MathDB
Sequence inequality

Source: Moldova TST 2014, Second Day, Problem 1

March 30, 2014
inequalitiesalgebra proposedalgebra

Problem Statement

Consider n≥2n \geq 2 positive numbers 0<x1≤x2≤...≤xn0<x_1 \leq x_2 \leq ... \leq x_n, such that x1+x2+...+xn=1x_1 + x_2 + ... + x_n = 1. Prove that if xn≤23x_n \leq \dfrac{2}{3}, then there exists a positive integer 1≤k≤n1 \leq k \leq n such that 13≤x1+x2+...+xk<23\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}.
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