7

Part of 2011 JBMO Shortlist

Problems(2)

Inequality with abc=1

Source: JBMO 2011 Shortlist A7

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a,b,c

be positive reals such that

abc=1

.Prove the inequality

\sum\frac{2a^2+\frac{1}{a}}{b+\frac{1}{a}+1}\geq 3

dimensions of max of squares filling a rectangle by half

Source: JBMO 2011 Shortlist C7

Consider a rectangle whose lengths of sides are natural numbers. If someone places as many squares as possible, each with area

3

, inside of the given rectangle, such that the sides of the squares are parallel to the rectangle sides, then the maximal number of these squares fill exactly half of the area of the rectangle. Determine the dimensions of all rectangles with this property.

JBMOcombinatoricscombinatorial geometry