Difference between revisions of "1999 USAMO Problems/Problem 4"

Latest revision as of 08:47, 20 July 2016

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 $1\le k <n$ . 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 $k<n$ , $4(n-k) > 4$ . 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.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 10: / Line 10: @@
 == Solution ==
+=== Solution 1 ===
 First, suppose all the <math>a_i</math> are positive.  Then
 <cmath> \max(a_1, \dotsc, a_n) \ge \sqrt{\frac{a_1^2 +
@@ Line 19: / Line 20: @@
 <cmath> \sum_{i=k+1}^n -a_i \le 2k-n . </cmath>
 Since <math>-a_i</math> is a positive real for all <math>k+1 \le i \le n</math>, it follows that
-<cmath> \sum_{i=k+1}^n a_i^2 \le \left( \sum_{i=k+1}^n} -a_i \right)^2 \le (2k-n)^2 . </cmath>
+<cmath> \sum_{i=k+1}^n a_i^2 \le \left( \sum_{i=k+1}^n-a_i \right)^2 \le (2k-n)^2 . </cmath>
 Then
 <cmath> \begin{align*}
@@ Line 27: / Line 28: @@
 \end{align*} </cmath>
 Since <math>k<n</math>, <math>4(n-k) > 4</math>.  It follows that <math>\max(a_1, \dotsc, a_n) \ge \sqrt{4} = 2</math>, as desired.  <math>\blacksquare</math>
+===Solution 2===
+Assume the contrary and suppose each <math>a_i</math> is less than 2.
+Without loss of generality let <math>a_1 \le a_2 \le a_3 \le ... \le a_n < 2</math>, and let <math>k</math> be the largest integer such that <math>a_k \le 0</math> and <math>0 \le a_{k+1}</math> if it exists, or 0 if all the <math>a_i</math> are non-negative. If <math>k = 0</math>, then (as <math>n \ge 4</math>) <math>\sum_{i=1}^n a_i^2 < \sum_{i=1}^n 4 = 4n \le n^2</math>, a contradiction. Hence, assume <math>n \ge k \ge 1</math>. Then
+<cmath>\sum_{i=1}^k a_i \ge n - \sum_{i=k+1}^n a_i > n - 2(n-k) = 2k - n.</cmath>
+Because <math>a_i \le 0</math> for <math>i \le k</math>, both sides of the inequality are non-positive, so squaring flips the sign. But we also know that <math>a_i a_j \ge 0</math> for <math>i, j \le k</math>, so
+<cmath>\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,</cmath>
+which results in
+<cmath>\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,</cmath>
+a contradiction to our given condition. The proof is complete.
 {{alternate solutions}}

1999 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
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