Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation

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Problem A. 447. (February 2008)

A. 447. Prove that the inequality $\sum\limits_{i=1}^n\ \sum\limits_{j=1}^n \frac{a_ia_j}{i+j}\ge 0$ is true for all real numbers a₁,a₂,...,a_n.

(5 pont)

Deadline expired on March 17, 2008.

Solution.

$\sum_{i=1}^n \sum_{j=1}^n\frac{a_ia_j}{i+j} = \sum_{i=1}^na_i \sum_{j=1}^n a_j \int_0^1 t^{i+j-1} \mathrm{d}t =$

$=\int_0^1 \left(\sum_{i=1}^na_it^{i-1}\right) \left(\sum_{j=1}^na_jt^{j-1}\right) t\mathrm{d}t = \int_0^1 \left(\sum_{i=1}^na_it^{i-1}\right)^2 t\mathrm{d}t \ge 0.$

Statistics:

7 students sent a solution.

5 points: Blázsik Zoltán, Lovász László Miklós, Nagy 235 János, Nagy 314 Dániel, Tomon István, Wolosz János.

0 point: 1 student.

Problems in Mathematics of KöMaL, February 2008

7 students sent a solution.
5 points:	Blázsik Zoltán, Lovász László Miklós, Nagy 235 János, Nagy 314 Dániel, Tomon István, Wolosz János.
0 point:	1 student.