Problem 1
For each positive integer n let $$ x_n=p_1+.....+p_n $$ where $$ p_{1},.....,p_{n} $$ are the first n primes. Prove that for each positive integer n, there is an integer $$ k_{n} $$ such that $$ x_{n}<k_n^2<x_{n+1} $$ .
Problem 2
Find, with justification, all positive real numbers a, b, c satisfying the system of equations: $$ a\sqrt{b}=a+c,b\sqrt{c}=b+a,c\sqrt{a}=c+b $$
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