MathDB

9

Part of 2014 JBMO Shortlist

Problems(1)

Σ|x_n-y_n| \leq 2- min {x_i/y_i} - min {y_i/x_i}, if Σx_i=Σy_i=1

Source: JBMO Shortlist 2014 A9

4/24/2019
Let nn a positive integer and let x1,…,xn,y1,…,ynx_1, \ldots, x_n, y_1, \ldots, y_n real positive numbers such that x1+…+xn=y1+…+yn=1x_1+\ldots+x_n=y_1+\ldots+y_n=1. Prove that:
∣x1−y1∣+…+∣xn−yn∣≤2−min1≤i≤n  xiyi−min1≤i≤n  yixi|x_1-y_1|+\ldots+|x_n-y_n|\leq 2-\underset{1\leq i\leq n}{min} \;\dfrac{x_i}{y_i}-\underset{1\leq i\leq n}{min} \;\dfrac{y_i}{x_i}
algebraminimuminequalitiespositive real