Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.5 AIME Prove it Philippines

Problem:
A function $f(x)$ satisfies
$(2-x) f(x)-2 f(3-x)=-x^{3}+5 x-18$
for all real numbers $x$ . Solve for $f(0)$ .

Solution

Solution:
Set $x=0$ and $x=3$ to obtain the system
$\begin{cases} 2 f(0)-2 f(3) = -18 \\ -f(3)-2 f(0) = -30 \end{cases}$
We can solve for $f(0)$ from the system and obtain $f(0)=7$ .

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