Problem - 3265
Let $P(x)$ be a monic polynomial of degree 3. (Monic here means that the coefficient of $x^3$ is 1.) Suppose that the remainder when $P(x)$ is divided by $x^2 - 5x+6$ equals 2 times the remainder when $P(x)$ is divided by $x^2 - 5x + 4$. If $P(0) = 100$, what is $P(5)$?
There exist numbers $a, b, c,$ and $d$ such that $$(x + c)(x^2 - 5x + 4) + ax + b = P(x) = (x + d)(x^2 - 5x + 6) + 2ax + 2b$$
Then $4c + b = 100 = 6d + 2b$. Now $P(5) = 4(5 + c) + 5a + b = 120 + 5a$ and $P(5) = 6(5 + d) + 10a + 2b = 130 + 10a$. Since these are equal, $a = -2$ and $P(5) = \boxed{110}$.