algebra , algebraic identities

$(USS 2)$ Prove that for $a > b^2,$ the identity ${\sqrt{a-b\sqrt{a+b\sqrt{a-b\sqrt{a+\cdots}}}}=\sqrt{a-\frac{3}{4}b^2}-\frac{1}{2}b}$

2007 AIME Problems, 8

Tags: polynomial , quadratic , number theory , greatest common divisor , algorithm , least common multiple , algebra

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?

2021 Switzerland - Final Round, 4

Tags: algebra , inequality , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2011 QEDMO 9th, 2

Tags: polynomial , algebra , cubic

Let $a,b,c$ be the three different solutions of $x^3-x-1 = 0$. Compute $a^4+b^5+c^6-c$.

2017 MIG, 5

Tags:

Based on the fact that only $10\%$ of the total volume of an iceberg can be seen above the surface of the water, if $500\text{ m}^3$ of a certain iceberg is seen, then what is the total volume of the iceberg? $\textbf{(A) } 5\qquad\textbf{(B) } 50\qquad\textbf{(C) } 500\qquad\textbf{(D) } 5{,}000\qquad\textbf{(E) } \text{imposssible to determined}$

2019 AMC 12/AHSME, 24

Tags: complex plane , area , geometry

Let $\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\omega+c\omega^2,$ where $0\leq a \leq 1,0\leq b\leq 1,$ and $0\leq c\leq 1.$ What is the area of $S$? $\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi$

2010 Abels Math Contest (Norwegian MO) Final, 2b

Tags: inequalities , algebra

Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: divisor , number theory

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

2015 BMT Spring, 13

Tags: combinatorics , geometry

There exist right triangles with integer side lengths such that the legs differ by $ 1$. For example, $3-4-5$ and $20-21-29$ are two such right triangles. What is the perimeter of the next smallest Pythagorean right triangle with legs differing by $ 1$?

2014 CentroAmerican, 1

Tags: combinatorics

Using squares of side 1, a stair-like figure is formed in stages following the pattern of the drawing. For example, the first stage uses 1 square, the second uses 5, etc. Determine the last stage for which the corresponding figure uses less than 2014 squares. [img]http://www.artofproblemsolving.com/Forum/download/file.php?id=49934[/img]

2005 Postal Coaching, 19

Tags: function , algebra , polynomial

Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.

2024 China National Olympiad, 2

Tags: inequalities , algebra

Find the largest real number $c$ such that $$\sum_{i=1}^{n}\sum_{j=1}^{n}(n-|i-j|)x_ix_j \geq c\sum_{j=1}^{n}x^2_i$$ for any positive integer $n $ and any real numbers $x_1,x_2,\dots,x_n.$

2010 Brazil Team Selection Test, 3

Tags: cyclic quadrilateral , projective geometry , tangent , power of a point , geometry

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

2015 Singapore MO Open, 4

Tags: number theory , fibonacci sequence

Let $f_0, f_1,...$ be the Fibonacci sequence: $f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2}$ if $n \geq 2$. Determine all possible positive integers $n$ so that there is a positive integer $a$ such that $f_n \leq a \leq f_{n+1}$ and that $a( \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n} )$ is an integer.

2012 China Team Selection Test, 1

Tags: inequalities , complex number , triangle inequality , algebra

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$. Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$, $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$. Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$.

1984 Canada National Olympiad, 4

Tags: trigonometry , geometry

An acute triangle has unit area. Show that there is a point inside the triangle whose distance from each of the vertices is at least $\frac{2}{\sqrt[4]{27}}$.

2011 Argentina National Olympiad Level 2, 4

Tags: combinatorics

Each face of a regular tetrahedron with edge length $2011$ is divided into $2011^2$ equilateral triangles of side length $1$, created by drawing lines parallel to each edge. Bruno and Mariano take turns marking one of the unit triangles. Except for the first move, every triangle marked must share at least one point with the triangle marked in the previous move. Bruno plays first. The game ends when a player cannot make a move, and that player loses. Determine which of the two players has a winning strategy and describe the strategy.