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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{{solution}}
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==See Also==
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{{IMO box|year=1963|num-b=First Question|num-a=2}}

Revision as of 15:01, 17 February 2018

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

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See Also

1963 IMO (Problems) • Resources
Preceded by
Problem First Question
1 2 3 4 5 6 Followed by
Problem 2
All IMO Problems and Solutions
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