## Problem 1

The sum of 5 real numbers is 8 and the sum of their squares is 16 . What is the largest possible value for one of the numbers?


## Solution

Answer: $16 / 5$.

Let the numbers be $\mathrm{v}, \mathrm{w}, \mathrm{x}, \mathrm{y}, \mathrm{z}$. We have $(\mathrm{v}-6 / 5)^{2}+(\mathrm{w}-6 / 5)^{2}+\ldots+(\mathrm{z}-6 / 5)^{2}=\left(\mathrm{v}^{2}+\ldots+\mathrm{z}^{2}\right)-12 / 5(\mathrm{v}+\ldots$ $+z)+36 / 5=16-96 / 5+36 / 5=4$. Hence $|v-6 / 5| \leq 2$, so $v \leq 16 / 5$. This value can be realized by putting $v=$ $16 / 5$ and setting the other numbers to $6 / 5$.

