Olympiad problems

2006 Vietnam Team Selection Test, 1

Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds: \[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \] When does the equality occur?

2005 Poland - Second Round, 3

Prove that if the real numbers $a,b,c$ lie in the interval $[0,1]$, then \[\frac{a}{bc+1}+\frac{b}{ac+1}+\frac{c}{ab+1}\le 2\]

inequalities convex-concave inequalities algebra

2017 China National Olympiad, 6

Given an integer $n \geq2$ and real numbers $a,b$ such that $0<a<b$. Let $x_1,x_2,\ldots, x_n\in [a,b]$ be real numbers. Find the maximum value of $$\frac{\frac{x^2_1}{x_2}+\frac{x^2_2}{x_3}+\cdots+\frac{x^2_{n-1}}{x_n}+\frac{x^2_n}{x_1}}{x_1+x_2+\cdots +x_{n-1}+x_n}.$$

prc sqing orz inequalities inequality convex-concave inequalities Cauchy-Schwarz inequality algebra parity rearrangement inequality

1977 USAMO, 5

If $ a,b,c,d,e$ are positive numbers bounded by $ p$ and $ q$, i.e, if they lie in $ [p,q], 0 < p$, prove that \[ (a \plus{} b \plus{} c \plus{} d \plus{} e)\left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \plus{} \frac {1}{d} \plus{} \frac {1}{e}\right) \le 25 \plus{} 6\left(\sqrt {\frac {p}{q}} \minus{} \sqrt {\frac {q}{p}}\right)^2\] and determine when there is equality.

calculus function inequalities convex-concave inequalities AM-GM inequality Cauchy-Schwarz inequality Jensen's inequality