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

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Problem A. 425. (April 2007)

A. 425. Let n $\ge$ 2 and let $a_1,a_2,\ldots,a_n, x_1,x_2,\ldots,x_n$ be positive real numbers such that a₁+...+a_n=x₁+...+x_n=1. Prove that

$2\sum_{1\le i<j\le n}x_ix_j \le \frac{n-2}{n-1} + \sum_{i=1}^n \frac{a_ix_i^2}{1-a_i}\,.$

Polish competition problem

(5 pont)

Deadline expired on May 15, 2007.

Statistics:

16 students sent a solution.

5 points: Bogár 560 Péter, Dobribán Edgár, Gyenizse Gergő, Honner Balázs, Hujter Bálint, Kiarash Adl, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Lovász László Miklós, Nagy 224 Csaba, Nagy 235 János, Tomon István, Varga Bonbien.

4 points: Kornis Kristóf, Nagy 314 Dániel.

Unfair, not evaluated: 1 solutions.

Problems in Mathematics of KöMaL, April 2007

16 students sent a solution.
5 points:	Bogár 560 Péter, Dobribán Edgár, Gyenizse Gergő, Honner Balázs, Hujter Bálint, Kiarash Adl, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Lovász László Miklós, Nagy 224 Csaba, Nagy 235 János, Tomon István, Varga Bonbien.
4 points:	Kornis Kristóf, Nagy 314 Dániel.
Unfair, not evaluated:	1 solutions.