Math question related to functional equations

Question

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

What have you determined about this function so far? – John Wayland Bales Apr 11 '18 at 02:49 — John Wayland Bales, Apr 11 '18 at 02:49

score 2 · Answer 1 · answered Apr 11 '18 at 02:50

2

Let $x=10, y=5$. Then the condition becomes $$f(10)+f(25)+250=f(25)+201$$ Which implies that $f(10)=-49$.

answered Apr 11 '18 at 02:50

pwerth

1

@Will Jagy and pwerth: Putting $x=2y$ the equation gives $f(2y)=-2y^2+1$ since in this case $2x+y=3x-y$. Therefore the solution above is the only one. One should however check that it is a solution. – Jens Schwaiger Apr 11 '18 at 04:20
@JensSchwaiger that's a good trick. The one thing I completed was assume $f(w) = a w^2 + b w + c,$ find the coefficients and confirm that it worked. I've never taught this kind of contest problem, i just know quadratic forms, so the problem caught my eye – Will Jagy Apr 11 '18 at 15:48

score 1 · Accepted Answer · answered Apr 11 '18 at 03:02

1

meanwhile, there really is such a function, $$ f(w) = -\frac{1}{2} w^2 + 1 $$ works

answered Apr 11 '18 at 03:02

Will Jagy

2 Answers2