Difference between revisions of "2016 AIME II Problems/Problem 15"
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Revision as of 21:02, 17 March 2016
For let
and
. Let
be positive real numbers such that
and $\sum_{i \leq i < j \leq 216 x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}. The maximum possible value of$ (Error compiling LaTeX. Unknown error_msg)x_2=\dfrac{m}{n}
m
n
m+n$.
==Solution==
Replace$ (Error compiling LaTeX. Unknown error_msg)\sum x_ix_j\frac12\left(\left(\sum x_i\right)^2-\sum x_i^2\right)
\sum\frac{x_i^2}{1-a_i}=\frac{1}{215}
\sum 1-a_i=215
\left(\sum 1-a_i\right)\left(\sum\frac{x_i^2}{1-a_i}\right)=1=\left(\sum x_i\right)^2
x_i=c(1-a_i)
c
\sum x_i=1
c=\frac{1}{215}
x_2=\frac{3}{860}$.