This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 10

2022 District Olympiad, P2
Tags: inequality , cyclic inequality , inequalities , algebra
$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$ $b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$

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$.

2017 Vietnamese Southern Summer School contest, Problem 2
Tags: cyclic inequality , fraction , inequalities
Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that: $$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$

2013 India Regional Mathematical Olympiad, 3
Tags: cyclic inequality
Given real numbers $a,b,c,d,e>1$. Prove that \[ \frac{a^2}{c-1}+\frac{b^2}{d-1}+\frac{c^2}{e-1}+\frac{d^2}{a-1}+\frac{e^2}{b-1} \ge 20 \]

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]

2004 All-Russian Olympiad, 4
Tags: cyclic inequality , n-variable inequality , induction , inequalities
Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality \[ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. \]

2019 Jozsef Wildt International Math Competition, W. 49
Tags: cyclic inequality , inequalities
Let $a$, $b$, $c \in (0,+\infty)$ . Then the following inequality is true:$$\sqrt{(a+b)(b+c)}+\sqrt{(b+c)(c+a)}+\sqrt{(c+a)(a+b)}+a+b+c\leq \left(ab+bc+ca\right)\left(\frac{1}{\sqrt{ab}}+\frac{1}{\sqrt{bc}}+\frac{1}{\sqrt{ca}}\right)$$

2014 IMAC Arhimede, 6
Tags: inequalities , cyclic inequality , four variables , algebra
If $a, b, c, d$ are positive numbers, prove that $$\sum_{cyclic}\frac{a-\sqrt[3]{bcd}}{a+3(b+c+d)}\ge 0$$

2005 Iran MO (3rd Round), 1
Tags: inequalities , 3-variable inequality , cyclic inequality , cauchy schwarz
Suppose $a,b,c\in \mathbb R^+$. Prove that :\[\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)\]

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$.