Prove Nesbitt’s inequality, which states that 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 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 \[\frac{1}{a_1} + ... + \frac{1}{a_n} < 1+ \frac{n}{1951}.\]