Problem - 2815
Let $x$ be a positive number. Denote by $[x]$ the integer part of $x$ and by $\{x\}$ the decimal part of $x$. Find the sum of all positive numbers satisfying $5\{x\} + 0.2[x] = 25$.
The given equation may be rewritten as $\{x\} = \frac{125−[x]}{25}$. From $0\le\{x\}<1$, we have $100 < [x]\le 125$. For each solution $x, x = [x] + \{x\} = 5+ \frac{24}{25} [x]$. It follows that the desired sum is $5(25) + (24/25)(101+ 102 + 103 + \cdots + 125) = \boxed{2837}$.