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