1982 USAMO Problems/Problem 2
Let with real. It is known that if ,
for , or . Determine all other pairs of integers if any, so that holds for all real numbers such that .
This problem needs a solution. If you have a solution for it, please help us out by. Claim Both can not be even.
by equating cofficient of on LHS and RHS ,get
So we have, and .
So we have .
Now since it will true for all real . So choose .
and so .
This is contradiction !! So, atlest one of must be odd. WLOG assume is odd and m is even . The cofficient of in is
The cofficient of in is .
Now choose .
Since holds for all real .
We have ,.
Clearly holds for . Even one can say that for , .
So our answer is .
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