sum a_ia_j(1 - a_ia_j) >= 0 if sum a_1 = sum a_i^2, for a_i>0

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sum a_ia_j(1 - a_ia_j) >= 0 if sum a_1 = sum a_i^2, for a_i>0

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Source: 2015 Saudi Arabia IMO TST II p3

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#1 h Jul 24, 2020, 3:36 PM

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Let $a_1, a_2, ...,a_n$ be positive real numbers such that $a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2$ Prove that $\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge 0$ Võ Quốc Bá Cẩn.

This post has been edited 1 time. Last edited by parmenides51, Jul 24, 2020, 3:36 PM

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NaPrai

#3 h Jul 24, 2020, 4:25 PM • 1 Y

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We may write the inequality as $\sum_{1 \le i<j \le n} a_ia_j \ge \sum_{1 \le i<j \le n} a_i^2a_j^2.$ From the identities $\sum_{1 \le i < j \le n} a_ia_j = \frac{1}{2}\left(\left(\sum_{i=1}^n a_i\right)^2 - \sum_{i=1}^n a_i^2\right), \ \sum_{1 \le i < j \le n} a_i^2a_j^2 = \frac{1}{2}\left(\left(\sum_{i=1}^n a_i^2\right)^2 - \sum_{i=1}^n a_i^4\right)$ and the condition $\sum_{i=1}^n a_i = \sum_{i=1}^n a_i^2$ , the inequality is equivalent to $\sum_{i=1} a_i^4 \ge \sum_{i=1} a_i^2.$ By Hölder's inequality, we have $\left(\sum_{i=1}^n a_i^4\right)\left(\sum_{i=1}^n a_i\right)^2 \ge \left(\sum_{i=1}^n a_i^2\right)^3.$ From $\sum_{i=1}^n a_i = \sum_{i=1}^n a_i^2$ , the inequality becomes $\sum_{i=1} a_i^4 \ge \sum_{i=1} a_i^2$ as we want.

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#4 h Jul 24, 2020, 6:33 PM

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NaPrai wrote:

We may write the inequality as $\sum_{1 \le i<j \le n} a_ia_j \ge \sum_{1 \le i<j \le n} a_i^2a_j^2.$ From the identities $\sum_{1 \le i < j \le n} a_ia_j = \frac{1}{2}\left(\left(\sum_{i=1}^n a_i\right)^2 - \sum_{i=1}^n a_i^2\right), \ \sum_{1 \le i < j \le n} a_i^2a_j^2 = \frac{1}{2}\left(\left(\sum_{i=1}^n a_i^2\right)^2 - \sum_{i=1}^n a_i^4\right)$ and the condition $\sum_{i=1}^n a_i = \sum_{i=1}^n a_i^2$ , the inequality is equivalent to $\sum_{i=1} a_i^4 \ge \sum_{i=1} a_i^2.$ By Hölder's inequality, we have $\left(\sum_{i=1}^n a_i^4\right)\left(\sum_{i=1}^n a_i\right)^2 \ge \left(\sum_{i=1}^n a_i^2\right)^3.$ From $\sum_{i=1}^n a_i = \sum_{i=1}^n a_i^2$ , the inequality becomes $\sum_{i=1} a_i^4 \ge \sum_{i=1} a_i^2$ as we want.

Your solution is awesome! I am kinda new in olympiad inequalities and summation signs really scare me, i would be really thankful if you could share any resource that you used to improve your skills in this topic

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