Difference between revisions of "2013 Canadian MO Problems/Problem 1"

Revision as of 00:32, 27 November 2023

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.

Let $F(x)=(x+1)P(x-1)-(x-1)P(x)$

$P(x)=\sum_{i=0}^{n}c_ix^i$

$F(x)=(x+1)\sum_{i=0}^{n}(x-1)^ic_i-(x-1)\sum_{i=0}^{n}c_ix^i$

$F(x)=\sum_{i=0}^{n}x(x-1)^ic_i+\sum_{i=0}^{n}(x-1)^ic_i-\sum_{i=0}^{n}c_ix^{i+1}+\sum_{i=0}^{n}c_ix^i$

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 12: / Line 12: @@
 <math>F(x)=(x+1)\sum_{i=0}^{n}(x-1)^ic_i-(x-1)\sum_{i=0}^{n}c_ix^i</math>
-<math>F(x)=\sum_{i=0}^{n}x(x-1)^ic_i+\sum_{i=0}^{n}(x-1)^ic_i-\sum_{i=0}^{n}c_ix^{i+1}+sum_{i=0}^{n}c_ix^i</math>
+<math>F(x)=\sum_{i=0}^{n}x(x-1)^ic_i+\sum_{i=0}^{n}(x-1)^ic_i-\sum_{i=0}^{n}c_ix^{i+1}+\sum_{i=0}^{n}c_ix^i</math>
 ~Tomas Diaz. orders@tomasdiaz.com
 {{alternate solutions}}