Let \(a,b,c >0\) be positive real numbers with \(abc \leq 1\). Prove that \[\frac{a}{c} + \frac{b}{a} + \frac{c}{b} \geq a+b+c.\]
Let \(a,b,c >0\) be positive real numbers. Prove that \[(1+a)(1+b)(1+c)\geq 8\sqrt{abc}.\]
For a natural number \(n\) prove that \(n! \leq (\frac{n+1}{2})^n\), where \(n!\) is the factorial \(1\times 2\times 3\times ... \times n\).
Prove the \(AM-GM\) inequality for \(n=2\). Namely for two non-negative real numbers \(a\) and \(b\) we have \(2\sqrt{ab} \leq a+b\).
Prove the Cauchy-Schwartz inequality: for a natural number \(n\) and real numbers \(a_1\), \(a_2\), ..., \(a_n\) and \(b_1\), \(b_2\), ..., \(b_n\) we have \[(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...+b_n^2).\]
Prove the \(GM-HM\) inequality for positive real numbers \(a_1\), \(a_2\), ..., \(a_n\): \[\sqrt[n]{a_1a_2...a_n} \geq \frac{n}{\frac{1}{a_1} + ... \frac{1}{a_n}}.\]
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 \(a_1\), \(a_2\), ..., \(a_n\): \[\frac{a_1+a_2+...+a_n}{n}\geq \sqrt[n]{a_1a_2...a_n}.\]
For non-negative real numbers \(a,b,c\) prove that \[a^3+b^3+c^3 \geq \frac{(a+b+c)(a^2+b^2+c^2)}{3}\geq a^2b+b^2c+c^2a.\]