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

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Found problems: 177

2024 Junior Balkan MO, 1
Tags: JBMO , Balkan , inequalities , algebra
Let $a, b, c$ be positive real numbers such that $$a^2 + b^2 + c^2 = \frac{1}{4}.$$ Prove that $$\frac{1}{\sqrt{b^2 + c^2}} + \frac{1}{\sqrt{c^2 + a^2}} + \frac{1}{\sqrt{a^2 + b^2}} \le \frac{\sqrt{2}}{(a + b)(b + c)(c + a)}.$$ [i]Proposed by Petar Filipovski, Macedonia[/i]

2009 JBMO Shortlist, 3
Tags: JBMO , algebra , system of equations
Find all values of the real parameter $a$, for which the system $(|x| + |y| - 2)^2 = 1$ $y = ax + 5$ has exactly three solutions