Olympiad problems

2007 ISI B.Math Entrance Exam, 7

Let $ 0\leq \theta\leq \frac{\pi}{2}$ . Prove that $\sin \theta \geq \frac{2\theta}{\pi}$.

1990 IMO Longlists, 5

Tags: inequalities , inequalities proposed

Let $x,y,z$ be positive reals and $x \geq y \geq z$. Prove that \[\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2\]

2010 Romania Team Selection Test, 1

Tags: inequalities , IMO Shortlist , inequalities proposed

Let $n$ be a positive integer and let $x_1, x_2, \ldots, x_n$ be positive real numbers such that $x_1x_2 \cdots x_n = 1$. Prove that \[\displaystyle\sum_{i=1}^n x_i^n (1 + x_i) \geq \dfrac{n}{2^{n-1}} \prod_{i=1}^n (1 + x_i).\] [i]IMO Shortlist[/i]

2014 Iran Team Selection Test, 5

Tags: inequalities , inequalities proposed

if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$ prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]

2007 Irish Math Olympiad, 2

Tags: inequalities , inequalities proposed

Suppose that $ a,b,$ and $ c$ are positive real numbers. Prove that: $ \frac{a\plus{}b\plus{}c}{3} \le \sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{3}} \le \frac {\frac{ab}{c}\plus{}\frac{bc}{a}\plus{}\frac{ca}{b}}{3}$. For each of the inequalities, find the conditions on $ a,b,$ and $ c$ such that equality holds.

1996 Baltic Way, 15

Tags: inequalities , inequalities proposed

For which positive real numbers $a,b$ does the inequality \[x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a\] hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?

2006 Turkey Team Selection Test, 3

Tags: inequalities , inequalities proposed , algebra

If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that \[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]

2002 JBMO ShortLists, 3

Tags: inequalities , inequalities proposed

Let $ a,b,c$ be positive real numbers such that $ abc\equal{}\frac{9}{4}$. Prove the inequality: $ a^3 \plus{} b^3 \plus{} c^3 > a\sqrt {b \plus{} c} \plus{} b\sqrt {c \plus{} a} \plus{} c\sqrt {a \plus{} b}$ Jury's variant: Prove the same, but with $ abc\equal{}2$

2001 India IMO Training Camp, 1

Tags: inequalities , inequalities proposed

Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+ y+z)$.

2012 Korea National Olympiad, 3

Tags: induction , inequalities , inequalities proposed

Let $ \{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \} $ . Find the maximum value of \[ \sum_{n=1}^{10}(na_n ^2 - n^2 a_n ) \]

2009 Indonesia TST, 3

Tags: calculus , derivative , geometry , 3D geometry , function , inequalities proposed , inequalities

Let $ x,y,z$ be real numbers. Find the minimum value of $ x^2\plus{}y^2\plus{}z^2$ if $ x^3\plus{}y^3\plus{}z^3\minus{}3xyz\equal{}1$.

2014 Mexico National Olympiad, 5

Tags: inequalities , inequalities proposed , Holder

Let $a, b, c$ be positive reals such that $a + b + c = 3$. Prove: \[ \frac{a^2}{a + \sqrt[3]{bc}} + \frac{b^2}{b + \sqrt[3]{ca}} + \frac{c^2}{c + \sqrt[3]{ab}} \geq \frac{3}{2} \] And determine when equality holds.

2013 Hong kong National Olympiad, 1

Tags: inequalities , function , inequalities proposed , China , High school olympiad

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=1$. Prove that \[\sqrt[4]{\frac{\sqrt{3}}{a}+6\sqrt{3}b}+\sqrt[4]{\frac{\sqrt{3}}{b}+6\sqrt{3}c}+\sqrt[4]{\frac{\sqrt{3}}{c}+6\sqrt{3}a}\le\frac{1}{abc}\] When does inequality hold?

2006 Moldova Team Selection Test, 3

Tags: function , inequalities , geometry , circumcircle , trigonometry , abstract algebra , inequalities proposed

Let $a,b,c$ be sides of a triangle and $p$ its semiperimeter. Show that $a\sqrt{\frac{(p-b)(p-c)}{bc}}+b \sqrt{\frac{(p-c)(p-a)}{ac}}+c\sqrt{\frac{(p-a)(p-b)}{ab}}\geq p$

2010 Iran MO (3rd Round), 5

Tags: inequalities , inequalities proposed , Iran

$x,y,z$ are positive real numbers such that $xy+yz+zx=1$. prove that: $3-\sqrt{3}+\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge(x+y+z)^2$ (20 points) the exam time was 6 hours.

2010 Lithuania National Olympiad, 1

Tags: inequalities proposed , inequalities

$a,b$ are real numbers such that: \[ a^3+b^3=8-6ab. \] Find the maximal and minimal value of $a+b$.

2011 India Regional Mathematical Olympiad, 6

Tags: inequalities , inequalities proposed , india , 2011

Find the largest real constant $\lambda$ such that \[\frac{\lambda abc}{a+b+c}\leq (a+b)^2+(a+b+4c)^2\] For all positive real numbers $a,b,c.$

2006 Iran Team Selection Test, 4

Tags: inequalities , induction , function , triangle inequality , inequalities proposed

Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that \[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]

2002 Iran Team Selection Test, 6

Tags: ceiling function , inequalities proposed , inequalities

Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.

2009 Baltic Way, 4

Tags: inequalities , inequalities proposed

Determine all integers $ n>1$ for which the inequality \[ x_1^2\plus{}x_2^2\plus{}\ldots\plus{}x_n^2\ge(x_1\plus{}x_2\plus{}\ldots\plus{}x_{n\minus{}1})x_n\] holds for all real $ x_1,x_2,\ldots,x_n$.

1989 IberoAmerican, 3

Tags: inequalities , inequalities proposed

Let $a,b$ and $c$ be the side lengths of a triangle. Prove that: \[\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}\]

2009 Baltic Way, 2

Tags: inequalities , algebra , polynomial , inequalities proposed

Let $ a_1,a_{2},\ldots ,a_{100}$ be nonnegative integers satisfying the inequality \[a_1\cdot (a_1-1)\cdot\ldots\cdot (a_1-20)+a_2\cdot (a_2-1)\cdot\ldots\cdot (a_2-20)+\\ \ldots+a_{100}\cdot (a_{100}-1)\cdot\ldots\cdot (a_{100}-20)\le 100\cdot 99\cdot 98\cdot\ldots\cdot 79.\] Prove that $a_1+a_2+\ldots+a_{100}\le 9900$.

1999 Akdeniz University MO, 3

Tags: inequalities , algebra , inequalities proposed

Let $a$,$b$,$c$ and $d$ positive reals. Prove that $$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$

2014 District Olympiad, 1

Tags: inequalities , inequalities proposed

[list=a] [*]Prove that for any real numbers $a$ and $b$ the following inequality holds: \[ \left( a^{2}+1\right) \left( b^{2}+1\right) +50\geq2\left( 2a+1\right)\left( 3b+1\right)\] [*]Find all positive integers $n$ and $p$ such that: \[ \left( n^{2}+1\right) \left( p^{2}+1\right) +45=2\left( 2n+1\right)\left( 3p+1\right) \][/list]

2007 China Team Selection Test, 1

Tags: inequalities , inequalities proposed

Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that \[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]