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

1995 IMO Shortlist, 4
Tags: function , inequalities , IMO Shortlist , optimization , system of equations , 133 , 109
Find all of the positive real numbers like $ x,y,z,$ such that : 1.) $ x \plus{} y \plus{} z \equal{} a \plus{} b \plus{} c$ 2.) $ 4xyz \equal{} a^2x \plus{} b^2y \plus{} c^2z \plus{} abc$ Proposed to Gazeta Matematica in the 80s by VASILE CÎRTOAJE and then by Titu Andreescu to IMO 1995.

1998 Belarus Team Selection Test, 2
Tags: inequalities , Ring Theory , inequalities proposed , 109 , Belarus , algebra
Let $a$, $b$, $c$ be real positive numbers. Show that \[\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq \frac{a+b}{b+c}+\frac{b+c}{a+b}+1\]

2008 Ukraine Team Selection Test, 3
Tags: inequalities , inequalities proposed , 109
For positive $ a, b, c, d$ prove that $ (a \plus{} b)(b \plus{} c)(c \plus{} d)(d \plus{} a)(1 \plus{} \sqrt [4]{abcd})^{4}\geq16abcd(1 \plus{} a)(1 \plus{} b)(1 \plus{} c)(1 \plus{} d)$