This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3

2010 Iran Team Selection Test, 9
Tags: concavity , induction , inequalities
Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers: \[\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2\]

2021 IMO Shortlist, A4
Tags: n-variable inequality , convexity , concavity , inequalities , algebra
Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$

2021 IMO, 2
Tags: inequalities , n-variable inequality , convexity , concavity , algebra
Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$