Difference between revisions of "2014 USAMO Problems/Problem 2"
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Then we can let <math>k</math> be anything except 0, and get <math>f(x)</math> is 0 for all <math>x \equiv 3 \pmod{4}</math> except <math>-m^2</math>. Also since <math>x^2f(-x) = f(x)^2</math>, we have <math>f(x) = 0 \Rightarrow f(-x) = 0</math>, so <math>f(x)</math> is 0 for all <math>x \equiv 1 \pmod{4}</math> except <math>m^2</math>. So <math>f(x)</math> is 0 for all <math>x</math> except <math>\pm m^2</math>. Since <math>f(m) \neq 0</math>, <math>m = \pm m^2</math>. Squaring, <math>m^2 = m^4</math> and dividing by <math>m</math>, <math>m = m^3</math>. Since <math>f(m^3) = 0</math>, <math>f(m) = 0</math>, which is a contradiction for <math>m \neq 1</math>. However, if we plug in <math>x = 1</math> with <math>f(1) = 1</math> and <math>y</math> as an arbitrary large number with <math>f(y) = 0</math> into the original equation, we get <math>0 = 1</math> which is a clear contradiction, so our only solutions are <math>f(x) = 0</math> and <math>f(x) = x^2</math>. | Then we can let <math>k</math> be anything except 0, and get <math>f(x)</math> is 0 for all <math>x \equiv 3 \pmod{4}</math> except <math>-m^2</math>. Also since <math>x^2f(-x) = f(x)^2</math>, we have <math>f(x) = 0 \Rightarrow f(-x) = 0</math>, so <math>f(x)</math> is 0 for all <math>x \equiv 1 \pmod{4}</math> except <math>m^2</math>. So <math>f(x)</math> is 0 for all <math>x</math> except <math>\pm m^2</math>. Since <math>f(m) \neq 0</math>, <math>m = \pm m^2</math>. Squaring, <math>m^2 = m^4</math> and dividing by <math>m</math>, <math>m = m^3</math>. Since <math>f(m^3) = 0</math>, <math>f(m) = 0</math>, which is a contradiction for <math>m \neq 1</math>. However, if we plug in <math>x = 1</math> with <math>f(1) = 1</math> and <math>y</math> as an arbitrary large number with <math>f(y) = 0</math> into the original equation, we get <math>0 = 1</math> which is a clear contradiction, so our only solutions are <math>f(x) = 0</math> and <math>f(x) = x^2</math>. | ||
+ | ==Alternative Solution== | ||
+ | Given that the range of f consists entirely of integers, it is clear that the LHS must be an integer and <math>f(yf(y))</math> must also be an integer, therefore <math>\frac{f(x)^2}{x}</math> is an integer. If <math>x</math> divides <math>f(x)^2</math> for all integers <math>x \ne 0</math>, then <math>x</math> must be a factor of <math>f(x)</math>, therefore <math>f(0)=0</math>. Now, by setting <math>y=0</math> in the original equation, we have <math>xf(-x)=\frac{f(x)^2}{x}</math>. Assuming <math>x \ne 0</math>, we have <math>x^2f(-x)=f(x)^2</math>. Substituting in <math>-x</math> for <math>x</math> gives us <math>x^2f(x)=f(-x)^2</math>. Substituting in <math>\frac{f(x)^2}{x^2}</math> in for <math>f(-x)</math> in the second equation gives us <math>x^2f(x)=\frac{f(x)^4}{x^4}</math>, so <math>x^6f(x)=f(x)^4</math>. In particular, if <math>f(x) \ne 0</math>, then we have <math>f(x)^3=x^6</math>, therefore <math>f(x)=0, x^2</math> are the only possible solutions. | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] | ||
[[Category:Functional Equation Problems]] | [[Category:Functional Equation Problems]] |
Revision as of 10:13, 4 January 2019
Problem
Let be the set of integers. Find all functions
such that
for all
with
.
Solution
Note: This solution is kind of rough. I didn't want to put my 7-page solution all over again. It would be nice if someone could edit in the details of the expansions.
Lemma 1: .
Proof: Assume the opposite for a contradiction. Plug in
(because we assumed that
),
. What you get eventually reduces to:
which is a contradiction since the LHS is divisible by 2 but not 4.
Then plug in into the original equation and simplify by Lemma 1. We get:
Then:
Therefore, must be 0 or
.
Now either is
for all
or there exists
such that
. The first case gives a valid solution. In the second case, we let
in the original equation and simplify to get:
But we know that
, so:
Since
is not 0,
is 0 for all
(including 0). Now either
is 0 for all
, or there exists some
such that
. Then
must be odd. We can let
in the original equation, and since
is 0 for all
, stuff cancels and we get:
[b]for
.[/b]
Now, let
and we get:
Now, either both sides are 0 or both are equal to
. If both are
then:
which simplifies to:
Since
and
is odd, both cases are impossible, so we must have:
Then we can let
be anything except 0, and get
is 0 for all
except
. Also since
, we have
, so
is 0 for all
except
. So
is 0 for all
except
. Since
,
. Squaring,
and dividing by
,
. Since
,
, which is a contradiction for
. However, if we plug in
with
and
as an arbitrary large number with
into the original equation, we get
which is a clear contradiction, so our only solutions are
and
.
Alternative Solution
Given that the range of f consists entirely of integers, it is clear that the LHS must be an integer and must also be an integer, therefore
is an integer. If
divides
for all integers
, then
must be a factor of
, therefore
. Now, by setting
in the original equation, we have
. Assuming
, we have
. Substituting in
for
gives us
. Substituting in
in for
in the second equation gives us
, so
. In particular, if
, then we have
, therefore
are the only possible solutions.