2

Part of 2011 Spain Mathematical Olympiad

Problems(2)

Ineq: √( (ab+bc+ca)/(a^2+b^2+c^2) ) + Σ ( a/(b+c) ) ≥ 5/2

Source: Spanish Mathematical Olympiad, National Stage, 2011

a

b

c

be positive real numbers. Prove that

\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+\sqrt{\frac{ab+bc+ca}{a^2+b^2+c^2}}\ge\frac52

and determine when equality holds.

inequalitiesfunctioncalculusderivativeinequalities proposedalgebra

Rationals x & y are different colors if xy=1, x+y=0 or x+y=1

Source: Spanish Mathematical Olympiad, National Stage, 2011

Each rational number is painted either white or red. Call such a coloring of the rationals sanferminera if for any distinct rationals numbers

x

y

satisfying one of the following three conditions: [*]

xy=1

x+y=0

x+y=1

x

y

painted different colors. How many sanferminera colorings are there?

functionalgorithmgroup theoryabstract algebrainductionnumber theoryEuclidean algorithm