Home Competitions IR-TST 2011 Problem 10
Find the least value of k k k a , b , c , d ∈ R a,b,c,d \in \mathbb{R} a , b , c , d ∈ R ( a 2 + 1 ) ( b 2 + 1 ) ( c 2 + 1 ) + ( b 2 + 1 ) ( c 2 + 1 ) ( d 2 + 1 ) + ( c 2 + 1 ) ( d 2 + 1 ) ( a 2 + 1 ) + ( d 2 + 1 ) ( a 2 + 1 ) ( b 2 + 1 ) ≥ 2 ( a b + b c + c d + d a + a c + b d ) − k \begin{array} c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \ \ge 2( ab+bc+cd+da+ac+bd)-k \end{array} ( a 2 + 1 ) ( b 2 + 1 ) ( c 2 + 1 ) + ( b 2 + 1 ) ( c 2 + 1 ) ( d 2 + 1 ) + ( c 2 + 1 ) ( d 2 + 1 ) ( a 2 + 1 ) + ( d 2 + 1 ) ( a 2 + 1 ) ( b 2 + 1 ) ≥ 2 ( ab + b c + c d + d a + a c + b d ) − k
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