4

Part of 2012 Cuba MO

Problems(2)

a rectangle is formed of 3 × 7 by 21 bw pieces, vertices of rectangle

Source: 2012 Cuba MO 1.4

21

pieces, some white and some black, a rectangle is formed of

3 \times 7

. Prove that there are always four pieces of the same color located at the vertices of a rectangle.

geometrycombinatorial geometrycombinatoricsColoringrectangle

sum xz / (x^2 + xy + y^2 + 6z^2) <= 1/3

Source: 2012 Cuba MO 2.4

x, y, z

be positive reals. Prove that

\frac{xz}{x^2 + xy + y^2 + 6z^2} + \frac{zx}{z^2 + zy + y^2 + 6x^2} + \frac{xy}{x^2 + xz + z^2 + 6y^2} \le \frac13

algebrainequalities