Olympiad problems

2002 Vietnam National Olympiad, 1

Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?

2004 239 Open Mathematical Olympiad, 4

Tags: inequalities , inequalities unsolved

Let the sum of positive reals $a,b,c$ be equal to 1. Prove an inequality \[ \sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2 \]. [b]proposed by Fedor Petrov[/b]

2002 Tuymaada Olympiad, 2

Tags: inequalities , inequalities unsolved

Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove that \[ \frac{1+ab}{1+a} + \frac{1+bc}{1+b} + \frac{1+cd}{1+c} + \frac{1+da}{1+d} \geq 4 . \] [i]Proposed by A. Khrabrov[/i]

1992 Vietnam National Olympiad, 2

Tags: geometry , geometric transformation , rotation , rectangle , inequalities unsolved , inequalities

Let $H$ be a rectangle with angle between two diagonal $\leq 45^{0}$. Rotation $H$ around the its center with angle $0^{0}\leq x\leq 360^{0}$ we have rectangle $H_{x}$. Find $x$ such that $[H\cap H_{x}]$ minimum, where $[S]$ is area of $S$.

1972 IMO Longlists, 6

Tags: inequalities , trigonometry , inequalities unsolved

Prove the inequality \[(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1\] for all natural numbers $n \ge 2.$

1999 Austrian-Polish Competition, 2

Tags: inequalities unsolved , inequalities

Find the best possible $k,k'$ such that \[k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'\] for all positive reals $v,w,x,y,z$.

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 \[ \frac{ \prod\limits_{j=1}^{n} x_j } { \left( \sum\limits_{j=1}^{n} x_j \right)^n} \leq \frac{ \prod\limits_{j=1}^{n} (1-x_j) } { \left( \sum\limits_{j=1}^{n} (1 - x_j) \right)^n} \]

2007 Mongolian Mathematical Olympiad, Problem 4

Tags: inequalities , inequalities unsolved , algebra

Let $ a,b,c>0$. Prove that $ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$

2006 Bulgaria Team Selection Test, 2

Tags: inequalities , inequalities unsolved , algebra

Prove that if $a,b,c>0,$ then \[ \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}. \] [i] Nikolai Nikolov[/i]

1994 Polish MO Finals, 3

Tags: inequalities unsolved , inequalities

The distinct reals $x_1, x_2, ... , x_n$ ,($n > 3$) satisfy $\sum_{i=1}^n x_i = 0$, $\sum_{i=1}^n x_i ^2 = 1$. Show that four of the numbers $a, b, c, d$ must satisfy: \[ a + b + c + nabc \leq \sum_{i=1}^n x_i ^3 \leq a + b + d + nabd \].

2000 France Team Selection Test, 3

Tags: inequalities , inequalities unsolved

$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.

1990 Polish MO Finals, 2

Tags: inequalities , inequalities unsolved

Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that \[ \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 \] Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.

2010 Malaysia National Olympiad, 8

Tags: logarithms , inequalities , inequalities unsolved

Show that \[\log_{a}bc+\log_bca+\log_cab \ge 4(\log_{ab}c+\log_{bc}a+\log_{ca}b)\] for all $a,b,c$ greater than 1.

1989 IMO Longlists, 61

Tags: inequalities , calculus , derivative , function , inequalities unsolved

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

2000 All-Russian Olympiad, 2

Tags: inequalities , inequalities unsolved , n-variable inequality , abel formula

Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$. Prove that if $y_1 < y_2 < \cdots < y_n$, then \[ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. \]

2013 Pan African, 3

Tags: inequalities , inequalities unsolved

Let $x$, $y$, and $z$ be real numbers such that $x<y<z<6$. Solve the system of inequalities: \[\left\{\begin{array}{cc} \dfrac{1}{y-x}+\dfrac{1}{z-y}\le 2 \\ \dfrac{1}{6-z}+2\le x \\ \end{array}\right.\]

2005 China Team Selection Test, 2

Tags: Gauss , function , induction , search , inequalities unsolved , inequalities

Let $n$ be a positive integer, and $x$ be a positive real number. Prove that $$\sum_{k=1}^{n} \left( x \left[\frac{k}{x}\right] - (x+1)\left[\frac{k}{x+1}\right]\right) \leq n,$$ where $[x]$ denotes the largest integer not exceeding $x$.

1999 Czech and Slovak Match, 1

Tags: inequalities , search , inequalities unsolved

Leta,b,c are postive real numbers,proof that $ \frac{a}{b\plus{}2c}\plus{}\frac{b}{c\plus{}2a}\plus{}\frac{c}{a\plus{}2b}\geq1$

2006 Federal Competition For Advanced Students, Part 2, 2

Tags: inequalities unsolved , inequalities

Let $ a,b,c$ be positive real numbers. Show that $ 3(a \plus{} b \plus{} c) \ge 8 \sqrt [3]{abc} \plus{} \sqrt [3]{\frac {a^3 \plus{} b^3 \plus{} c^3}{3} }.$

2010 Contests, 1

Tags: inequalities unsolved , inequalities

Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.

2014 India Regional Mathematical Olympiad, 4

Tags: geometry , inradius , inequalities unsolved , inequalities

let $ABC$ be a right angled triangle with inradius $1$ find the minimum area of triangle $ABC$

2002 China Girls Math Olympiad, 5

Tags: inequalities , floor function , inequalities unsolved

There are $ n \geq 2$ permutations $ P_1, P_2, \ldots, P_n$ each being an arbitrary permutation of $ \{1,\ldots,n\}.$ Prove that \[ \sum^{n\minus{}1}_{i\equal{}1} \frac{1}{P_i \plus{} P_{i\plus{}1}} > \frac{n\minus{}1}{n\plus{}2}.\]

2012 Kyrgyzstan National Olympiad, 2

Tags: inequalities , inequalities unsolved

Given positive real numbers $ {a_1},{a_2},...,{a_n} $ with $ {a_1}+{a_2}+...+{a_n}= 1 $. Prove that $ \left({\frac{1}{{a_1^2}}-1}\right)\left({\frac{1}{{a_2^2}}-1}\right)...\left({\frac{1}{{a_n^2}}-1}\right)\geqslant{({n^2}-1)^n} $.

1993 India National Olympiad, 3

Tags: inequalities , LaTeX , induction , invariant , inequalities unsolved

If $a,b,c,d \in \mathbb{R}_{+}$ and $a+b +c +d =1$, show that \[ ab +bc +cd \leq \dfrac{1}{4}. \]

2011 Croatia Team Selection Test, 1

Tags: inequalities , function , inequalities unsolved

Let $a,b,c$ be positive reals such that $a+b+c=3$. Prove the inequality \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geq \frac{3}{2}.\]