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 \(1\times 2\times 3...\times n\).
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 \[(a_1b_1 + a_2b_2 + ...a_nb_n)^2 \leq (a_1^2+a_2^2+...+a_n^2)(b_1^2+b_2^2+...b_n^2).\]
Calculate the following squares in the shortest possible way (without
a calculator or any other device):
a) \(1001^2\) b) \(9998^2\) c) \(20003^2\) d) \(497^2\)
Real numbers \(x,y\) are such that \(x^2 +x \le y\). Show that \(y^2 +y \ge x\).