Olympiad problems

2003 China Team Selection Test, 1

$x$, $y$ and $z$ are positive reals such that $x+y+z=xyz$. Find the minimum value of: \[ x^7(yz-1)+y^7(zx-1)+z^7(xy-1) \]

2005 China Second Round Olympiad, 2

Tags: function , inequalities unsolved , inequalities , China , algebra , inequalities proposed , Inequality

Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]

2009 Croatia Team Selection Test, 1

Tags: inequalities , LaTeX , inequalities unsolved

Prove for all positive reals a,b,c,d: $ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$

2012 Indonesia TST, 3

Tags: inequalities unsolved , inequalities

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be positive reals such that \[a_1 + b_1 = a_2 + b_2 = \ldots + a_n + b_n\] and \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge n.\] Prove that \[\sqrt[n]{\dfrac{a_1a_2\ldots a_n}{b_1b_2\ldots b_n}} \ge \dfrac{a_1+a_2+\ldots+a_n}{b_1+b_2+\ldots+b_n}.\]

2002 China Team Selection Test, 3

Tags: inequalities , inequalities unsolved

$ n$ sets $ S_1$, $ S_2$ $ \cdots$, $ S_n$ consists of non-negative numbers. $ x_i$ is the sum of all elements of $ S_i$, prove that there is a natural number $ k$, $ 1<k<n$, and: \[ \sum_{i\equal{}1}^n x_i < \frac{1}{k\plus{}1} \left[ k \cdot \frac{n(n\plus{}1)(2n\plus{}1)}{6} \minus{} (k\plus{}1)^2 \cdot \frac{n(n\plus{}1)}{2} \right]\] and there exists subscripts $ i$, $ j$, $ t$, and $ l$ (at least $ 3$ of them are distinct) such that $ x_i \plus{} x_j \equal{} x_t \plus{} x_l$.

2005 MOP Homework, 3

Tags: inequalities , inequalities unsolved

Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}

2014 Vietnam National Olympiad, 2

Tags: inequalities , inequalities unsolved

Find the maximum of \[P=\frac{x^3y^4z^3}{(x^4+y^4)(xy+z^2)^3}+\frac{y^3z^4x^3}{(y^4+z^4)(yz+x^2)^3}+\frac{z^3x^4y^3}{(z^4+x^4)(zx+y^2)^3}\] where $x,y,z$ are positive real numbers.

1999 Croatia National Olympiad, Problem 2

Tags: inequalities , inequalities unsolved

How do I prove that, for every $a, b, c$ positive real numbers such that $a+b+c = 1$ the following inequality holds: $\frac{a^3}{a^2+b^2} +\frac{b^3}{b^2+c^2} +\frac {c^3}{c^2+a^2} \geq \frac{1}{2}$?

2006 Estonia Math Open Junior Contests, 10

Tags: inequalities , inequalities unsolved

Let a, b, c be positive integers. Prove that the inequality \[ (x\minus{}y)^a(x\minus{}z)^b(y\minus{}z)^c \ge 0 \] holds for all reals x, y, z if and only if a, b, c are even.

2013 IFYM, Sozopol, 8

Tags: inequalities unsolved , inequalities

Let $ x, y, z $ be positive real numbers. Prove that \[ \frac{2x^2 + xy}{(y+ \sqrt{zx} + z )^2} + \frac{2y^2 + yz}{(z+ \sqrt{xy} + x )^2} + \frac{2z^2 + zx}{(x+ \sqrt{yz} +y )^2} \ge 1 \]

2001 Federal Math Competition of S&M, Problem 3

Tags: inequalities , inequalities unsolved

Let $p_{1}, p_{2},...,p_{n}$, where $n>2$, be the first $n$ prime numbers. Prove that $\frac{1}{p_{1}^2}+\frac{1}{p_{2}^2}+...+\frac{1}{p_{n}^2}+\frac{1}{p_{1}p_{2}...p_{n}}<\frac{1}{2}$

1986 China Team Selection Test, 3

Tags: inequalities , quadratics , inequalities unsolved

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

2011 Morocco National Olympiad, 1

Tags: inequalities , trigonometry , inequalities unsolved

Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$. Prove that $2(x+y+z)\leq 3$.

1976 IMO Longlists, 34

Tags: inequalities unsolved , inequalities

Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas \[a_{n+1} = a_n + b_n,\] \[ b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,\] and the initial values $a_0 = b_0 = 1$. Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$.

2004 India IMO Training Camp, 1

Tags: inequalities , logarithms , function , inequalities unsolved

Let $x_1, x_2 , x_3, .... x_n$ be $n$ real numbers such that $0 < x_j < \frac{1}{2}$. Prove that \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

2009 China Team Selection Test, 3

Tags: induction , inequalities , inequalities unsolved

Let $ x_{1},x_{2},\cdots,x_{m},y_{1},y_{2},\cdots,y_{n}$ be positive real numbers. Denote by $ X \equal{} \sum_{i \equal{} 1}^{m}x,Y \equal{} \sum_{j \equal{} 1}^{n}y.$ Prove that $ 2XY\sum_{i \equal{} 1}^{m}\sum_{j \equal{} 1}^{n}|x_{i} \minus{} y_{j}|\ge X^2\sum_{j \equal{} 1}^{n}\sum_{l \equal{} 1}^{n}|y_{i} \minus{} y_{l}| \plus{} Y^2\sum_{i \equal{} 1}^{m}\sum_{k \equal{} 1}^{m}|x_{i} \minus{} x_{k}|$

1983 IMO Longlists, 72

Tags: trigonometry , inequalities , inequalities unsolved

Prove that for all $x_1, x_2,\ldots , x_n \in \mathbb R$ the following inequality holds: \[\sum_{n \geq i >j \geq 1} \cos^2(x_i - x_j ) \geq \frac{n(n-2)}{4}\]

2009 Serbia Team Selection Test, 2

Tags: inequalities , inequalities unsolved , Serbia

Let $ x,y,z$ be positive real numbers such that $ xy \plus{} yz \plus{} zx \equal{} x \plus{} y \plus{} z$. Prove the inequality $ \frac1{x^2 \plus{} y \plus{} 1} \plus{} \frac1{y^2 \plus{} z \plus{} 1} \plus{} \frac1{z^2 \plus{} x \plus{} 1}\le1$ When does the equality hold?

1998 China Team Selection Test, 3

Tags: inequalities , trigonometry , inequalities unsolved

For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions: [b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1$. [b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a$.

2013 Indonesia MO, 3

Tags: inequalities , inequalities unsolved

Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.

1991 USAMO, 4

Tags: inequalities , AMC , USA(J)MO , USAMO , inequalities unsolved , 1991

Let $a = \frac{m^{m+1} + n^{n+1}}{m^m + n^n}$, where $m$ and $n$ are positive integers. Prove that $a^m + a^n \geq m^m + n^n$.

2004 Switzerland Team Selection Test, 4

Tags: inequalities , inequalities unsolved

[i]Second Test, May 16[/i] Let $a$, $b$, and $c$ be positive real numbers such that $abc = 1$. Prove that $\frac{ab}{a^{5}+b^{5}+ab}+\frac{bc}{b^{5}+c^{5}+bc}+\frac{ca}{c^{5}+a^{5}+ca}\le 1$ . When does equality hold?

2004 Croatia Team Selection Test, 2

Tags: inequalities , inequalities unsolved

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

1997 China National Olympiad, 1

Tags: function , inequalities , inequalities unsolved

Let $x_1,x_2,\ldots ,x_{1997}$ be real numbers satisfying the following conditions: i) $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$ for $i=1,2,\ldots ,1997$; ii) $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$ . Determine (with proof) the maximum value of $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .

1992 Iran MO (2nd round), 2

Tags: inequalities , induction , inequalities unsolved

In the sequence $\{a_n\}_{n=0}^{\infty}$ we have $a_0=1$, $a_1=2$ and \[a_{n+1}=a_n+\dfrac{a_{n-1}}{1+a_{n-1}^2} \qquad \forall n \geq 1\] Prove that \[52 < a_{1371} < 65\]