39th Putnam 1978

Let a₁, a₂, ... , a_n be reals in the interval (0, π) with arithmetic mean μ. Show that ∏ (sin a_i)/a_i ≤ ( (sin μ)/μ )ⁿ.

Solution

Products are intractable, whereas sums are easy, so we take logs. Let f(x) = ln ( (sin x)/x) = ln sin x - ln x. The required relation is now that 1/n ∑ f(a_i) ≤ f(μ). This is true if the curve is concave, in other words if f ''(x) ≤ 0.

Differentiating, we have: f '(x) = cot x - 1/x, f ''(x) = - cosec²x + 1/x². But sin x ≤ x on [0, π], so f ''(x) ≤ 0 on [0, π].

39th Putnam 1978

© John Scholes
jscholes@kalva.demon.co.uk
30 Nov 1999