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Difference between revisions of "1971 IMO Problems/Problem 1"

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==Problem==
 
Prove that the following assertion is true for <math>n=3</math> and <math>n=5</math>, and that it is false for every other natural number <math>n>2:</math>
 
Prove that the following assertion is true for <math>n=3</math> and <math>n=5</math>, and that it is false for every other natural number <math>n>2:</math>
  
 
If <math>a_1, a_2,\cdots, a_n</math> are arbitrary real numbers, then <math>(a_1-a_2)(a_1-a_3)\cdots (a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots (a_2-a_n)+\cdots+(a_n-a_1)(a_n-a_2)\cdots (a_n-a_{n-1})\ge 0.</math>
 
If <math>a_1, a_2,\cdots, a_n</math> are arbitrary real numbers, then <math>(a_1-a_2)(a_1-a_3)\cdots (a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots (a_2-a_n)+\cdots+(a_n-a_1)(a_n-a_2)\cdots (a_n-a_{n-1})\ge 0.</math>
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==Solution==
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Take <math>a_1 < 0</math>, and the remaining <math>a_i = 0</math>. Then <math>E_n = a_1(n-1) < 0</math> for <math>n</math> even,
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so the proposition is false for even <math>n</math>.
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Suppose <math>n \ge 7</math> and odd. Take any <math>c > a > b</math>, and let <math>a_1 = a</math>, <math>a_2 = a_3 = a_4= b</math>,
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and <math>a_5 = a_6 = ... = a_n = c</math>.
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Then <math>E_n = (a - b)^3 (a - c)^{n-4} < 0</math>.
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So the proposition is false for odd <math>n \ge 7</math>.
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Assume <math>a_1 \ge a_2 \ge a_3</math>.
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Then in <math>E_3</math> the sum of the first two terms is non-negative, because <math>a_1 - a_3 \ge a_2 - a3</math>.
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The last term is also non-negative.
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Hence <math>E_3 \ge 0</math>, and the proposition is true for <math>n = 3</math>.
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It remains to prove <math>S_5</math>.
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Suppose <math>a_1 \ge a_2 \ge a_3 \ge a_4 \ge a_5</math>.
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Then the sum of the first two terms in <math>E_5</math> is
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<math>(a_1 - a_2){(a_1 - a_3)(a_1 - a_4)(a_1 - a_5) - (a_2 - a_3)(a_2 - a_4)(a_2 - a_5)} \ge 0</math>.
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The third term is non-negative (the first two factors are non-positive and the last two non-negative).
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The sum of the last two terms is:
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<math>(a_4 - a_5){(a_1 - a_5)(a_2 - a_5)(a_3 - a_5) - (a_1 - a_4)(a_2 - a_4)(a_3 - a_4)} \ge 0</math>.
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Hence <math>E_5 \ge 0</math>.
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This solution was posted and copyrighted by e.lopes. The original thread can be found here: [https://aops.com/community/p366761]
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==See Also==
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{{IMO box|year=1971|before=First Question|num-a=2}}

Latest revision as of 23:16, 18 July 2024

Problem

Prove that the following assertion is true for $n=3$ and $n=5$, and that it is false for every other natural number $n>2:$

If $a_1, a_2,\cdots, a_n$ are arbitrary real numbers, then $(a_1-a_2)(a_1-a_3)\cdots (a_1-a_n)+(a_2-a_1)(a_2-a_3)\cdots (a_2-a_n)+\cdots+(a_n-a_1)(a_n-a_2)\cdots (a_n-a_{n-1})\ge 0.$

Solution

Take $a_1 < 0$, and the remaining $a_i = 0$. Then $E_n = a_1(n-1) < 0$ for $n$ even, so the proposition is false for even $n$.

Suppose $n \ge 7$ and odd. Take any $c > a > b$, and let $a_1 = a$, $a_2 = a_3 = a_4= b$, and $a_5 = a_6 = ... = a_n = c$. Then $E_n = (a - b)^3 (a - c)^{n-4} < 0$. So the proposition is false for odd $n \ge 7$.

Assume $a_1 \ge a_2 \ge a_3$. Then in $E_3$ the sum of the first two terms is non-negative, because $a_1 - a_3 \ge a_2 - a3$. The last term is also non-negative. Hence $E_3 \ge 0$, and the proposition is true for $n = 3$.

It remains to prove $S_5$. Suppose $a_1 \ge a_2 \ge a_3 \ge a_4 \ge a_5$. Then the sum of the first two terms in $E_5$ is $(a_1 - a_2){(a_1 - a_3)(a_1 - a_4)(a_1 - a_5) - (a_2 - a_3)(a_2 - a_4)(a_2 - a_5)} \ge 0$. The third term is non-negative (the first two factors are non-positive and the last two non-negative). The sum of the last two terms is: $(a_4 - a_5){(a_1 - a_5)(a_2 - a_5)(a_3 - a_5) - (a_1 - a_4)(a_2 - a_4)(a_3 - a_4)} \ge 0$. Hence $E_5 \ge 0$.

This solution was posted and copyrighted by e.lopes. The original thread can be found here: [1]

See Also

1971 IMO (Problems) • Resources
Preceded by
First Question 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 2
All IMO Problems and Solutions
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