Mathematical and Physical Journal
for High Schools
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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,...,an.

(5 pont)

Deadline expired on March 17, 2008.


\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.


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