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5

Part of 2010 Thailand Mathematical Olympiad

Problems(2)

tennis, a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}

Source: 2010 Thailand Mathematical Olympiad day 1 p5

3/12/2021
In a round-robin table tennis tournament between 20102010 athletes, where each match ends with a winner and a loser, let a1,...,a2010a_1,... , a_{2010} denote the number of wins of each athlete, and let b1,..,b2010b_1, .., b_{2010} denote the number of losses of each athlete. Show that a12+a22+...+a20102=b12+b22+...+b20102a^2_1+a^2_2+...+a^2_{2010} =b^2_1 + b^2_2 + ... + b^2_{2010}.
combinatorics
f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010

Source: Thailand Mathematical Olympiad 2010 day 2 p5

8/17/2020
Determine all functions f:R×R→Rf : R \times R \to R satisfying the equation f(x−t,y)+f(x+t,y)+f(x,y−t)+f(x,y+t)=2010f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010 for all real numbers x,yx, y and for all nonzero tt
functional equationfunctionalalgebra