Mathematical and Physical Journal
for High Schools
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# 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=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|,

 (1)

Now we prove that

 (2)

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

### Statistics:

 6 students sent a solution. 5 points: Gyarmati Máté, Janzer Olivér, Omer Cerrahoglu. 4 points: Machó Bónis. 0 point: 2 students.

Problems in Mathematics of KöMaL, February 2012