Difference between revisions of "1975 IMO Problems/Problem 6"

Revision as of 10:47, 25 February 2017

Find all polynomials $P$ , in two variables, with the following properties:

(i) for a positive integer $n$ and all real $t, x, y$ $P(tx, ty) = t^nP(x, y)$ (that is, $P$ is homogeneous of degree $n$ ),

(ii) for all real $a, b, c$ , $P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,$

(iii) $P(1, 0) = 1.$

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-Find all polynomials P; in two variables, with the following properties:
+Find all polynomials <math>P</math>, in two variables, with the following properties:
-(i) for a positive integer n and all real t, x, y
-   P(tx, ty)=(t^n)P(x, y)
+(i) for a positive integer <math>n</math> and all real <math>t, x, y</math> <cmath>P(tx, ty) = t^nP(x, y)</cmath> (that is, <math>P</math> is homogeneous of degree <math>n</math>),
+(ii) for all real <math>a, b, c</math>, <cmath>P(b + c, a) + P(c + a, b) + P(a + b, c) = 0,</cmath>
+(iii) <cmath>P(1, 0) = 1.</cmath>