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Part of 2008 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Nice inequality

Source: Federation of Bosnia, 1. Grades 2008.

For arbitrary reals

x

y

z

prove the following inequality: x^{2} \plus{} y^{2} \plus{} z^{2} \minus{} xy \minus{} yz \minus{} zx \geq \max \{\frac {3(x \minus{} y)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} , \frac {3(y \minus{} z)^{2}}{4} \}

Easy inequality

Source: 0

a

b

c

are positive reals such that a^{2}\plus{}b^{2}\plus{}c^{2}\equal{}1 prove the inequality:
\frac{a^{5}\plus{}b^{5}}{ab(a\plus{}b)}\plus{} \frac {b^{5}\plus{}c^{5}}{bc(b\plus{}c)}\plus{}\frac {c^{5}\plus{}a^{5}}{ca(a\plus{}b)}\geq 3(ab\plus{}bc\plus{}ca)\minus{}2.

inequalitiesfunction

Source: 0

a

b

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.

inequalitieslogarithmscircumcircleHigh school olympiadBosniaalgebra