Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME Find the answer

The function $f$ satisfies $f(x)+f(2 x+y)+5 x y=f(3 x-y)+2 x^{2}+1$ for all real numbers $x, y$ . Determine the value of $f(10)$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Setting $x=10$ and $y=5$ gives $f(10)+f(25)+250=f(25)+200+1$ , from which we get $f(10)=-49$ . Remark: By setting $y=\frac{x}{2}$ , we see that the function is $f(x)=-\frac{1}{2} x^{2}+1$ , and it can be checked that this function indeed satisfies the given equation.

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