Problem A. 556. (February 2012)
A. 556. Prove that for arbitrary real numbers there exist a real t such that
Deadline expired on March 12, 2012.
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|,
Now we prove that
for all 1kn.
We prove (2) termwise. For each index 1in,
In the interval [ak-1,ak] the function sin (x-ai) has constant sign: it is nonnegative for ik-1, and nonpositive for ik. Multiplying by (-1) for i<k and summing up we obtain (2).
Combining (1) and (2), and applying Jensen's inequality to the tangent function (which is convex in [0,/2), we get