Olympiad problems

2002 Rioplatense Mathematical Olympiad, Level 3, 4

Let $a, b$ and $c$ be positive real numbers. Show that $\frac{a+b}{c^2}+ \frac{c+a}{b^2}+ \frac{b+c}{a^2}\ge \frac{9}{a+b+c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

2007 China Team Selection Test, 3

Tags: cyclic inequality , inequalities , square root inequality , 3-variable inequality

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.

2014 Federal Competition For Advanced Students, P2, 5

Tags: algebra , inequalities , three variable inequality , 3-variable inequality

Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?

2012 Kyiv Mathematical Festival, 2

Tags: 3-variable inequality , inequality , inequalities , algebra

Positive numbers $x, y, z$ satisfy $x + y + z \le 1$. Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 8$.

2002 Junior Balkan MO, 4

Tags: rearrangement inequality , 3-variable inequality , cyclic inequality , inequalities , function

Prove that for all positive real numbers $a,b,c$ the following inequality takes place \[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \] [i]Laurentiu Panaitopol, Romania[/i]

2005 Germany Team Selection Test, 2

Tags: function , 3-variable inequality , nesbitt , inequalities

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

2005 Georgia Team Selection Test, 10

Tags: 3-variable inequality , inequalities , algebra

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

2009 Dutch IMO TST, 3

Tags: 3-variable inequality , three variable inequality , inequalities

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

2015 FYROM JBMO Team Selection Test, 3

Tags: 3-variable inequality , inequalities

Let $a, b$ and $c$ be positive real numbers. Prove that $\prod_{cyc}(16a^2+8b+17)\geq2^{12}\prod_{cyc}(a+1)$.

2013 ELMO Shortlist, 5

Tags: derivative , function , inequalities , logarithm , 3-variable inequality , calculus

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2013 ELMO Problems, 2

Tags: function , inequalities , calculus , derivative , 3-variable inequality , logarithm

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2005 Georgia Team Selection Test, 10

Tags: algebra , 3-variable inequality , inequalities

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

2012 India Regional Mathematical Olympiad, 8

Tags: inequalities , inequality , 3-variable inequality

Let $x, y, z$ be positive real numbers such that $2(xy + yz + zx) = xyz$. Prove that $\frac{1}{(x-2)(y-2)(z-2)} + \frac{8}{(x+2)(y+2)(z+2)} \le \frac{1}{32}$

2014 India Regional Mathematical Olympiad, 2

Tags: inequality , 3-variable inequality , inequalities

Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.

2015 Saudi Arabia JBMO TST, 1

Tags: inequalities , 3-variable inequality , three variable inequality , algebraic inequality , inequality

Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

2013 ELMO Shortlist, 5

Tags: calculus , derivative , 3-variable inequality , inequalities , function , logarithm

Let $a,b,c$ be positive reals satisfying $a+b+c = \sqrt[7]{a} + \sqrt[7]{b} + \sqrt[7]{c}$. Prove that $a^a b^b c^c \ge 1$. [i]Proposed by Evan Chen[/i]

2005 Iran MO (3rd Round), 1

Tags: inequalities , 3-variable inequality , cyclic inequality , cauchy schwarz

Suppose $a,b,c\in \mathbb R^+$. Prove that :\[\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\]

2012 JBMO TST - Turkey, 4

Tags: function , inequalities , 3-variable inequality

Find the greatest real number $M$ for which \[ a^2+b^2+c^2+3abc \geq M(ab+bc+ca) \] for all non-negative real numbers $a,b,c$ satisfying $a+b+c=4.$

2020 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequality , 3-variable inequality

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

2005 Germany Team Selection Test, 2

Tags: inequalities , function , 3-variable inequality , nesbitt

If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that \[\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).\]

2009 Dutch IMO TST, 3

Tags: inequalities , 3-variable inequality , three variable inequality

Let $a, b$ and $c$ be positive reals such that $a + b + c \ge abc$. Prove that $a^2 + b^2 + c^2 \ge \sqrt3 abc$.

2007 China Team Selection Test, 3

Tags: inequalities , 3-variable inequality , cyclic inequality , square root inequality

Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$ for all positive $ x$, $ y$, $ z$.