Question 80
A polynomial in two variables $x$, $y$ is called homogeneous of degree $n \geq 0$ if all its terms are constant multiples of $x^k y^{n-k}$ for some integer $k$ with $0 \leq k \leq n$. Fix $n \geq 1$ and suppose that $P(x, y)$ and $Q(x, y)$ are two homogeneous polynomials of degree $n$, such that $P(k, n-k) = Q(n-k, k)$ for all integers $k$ satisfying $0 \leq k \leq n$. Prove that $P(x, y) = Q(y, x)$ for all $x$ and $y$.