# 1999 USAMO Problems/Problem 4

## Problem

Let $a_{1}, a_{2}, \dots, a_{n}$ ($n > 3$) be real numbers such that $$a_{1} + a_{2} + \cdots + a_{n} \geq n \qquad \mbox{and} \qquad a_{1}^{2} + a_{2}^{2} + \cdots + a_{n}^{2} \geq n^{2}.$$ Prove that $\max(a_{1}, a_{2}, \dots, a_{n}) \geq 2$.

## Solution

### Solution 1

First, suppose all the $a_i$ are positive. Then $$\max(a_1, \dotsc, a_n) \ge \sqrt{\frac{a_1^2 + \dotsb + a_n^2}{n}} \ge \sqrt{n} \ge 2 .$$ Suppose, on the other hand, that without loss of generality, $$a_1 \ge a_2 \ge \dotsb \ge a_k \ge 0 > a_{k+1} \ge \dotsb \ge a_n,$$ with $1\le k . If $a_1 >2$ we are done, so suppose that $a_1 \le2$. Then $\sum_{i=1}^k a_i \le 2k$, so $$\sum_{i=k+1}^n -a_i \le 2k-n .$$ Since $-a_i$ is a positive real for all $k+1 \le i \le n$, it follows that $$\sum_{i=k+1}^n a_i^2 \le \left( \sum_{i=k+1}^n-a_i \right)^2 \le (2k-n)^2 .$$ Then \begin{align*} \max(a_1, \dotsc, a_n)^2 &\ge \sum_{i=1}^k a_i^2 /k \\ &\ge \left( n^2 - \sum_{i=k+1}^n a_i^2 \right)/k \\ &\ge \frac{n^2 - (2k-n)^2}{k} = 4(n-k). \end{align*} Since $k, $4(n-k) > 4$. It follows that $\max(a_1, \dotsc, a_n) \ge \sqrt{4} = 2$, as desired. $\blacksquare$

### Solution 2

Assume the contrary and suppose each $a_i$ is less than 2.

Without loss of generality let $a_1 \le a_2 \le a_3 \le ... \le a_n < 2$, and let $k$ be the largest integer such that $a_k \le 0$ and $0 \le a_{k+1}$ if it exists, or 0 if all the $a_i$ are non-negative. If $k = 0$, then (as $n \ge 4$) $\sum_{i=1}^n a_i^2 < \sum_{i=1}^n 4 = 4n \le n^2$, a contradiction. Hence, assume $n \ge k \ge 1$. Then $$\sum_{i=1}^k a_i \ge n - \sum_{i=k+1}^n a_i > n - 2(n-k) = 2k - n.$$ Because $a_i \le 0$ for $i \le k$, both sides of the inequality are non-positive, so squaring flips the sign. But we also know that $a_i a_j \ge 0$ for $i, j \le k$, so $$\sum_{i=1}^k a_i^2 \le \left(\sum_{i=1}^k a_i \right)^2 < (2k - n)^2 = 4k^2 - 4kn + n^2,$$ which results in $$\sum_{i=1}^n a_i^2 = \sum_{i=1}^k a_i^2 + \sum_{i=k+1}^n a_i^2 < 4k^2 - 4kn + n^2 + 4(n-k) = n^2 - 4(k-1)(n-k) \le n^2,$$ a contradiction to our given condition. The proof is complete.