Home > Analysis, Inequalities, Problem Solving > Traian Lalescu contest 2009 Problem 3

## Traian Lalescu contest 2009 Problem 3

Suppose $n \in \Bbb{N}, \ n \geq 2$ and $x_1,...,x_n >0,\ x_1+x_2+...+x_n=1$.Prove that

$\displaystyle \sum_{k=1}^n \frac{x_k}{1+k(x_1^2+x_2^2+...+x_k^2)}< \frac{\pi}{4}$, and the constant $\pi/4$ is the best possible.

Traian Lalescu contest 2009

I don’t have a solution for the inequality, but if one can show it then we can conclude $\pi/4$ is the optimal constant by picking $x_1 = x_2 = x_3 = ... = x_n = 1/n$, which makes the sum a Riemann integral for $\int_0^1 1/(1+x^2) dx = \pi/4$, so we can make the sum arbitrarily close to $\pi/4$ by picking sufficiently large $n$.