Home > Algebra, IMO, Inequalities, Olympiad, Problem Solving > IMC 2010 Day 1 Problem 5

IMC 2010 Day 1 Problem 5


Suppose that a,b,c are real numbers in the interval [-1,1] such that 1+2abc \geq a^2+b^2+c^2. Prove that 1+2(abc)^n\geq a^{2n}+b^{2n}+c^{2n} for all positive integers n.
IMC 2010 Day 1 Problem 5

Official Solution:
The conditions of the problem imply that \displaystyle \begin{pmatrix} 1 & a & b \\ a&1& c \\ b&c&1 \end{pmatrix} is positive definite.

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