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China Team Selection Test 2014 TST 1 Day 2 Q4

Source: China Nanjing , 13 Mar 2014

March 13, 2014
inductioninequalities proposedinequalitiesChina TST

Problem Statement

For any real numbers sequence {xn}\{x_n\} ,suppose that {yn}\{y_n\} is a sequence such that: y1=x1,yn+1=xn+1−(∑i=1nxi2)12y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}} (n≥1){(n \ge 1}) . Find the smallest positive number λ\lambda such that for any real numbers sequence {xn}\{x_n\} and all positive integers mm , have 1m∑i=1mxi2≤∑i=1mλm−iyi2.\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} . (High School Affiliated to Nanjing Normal University )
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