Let \(a,b,c >0\) be positive real numbers with \(abc \leq 1\). Prove that \[\frac{a}{c} + \frac{b}{a} + \frac{c}{b} \geq a+b+c.\]
From IMO 1999. Let \(n\geq 2\) be an integer. Determine the least possible constant \(C\) such that the inequality \[\sum_{1\leq i<j\leq n} x_ix_j(x_i^2 + x_j^2) \leq C(\sum_{1\leq i\leq n}x_i)^4\] holds for all non-negative real numbers \(x_i\). For this constant \(C\) find out when the equality holds.
Proposed by USA for IMO 1993. For positive real numbers \(a,b,c,d\) prove that \[\frac{a}{b+2c+3d} + \frac{b}{c+2d+3a} + \frac{c}{d+2a+3b} + \frac{d}{a+2b+3c} \geq \frac{2}{3}.\]
Prove the \(AM-GM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\frac{a_1+a_2+...+a_n}{n}\geq \sqrt[n]{a_1a_2...a_n}.\]
Due to Paul Erdős. Each of the positive integers \(a_1\), \(a_2\), ..., \(a_n\) is less than \(1951\). The least common multiple of any two of these integers is greater than \(1951\). Prove that \[\frac{1}{a_1} + ... + \frac{1}{a_n} < 1+ \frac{n}{1951}.\]