The 1968 Putnam Competition included a beautiful one-line proof that π is less than 22/7, its common Diophantine approximation:

\displaystyle 0 \enspace \textless \int_{0}^{1}\frac{x^{4}\left ( 1 - x \right )^{4}}{1 + x^{2}} \: dx = \frac{22}{7} - \pi .

The integral must be positive, because the integrand’s denominator is positive and its numerator is the product of two non-negative numbers. But it evaluates to 22/7 – π — and if that expression is positive, then 22/7 must be greater than π.

University of St Andrews mathematician G.M. Phillips wrote, “Who will say that mathematics is devoid of humour?”