This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3

2009 Bosnia Herzegovina Team Selection Test, 1
Tags: induction , combinatorics unsolved , combinatorics , Game Theory , Bosnia , 2009
Given an $1$ x $n$ table ($n\geq 2$), two players alternate the moves in which they write the signs + and - in the cells of the table. The first player always writes +, while the second always writes -. It is not allowed for two equal signs to appear in the adjacent cells. The player who can’t make a move loses the game. Which of the players has a winning strategy?

2008 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: inequalities , logarithms , circumcircle , High school olympiad , Bosnia , algebra
If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]

2010 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: algebra , Bosnia
For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$ $$a^3+b^3+c^3+d^3=0$$ Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero