Olympiad problems

2019 Saudi Arabia JBMO TST, 2

Let $a, b, c$ be positive reals so that $a^2+b^2+c^2=1$. Find the minimum value of $S=1/a^2+1/b^2+1/c^2-2(a^3+b^3+c^3)/abc$

2003 Mediterranean Mathematics Olympiad, 3

Tags: inequalities , blogs , inequalities unsolved , algebra , highschoolmath , High school olympiad

Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]

1992 India National Olympiad, 1

Tags: trigonometry , algebra , highschoolmath , trig identities

In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.

2007 Turkey Team Selection Test, 3

Tags: inequalities , function , inequalities proposed , algebra , highschoolmath

Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]

2005 Junior Balkan Team Selection Tests - Romania, 8

Tags: inequalities , inequalities proposed , highschoolmath

Let $a$, $b$, $c$ be three positive reals such that $(a+b)(b+c)(c+a)=1$. Prove that the following inequality holds: \[ ab+bc+ca \leq \frac 34 . \] [i]Cezar Lupu[/i]