1975 IMO Problems/Problem 6
Contents
Problem
Find all polynomials , in two variables, with the following properties:
(i) for a positive integer and all real (that is, is homogeneous of degree ),
(ii) for all real ,
(iii)
Solution 1
(i) If : Clearly no solution (ii) If : the identification yields directly (iii) If , is divisible by It is then easy to see that , of degree verifies all the equations.
The only solutions are thus
The above solution was posted and copyrighted by mathmanman. The original thread for this problem can be found here: [1]
Solution 2
is a factor of .
We may write
Thus
We may rewrite it as is a polynomial in of degree for any two fixed ,which has infinitely many zeroes,i.e,.Thus holds for all .In particular it holds for ,i.e, .Now consider the polynomial .Suppose that its not the zero polynomial.Then its degree is defined.With it can be wriiten as .But has infinitely many zeroes and this forces ,forcing to be a zero polynomial.Contradiction!.Thus and are identical polynomials.This forces .With we get .Thus
The above solution was posted and copyrighted by JackXD. The original thread for this problem can be found here: [2]
See Also
1974 IMO (Problems) • Resources | ||
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