Olympiad problems

PEN H Problems, 85

Find all integer solutions to $2(x^5 +y^5 +1)=5xy(x^2 +y^2 +1)$.

2012 China Team Selection Test, 1

Tags: number theory , induction , inequalities

Given an integer $n\ge 2$. Prove that there only exist a finite number of n-tuples of positive integers $(a_1,a_2,\ldots,a_n)$ which simultaneously satisfy the following three conditions: [list] [*] $a_1>a_2>\ldots>a_n$; [*] $\gcd (a_1,a_2,\ldots,a_n)=1$; [*] $a_1=\sum_{i=1}^{n}\gcd (a_i,a_{i+1})$,where $a_{n+1}=a_1$.[/list]

2020 Israel National Olympiad, 7

Tags: inequalities , geometry , circumcircle

Let $P$ be a point inside a triangle $ABC$, $d_a$, $d_b$ and $d_c$ be distances from $P$ to the lines $BC$, $AC$ and $AB$ respectively, $R$ be a radius of the circumcircle and $r$ be a radius of the inscribed circle for $\Delta ABC.$ Prove that: $$\sqrt{d_a}+\sqrt{d_b}+\sqrt{d_c}\leq\sqrt{2R+5r}.$$

2010 Contests, 4

Tags: algebra , function , polynomial , number theory , prime number , inequalities

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

2013 China Western Mathematical Olympiad, 2

Tags: inequalities , induction

Let the integer $n \ge 2$, and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right] $.Prove that\[\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.\]

2011 Serbia National Math Olympiad, 3

Tags: inequalities , linear algebra , function , combinatorics , matrix

Set $T$ consists of $66$ points in plane, and $P$ consists of $16$ lines in plane. Pair $(A,l)$ is [i]good[/i] if $A \in T$, $l \in P$ and $A \in l$. Prove that maximum number of good pairs is no greater than $159$, and prove that there exits configuration with exactly $159$ good pairs.

1993 IMO Shortlist, 8

Tags: inequalities , ratio , trigonometry , geometric inequality , circumcircle , geometry

The vertices $D,E,F$ of an equilateral triangle lie on the sides $BC,CA,AB$ respectively of a triangle $ABC.$ If $a,b,c$ are the respective lengths of these sides, and $S$ the area of $ABC,$ prove that \[ DE \geq \frac{2 \cdot \sqrt{2} \cdot S}{\sqrt{a^2 + b^2 + c^2 + 4 \cdot \sqrt{3} \cdot S}}. \]

2003 Germany Team Selection Test, 3

Tags: floor function , least common multiple , number theory , inequalities , ceiling function

Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$

2004 Romania Team Selection Test, 12

Tags: inequalities , induction , n-variable inequality

Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have \[ \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 . \] [i]Gabriel Dospinescu[/i]

2017 China Girls Math Olympiad, 5

Tags: inequalities , n-variable inequality

Let $0=x_0<x_1<\cdots<x_n=1$ .Find the largest real number$ C$ such that for any positive integer $ n $ , we have $$\sum_{k=1}^n x^2_k (x_k - x_{k-1})>C$$

2002 National Olympiad First Round, 5

Tags: inequalities , triangle inequality , geometry

The lengths of two altitudes of a triangles are $8$ and $12$. Which of the following cannot be the third altitude? $ \textbf{a)}\ 4 \qquad\textbf{b)}\ 7 \qquad\textbf{c)}\ 8 \qquad\textbf{d)}\ 12 \qquad\textbf{e)}\ 23 $

1987 AIME Problems, 4

Tags: analytic geometry , geometry , inequalities

Find the area of the region enclosed by the graph of $|x-60|+|y|=|x/4|.$

2013 ELMO Shortlist, 3

Tags: induction , inequalities , number theory

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2000 Romania National Olympiad, 2b

Tags: geometry , inequalities , geometric inequality

If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

2014 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: algebra , inequalities , inequality

Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$

2021 China Team Selection Test, 5

Tags: combinatorics , algebra , inequalities

Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.

VMEO II 2005, 12

Tags: inequalities , algebra

a) Find all real numbers $k$ such that there exists a positive constant $c_k$ satisfying $$(x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k$$ for all positive real numbers. b) Given the numbers $k$ found, determine the largest number $c_k$.

2014 Turkey EGMO TST, 4

Tags: inequalities

Let $x,y,z$ be positive real numbers such that $x+y+z \ge x^2+y^2+z^2$. Show that; $$\dfrac{x^2+3}{x^3+1}+\dfrac{y^2+3}{y^3+1}+\dfrac{z^2+3}{z^3+1}\ge6$$

2018 Balkan MO Shortlist, G3

Tags: geometry , inequalities , geometric inequality , semiperimeter , maximum value , minimum

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

2006 Switzerland Team Selection Test, 1

Tags: inequalities

Let $a,b,c \in \mathbb{R^+}$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show $\sqrt{ab+c} + \sqrt{bc+a} + \sqrt{ca+b} \ge \sqrt{abc} + \sqrt{a} + \sqrt{b} + \sqrt{c}$. :D

2012 Turkmenistan National Math Olympiad, 3

Tags: logarithm , inequalities

Prove that : $\frac{1}{(\log_{bc} a)^n}+\frac{1}{(\log_{ac} b)^n}+\frac{1}{(\log_{bc} a)^n}\geq 3\cdot2^{n}$ where $a,b,c>1$ and $n$ is natural number.

1976 AMC 12/AHSME, 23

Tags: binomial coefficient , inequalities

For integers $k$ and $n$ such that $1\le k<n$, let $C^n_k=\frac{n!}{k!(n-k)!}$. Then $\left(\frac{n-2k-1}{k+1}\right)C^n_k$ is an integer $\textbf{(A) }\text{for all }k\text{ and }n\qquad$ $\textbf{(B) }\text{for all even values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(C) }\text{for all odd values of }k\text{ and }n,\text{ but not for all }k\text{ and }n\qquad$ $\textbf{(D) }\text{if }k=1\text{ or }n-1,\text{ but not for all odd values }k\text{ and }n\qquad$ $\textbf{(E) }\text{if }n\text{ is divisible by }k,\text{ but not for all even values }k\text{ and }n$

2006 Austrian-Polish Competition, 5

Tags: inequalities

Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: \[\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}\]

2004 Putnam, B2

Tags: function , logarithm , probability , inequalities , combinatorics

Let $m$ and $n$ be positive integers. Show that $\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$

2005 Taiwan TST Round 1, 1

Tags: algebra , function , triangle inequality , inequalities , polynomial , quadratic , complex number

Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$ My solution was nearly complete...