Olympiad problems

2004 China Western Mathematical Olympiad, 4

Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that \[ 1 < \frac {a}{\sqrt {a^{2} \plus{} b^{2}}} \plus{} \frac {b}{\sqrt {b^{2} \plus{} c^{2}}} \plus{} \frac {c}{\sqrt {c^{2} \plus{} a^{2}}}\leq\frac {3\sqrt {2}}{2} \]

2019 District Olympiad, 1

Let $n \in \mathbb{N}, n \ge 2$ and the positive real numbers $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ such that $a_1+a_2+…+a_n=b_1+b_2+…+b_n=S.$ $\textbf{a)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}.$ $\textbf{b)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.$

identity inequality cauchy inequality algebra Cauchy-Schwarz inequality inequalities

1997 Iran MO (3rd Round), 4

Let $x, y, z$ be real numbers greater than $1$ such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2$. Prove that \[\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}\leq \sqrt{x+y+z}.\]

inequalities cauchy inequality AM-GM inequality Cauchy-Schwarz inequality algebra

2008 Iran MO (3rd Round), 4

Let $ x,y,z\in\mathbb R^{\plus{}}$ and $ x\plus{}y\plus{}z\equal{}3$. Prove that: \[ \frac{x^3}{y^3\plus{}8}\plus{}\frac{y^3}{z^3\plus{}8}\plus{}\frac{z^3}{x^3\plus{}8}\geq\frac19\plus{}\frac2{27}(xy\plus{}xz\plus{}yz)\]

inequalities cauchy inequality AM-GM inequality