Problem A. 556. (February 2012)
A. 556. Prove that for arbitrary real numbers there exist a real t such that
(5 pont)
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=a_{1}, ..., t=a_{n} proves the statement.
The role of a_{1},...,a_{n} is symmetric and the function sin x is periodic by , so without loss of generality we may assume . Define a_{0}=0 too; then f(a_{0})=f(a_{n}).
If , then , and the statement is trivial. In the rest of the solution we assume a_{1}< as well; then .
By the periodicity of sin x,
 (1) 
Now we prove that
 (2) 
for all 1kn.
We prove (2) termwise. For each index 1in,
In the interval [a_{k1},a_{k}] the function sin (xa_{i}) has constant sign: it is nonnegative for ik1, 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
Statistics:
