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Philippine Mathematical Olympiad

Algebra Difficulty 5.2 AIME, harder Prove it Philippines

Problem:

We say that the constant $a$ is a fixed point of a function $f$ if $f(a)=a$ . Find all values of $c$ such that $f(x)=x^{2}-2$ and $g(x)=2x^{2}-c$ share a common fixed point.

Solution

Solution:

Note that if $f(x)=x$ , then $x^{2}-x-2=0$ , or $(x-2)(x+1)=0$ . Thus, the fixed points of $f$ are $2$ and $-1$ , and so we wish for either of these to be a fixed point of $g$ .

If $2$ is a fixed point of $g$ , we must have $g(2)=2=8-c$ , and so $c=6$ .

If $-1$ is a fixed point of $g$ , then $2-c=-1$ , and $c=3$ .

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