Bogognov i.i.

Each segment with endpoints at the vertices of a regular 100-gon is painted red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers are placed at the vertices such that the sum of their squares equals 1, and on the segments - the products of the numbers at the endpoints. Then, the sum of the numbers on the red segments is subtracted from the sum of the numbers on the blue segments. What is the largest number that could result?
Let the numbers $x_{1}, ..., x_{100}$ be placed at the vertices around a circle, $k = x_{1}x_{2} + x_{1}x_{4} + ... + x_{99}x_{100}$ - the sum of the red numbers, $s = x_{1}x_{3} + x_{1} x_{-}{5}+\ldots+x_{-}{98} x_{-}{100} $$ - the sum of the blue numbers. Note that

$0 \leq $(x_{-}{1}-x_{-}{2}+x_{-}{3}-x_{-}{4}+\ldots+x_{-}{99}-x_{-}{100}) \wedge 2=1-2 k+2 s $$$. Therefore, $k-s \leqslant \frac{1}{2} $. Equality is achieved when the expression in parentheses equals zero, for example, when $ x^{2}_{1}= x_{2}=\ldots= x_{100}= \frac{1}{10} $.

## Answer