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, 2007

Algebra Difficulty 5.9 AIME, harder Prove it Japan

Find one of the polynomials $f(x, y, z)$ whose degree is $3$ , with real coefficients, that satisfy the following conditions.
• $f(x, y, z) + x$ is divisible by $y + z$
• $f(x, y, z) + y$ is divisible by $z + x$
• $f(x, y, z) + z$ is divisible by $x + y$
A polynomial $P(x, y, z)$ is divisible by a polynomial $Q(x, y, z)$ means that there exists a polynomial $R(x, y, z)$ that satisfies $P(x, y, z) = Q(x, y, z)R(x, y, z)$ .

Solution

$f(x, y, z) = k(x + y)(y + z)(z + x) - x - y - z$ , ( $k \neq 0$ ) satisfies the conditions (in fact, only this form is a solution).

Put $g(x, y, z) = f(x, y, z) + x + y + z$ . Then the conditions are equivalent to the condition that $g(x, y, z)$ is divisible by $x + y$ , $y + z$ , $z + x$ . And the condition that the degree of $f(x, y, z)$ is $3$ and $f$ is with real coefficients are equivalent to the condition that the degree of $g(x, y, z)$ is $3$ and $g$ is with real coefficients.

Now $g(x, y, z) = k(x + y)(y + z)(z + x)$ , ( $k \neq 0$ ) satisfies all the conditions. Then $f(x, y, z) = k(x + y)(y + z)(z + x) - x - y - z$ , ( $k \neq 0$ ) satisfies all the conditions too.

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Source: MathNet, licensed CC-BY-4.0. Statement and solution reproduced as published; topic and difficulty added by this site.