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Part of 2007 Middle European Mathematical Olympiad

Problems(2)

Cyclic Inequality in 4 Variables with a+b+c+d=4

Source: MEMO Individual Competition, Question 1

a,b,c,d

be positive real numbers with a\plus{}b\plus{}c\plus{}d \equal{} 4. Prove that a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.

inequalitiesinequalities proposed

Cyclic Inequality in 4 variables with abcd=1

Source: MEMO Team Competition, Question 5

a,b,c,d

be real numbers which satisfy

\frac{1}{2}\leq a,b,c,d\leq 2

and abcd\equal{}1. Find the maximum value of \left(a\plus{}\frac{1}{b}\right)\left(b\plus{}\frac{1}{c}\right)\left(c\plus{}\frac{1}{d}\right)\left(d\plus{}\frac{1}{a}\right).

inequalitiesinequalities proposed