2024 USAJMO Problems/Problem 5
Contents
Problem
Find all functions that satisfy
for all
.
Solution 1
Plugging in as
\begin{equation}
f(x^2)=f(f(x))+f(0)
\end{equation}
Plugging in
as
or
Plugging in
as
but since
\begin{equation}
f(-y)+2yf(0)=f(y)
\end{equation}
Plugging in
instead of
in the given equation:
Replacing
and
:
The difference would be:
\begin{equation}
f(x^2-y^2)-f(y^2-x^2)+2y^2f(x)-2x^2f(y)=f(f(x))-f(x^2)-f(f(y))-f(y^2)
\end{equation}
The right-hand side would be
by
Also,
by
So,
is reduced to:
Regrouping and dividing by 2:
Because this holds for all x and y,
is a constant. So,
.
This function must be even, so
.
So, along with
,
for all
, so
, and
.
Plugging in
for
in the original equation, we get:
So,
or
All of these solutions work, so the solutions are
.
-codemaster11
See Also
2024 USAJMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
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