Solution. Define the function and let . We will show that . Then it follows that at least one of the choices t=a1, ..., t=an proves the statement.
The role of a1,...,an is symmetric and the function |sin x| is periodic by , so without loss of generality we may assume . Define a0=0 too; then f(a0)=f(an).
If , then , and the statement is trivial. In the rest of the solution we assume a1< as well; then .
By the periodicity of |sin x|,
We prove (2) termwise. For each index 1in,
Combining (1) and (2), and applying Jensen's inequality to the tangent function (which is convex in [0,/2), we get