Mock AIME 1 2007-2008 Problems/Problem 13

Revision as of 19:16, 2 April 2008 by Azjps (talk | contribs) (<blank edit> solution incomplete, rigor necessary. Could someone provide?)

Problem

Let $F(x)$ be a polynomial such that $F(6) = 15$ and \[\frac{F(3x)}{F(x+3)} = 9-\frac{48x+54}{x^2+5x+6}\] for $x \in \mathbb{R}$ such that both sides are defined. Find $F(12)$.

Solution

Combining denominators and simplifying, \[\frac{F(3x)}{F(x+3)} = \frac{9(x^2+5x+6)-48x-54}{x^2+5x+6} = \frac{9x^2 - 3x}{x^2 + 5x + 6}= \frac{3x(3x-1)}{(x+3)(x+2)}\] It becomes obvious that $F(x) = ax(x-1)$, for some constant $a$, matches the definition of the polynomial. To prove that $F(x)$ must have this form, note that \[(x+3)(x+2)F(3x) = 3x(3x-1)F(x+3)\] (rigor needed)

By the given, $F(6) = a(6)(5) = 15 \Longrightarrow a = \frac 12$. Thus, $F(12) = \frac{1}{2}(12)(11) = \boxed{066}$.

Template:Incomplete

See also

Mock AIME 1 2007-2008 (Problems, Source)
Preceded by
Problem 12
Followed by
Problem 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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