Prove the Nesbitt’s inequality, for positive real numbers \(a,b,c\) we have: \[\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}\geq \frac{3}{2}.\]
Due to Paul Erdos. 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 \[\frac{1}{a_1} + ... + \frac{1}{a_n} < 1+ \frac{n}{1951}.\]