MathDB

2

Part of 2014 Korea Junior Math Olympiad

Problems(1)

s/x_1+t/x_2+s/x_3+t/x_4+...+s/x_{2n-1}+t/x_{2n} > 2n^2/(n+1)

Source: KJMO 2014 p2

5/2/2019
Let there be 2n2n positive reals a1,a2,...,a2na_1,a_2,...,a_{2n}. Let s=a1+a3+...+a2n−1s = a_1 + a_3 +...+ a_{2n-1}, t=a2+a4+...+a2nt = a_2 + a_4 + ... + a_{2n}, and xk=ak+ak+1+...+ak+n−1x_k = a_k + a_{k+1} + ... + a_{k+n-1} (indices are taken modulo 2n2n). Prove that sx1+tx2+sx3+tx4+...+sx2n−1+tx2n>2n2n+1\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}
algebraSuminequalitiesInequality