Olympiad problems

1969 Poland - Second Round, 1

Prove that if the real numbers $ a, b, c, d $ satisfy the equations $$ \; a^2 + b^2 = 1,\; c^2 + d^2 = 1, \; ac + bd = -\frac{1}{2},$$ then $$a^2 + ac + c^2 = b^2 + bd + d^2.$$

2018 Iran MO (3rd Round), 1

Tags: inequalities

For positive real numbers$a,b,c$such that $ab+ac+bc=1$ prove that: $\prod\limits_{cyc} (\sqrt{bc}+\frac{1}{2a+\sqrt{bc}}) \ge 8abc$

2004 Romania National Olympiad, 4

Tags: geometry

Let $\displaystyle \left( P_n \right)_{n \geq 1}$ be an infinite family of planes and $\displaystyle \left( X_n \right)_{n \geq 1}$ be a family of non-void, finite sets of points such that $\displaystyle X_n \subset P_n$ and the projection of the set $\displaystyle X_{n+1}$ on the plane $\displaystyle P_n$ is included in the set $X_n$, for all $n$. Prove that there is a sequence of points $\displaystyle \left( p_n \right)_{n \geq 1}$ such that $\displaystyle p_n \in P_n$ and $p_n$ is the projection of $p_{n+1}$ on the plane $P_n$, for all $n$. Does the conclusion of the problem remain true if the sets $X_n$ are infinite? [i]Claudiu Raicu[/i]

2011 Indonesia TST, 1

Tags: inequalities , geometric inequalities , algebra

Let $a, b, c$ be the sides of a triangle with $abc = 1$. Prove that $$\frac{\sqrt{b + c -a}}{a}+\frac{\sqrt{c + a - b}}{b}+\frac{\sqrt{a + b - c}}{c} \ge a + b + c$$

2007 Cono Sur Olympiad, 1

Tags: number theory

Find all pairs $(x,y)$ of nonnegative integers that satisfy \[x^3y+x+y=xy+2xy^2.\]

2007 USAMO, 3

Tags: induction , combinatorics , set theory

Let $S$ be a set containing $n^{2}+n-1$ elements, for some positive integer $n$. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.

2019 Saudi Arabia JBMO TST, 1

Tags: algebra , sum

Real nonzero numbers $x, y, z$ are such that $x+y +z = 0$. Moreover, it is known that $$A =\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+ 1$$Determine $A$.

1994 Brazil National Olympiad, 1

Tags: geometry , 3d geometry , combinatorics

The edges of a cube are labeled from 1 to 12 in an arbitrary manner. Show that it is not possible to get the sum of the edges at each vertex the same. Show that we can get eight vertices with the same sum if one of the labels is changed to 13.

2011 Saudi Arabia Pre-TST, 2.4

Tags: similar triangles , geometry , median

Let $ABC$ be a triangle with medians $m_a$ , $m_b$, $m_c$. Prove that: (a) There is a triangle with side lengths $m_a$ ,$m_b$, $m_c$. (b) This triangle is similar to $ABC$ if and only if the squares of the side lengths of triangle $ABC$ form an arithmetical sequence.

1969 AMC 12/AHSME, 15

Tags: geometry

In a circle with center at $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$ a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is: $\textbf{(A) }\dfrac{3r^2}{16}\qquad \textbf{(B) }\dfrac{\pi r^2}{16}\qquad \textbf{(C) }\dfrac{\pi r^2\sqrt2}{8}\qquad \textbf{(D) }\dfrac{r^2\sqrt3}{32}\qquad \textbf{(E) }\dfrac{r^2\sqrt6}{48}$

2012 India PRMO, 10

Tags: geometry

$ABCD$ is a square and $AB = 1$. Equilateral triangles $AYB$ and $CXD$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $XY$?

2004 Turkey Junior National Olympiad, 1

Tags: geometry , incenter

Let $[AD]$ and $[CE]$ be internal angle bisectors of $\triangle ABC$ such that $D$ is on $[BC]$ and $E$ is on $[AB]$. Let $K$ and $M$ be the feet of perpendiculars from $B$ to the lines $AD$ and $CE$, respectively. If $|BK|=|BM|$, show that $\triangle ABC$ is isosceles.

2010 JBMO Shortlist, 1

Tags: polynomial , function , algebra

The real numbers $a$, $b$, $c$, $d$ satisfy simultaneously the equations \[abc -d = 1, \ \ \ bcd - a = 2, \ \ \ cda- b = 3, \ \ \ dab - c = -6.\] Prove that $a + b + c + d \not = 0$.

1994 IMC, 5

Tags: vector , induction

[b]problem 5.[/b] Let $x_1, x_2,\ldots, x_k$ be vectors of $m$-dimensional Euclidean space, such that $x_1+x_2+\ldots + x_k=0$. Show that there exists a permutation $\pi$ of the integers $\{ 1, 2, \ldots, k \}$ such that: $$\left\lVert \sum_{i=1}^n x_{\pi (i)}\right\rVert \leq \left( \sum_{i=1}^k \lVert x_i \rVert ^2\right)^{1/2}$$for each $n=1, 2, \ldots, k$. Note that $\lVert \cdot \rVert$ denotes the Euclidean norm. (18 points).

2018 Polish Junior MO First Round, 4

Tags: trapezoid , bisector , geometry

Let $ABCD$ be a trapezoid with bases $AB$ and $CD$. Bisectors of $AD$ and $BC$ intersect line segments $BC$ and $AD$ respectively in points $P$ and $Q$. Show that $\angle APD = \angle BQC$.

2011 AMC 10, 24

Tags: analytic geometry , calculus , integration , inequalities

A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

2007 Stanford Mathematics Tournament, 16

Tags: geometry , symmetry

Find the area of a square inscribed in an equilateral triangle, with one edge of the square on an edge of the triangle, if the side length of the triangle is $ 2\plus{}\sqrt{3}$.

2002 Brazil National Olympiad, 6

Tags: geometry , combinatorics

Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.

2017 CMIMC Geometry, 10

Tags: euler , geometry

Suppose $\triangle ABC$ is such that $AB=13$, $AC=15$, and $BC=14$. It is given that there exists a unique point $D$ on side $\overline{BC}$ such that the Euler lines of $\triangle ABD$ and $\triangle ACD$ are parallel. Determine the value of $\tfrac{BD}{CD}$. (The $\textit{Euler}$ line of a triangle $ABC$ is the line connecting the centroid, circumcenter, and orthocenter of $ABC$.)

2007 Germany Team Selection Test, 2

Tags: combinatorics , graph theory , extremal combinatorics , tournament graphs

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

2004 AIME Problems, 8

Tags:

How many positive integer divisors of $2004^{2004}$ are divisible by exactly $2004$ positive integers?

2019 China Girls Math Olympiad, 6

Tags: inequalities , algebra

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

1976 Canada National Olympiad, 1

Tags: ratio , geometric sequence , combinatorics

Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.

2012 Danube Mathematical Competition, 1

Tags: perfect square , number theory

a) Exist $a, b, c, \in N$, such that the numbers $ab+1,bc+1$ and $ca+1$ are simultaneously even perfect squares ? b) Show that there is an infinity of natural numbers (distinct two by two) $a, b, c$ and $d$, so that the numbers $ab+1,bc+1, cd+1$ and $da+1$ are simultaneously perfect squares.

1987 All Soviet Union Mathematical Olympiad, 456

Tags: combinatorics , algebra

Every evening uncle Chernomor (see Pushkin's tales) appoints either $9$ or $10$ of his 33 "knights" in the "night guard". When it can happen, for the first time, that every knight has been on duty the same number of times?