1971 IMO Problems/Problem 1
Prove that the following assertion is true for and , and that it is false for every other natural number
If are arbitrary real numbers, then
Take , and the remaining . Then for even, so the proposition is false for even .
Suppose and odd. Take any , and let , , and . Then . So the proposition is false for odd .
Assume . Then in the sum of the first two terms is non-negative, because . The last term is also non-negative. Hence , and the proposition is true for .
It remains to prove . Suppose . Then the sum of the first two terms in is . 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: . Hence .
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