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, 2021

Algebra Difficulty 5.5 AIME, harder Prove it United States

Problem:
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 $P(2,4,8)$ .

Note: $P(x, y, z)$ is a homogeneous degree 4 polynomial if it satisfies $P(k a, k b, k c) = k^{4} P(a, b, c)$ for all real $k, a, b, c$ .

Solution

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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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.