HMMT February

Algebra Difficulty 5.2 AIME, harder Find the answer

Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$ , and $c$ . If $P(1,2,3)=1$ , compute $P(2,4,8)$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Since $P(a, a, b)=0,(x-y)$ is a factor of $P$ , which means $(y-z)$ and $(z-x)$ are also factors by the symmetry of the polynomial. So, $\frac{P(x, y, z)}{(x-y)(y-z)(z-x)}$ is a symmetric homogeneous degree 1 polynomial, so it must be $k(x+y+z)$ for some real $k$ . So, the answer is $\frac{P(2,4,8)}{P(1,2,3)}=\frac{(2+4+8)(2-4)(4-8)(8-2)}{(1+2+3)(1-2)(2-3)(3-1)}=56$

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