Difference between revisions of "2010 IMO Problems/Problem 1"
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== Problem == | == Problem == | ||
− | Find all | + | Find all functions <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> such that for all <math>x,y\in\mathbb{R}</math> the following equality holds |
− | |||
<math>f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor </math> | <math>f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor </math> | ||
+ | where <math>\left\lfloor a\right\rfloor </math> is greatest integer not greater than <math>a.</math> | ||
− | + | ''Author: Pierre Bornsztein, France '' | |
== Solutions == | == Solutions == | ||
− | == Solution 1 == | + | === Solution 1 === |
Put <math>x=y=0</math>. Then <math>f(0)=0</math> or <math>\lfloor f(0) \rfloor=1</math>. | Put <math>x=y=0</math>. Then <math>f(0)=0</math> or <math>\lfloor f(0) \rfloor=1</math>. | ||
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So, <math>f(x)=0, \ \forall x</math> or <math>f(x)=a, \ \forall x, \ a \in [1,2)</math>. ( By socrates[http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=10045&]) | So, <math>f(x)=0, \ \forall x</math> or <math>f(x)=a, \ \forall x, \ a \in [1,2)</math>. ( By socrates[http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=10045&]) | ||
− | == Solution 2 == | + | === Solution 2 === |
Substituting <math>y=0</math> we have <math>f(0) = f(x) [f(0)]</math>. | Substituting <math>y=0</math> we have <math>f(0) = f(x) [f(0)]</math>. | ||
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Then the only solutions are <math>f(x)=0</math> or <math>f(x)=c</math> where <math>c \in [1,2)</math>.( By m.candales [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=35756&]) | Then the only solutions are <math>f(x)=0</math> or <math>f(x)=c</math> where <math>c \in [1,2)</math>.( By m.candales [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=35756&]) | ||
− | == Solution 3 == | + | === Solution 3 === |
Let <math>y=0</math>, then <math>f(0)=f(x)\left\lfloor f(0)\right\rfloor</math>. | Let <math>y=0</math>, then <math>f(0)=f(x)\left\lfloor f(0)\right\rfloor</math>. | ||
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) | ) | ||
− | == Solution 4 == | + | === Solution 4 === |
Clearly <math>f(\left\lfloor x\right\rfloor y) = f(\left\lfloor \lfloor x \rfloor \right\rfloor y) = f(\lfloor x \rfloor)\left\lfloor f(y)\right\rfloor</math>, so <math>(f(x) - f(\lfloor x \rfloor))\left\lfloor f(y)\right\rfloor = 0</math> for all <math>x,y\in\mathbb{R}</math>. | Clearly <math>f(\left\lfloor x\right\rfloor y) = f(\left\lfloor \lfloor x \rfloor \right\rfloor y) = f(\lfloor x \rfloor)\left\lfloor f(y)\right\rfloor</math>, so <math>(f(x) - f(\lfloor x \rfloor))\left\lfloor f(y)\right\rfloor = 0</math> for all <math>x,y\in\mathbb{R}</math>. | ||
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Therefore <math>\lfloor f(0) \rfloor \neq 0</math>, and now <math>y=0</math> in the given functional equation yields <math>f(0) = f(x)\lfloor f(0) \rfloor</math> for all <math>x \in \mathbb{R}</math>, therefore <math>f(x) = c \neq 0</math> constant, with <math>\lfloor c \rfloor = \lfloor f(1) \rfloor = 1</math>, i.e. <math>c \in [1,2)</math> (which obviously checks).( By mavropnevma [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=64716&]) | Therefore <math>\lfloor f(0) \rfloor \neq 0</math>, and now <math>y=0</math> in the given functional equation yields <math>f(0) = f(x)\lfloor f(0) \rfloor</math> for all <math>x \in \mathbb{R}</math>, therefore <math>f(x) = c \neq 0</math> constant, with <math>\lfloor c \rfloor = \lfloor f(1) \rfloor = 1</math>, i.e. <math>c \in [1,2)</math> (which obviously checks).( By mavropnevma [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=64716&]) | ||
+ | == Video Solution == | ||
+ | |||
+ | https://youtu.be/rY5YgNnVn7g [Video Solution by little fermat] | ||
− | == See | + | == See Also == |
{{IMO box|year=2010|before=First question|num-a=2}} | {{IMO box|year=2010|before=First question|num-a=2}} | ||
− | [[Category:Olympiad | + | [[Category:Olympiad Algebra Problems]] |
+ | [[Category:Functional Equation Problems]] |
Latest revision as of 12:54, 3 April 2024
Contents
Problem
Find all functions such that for all
the following equality holds
where is greatest integer not greater than
Author: Pierre Bornsztein, France
Solutions
Solution 1
Put . Then
or
.
If
, putting
we get
, that is f is constant.
Substituing in the original equation we find
or
, where
.
If
, putting
we get
or
.
For , we set
to find
, which is a solution.
For , setting
yields
.
Putting to the original we get
.
However, from
we have
, so
which contradicts the fact
.
So, or
. ( By socrates[1])
Solution 2
Substituting we have
.
If
then
. Then
is constant. Let
. Then substituting that in (1) we have
, or
. Therefore
where
or
If then
.
Now substituting
we have
.
If
then
and substituting this in (1) we have
.
Then
.
Substituting
we get
. Then
, which is a contradiction
Therefore
. and then
for all
Then the only solutions are or
where
.( By m.candales [2])
Solution 3
Let , then
.
Case 1:
Then is a constant. Let
, then
. It is easy to check that this are solutions.
Case 2:
In this case we conclude that
Lemma:If is such that
,
Proof of the Lemma: If we have that
, as desired.
Let , so that we have:
, using the lemma.
If is not constant and equal to
, letting
be such that
implies that
.
Now it's enough to notice that any real number is equal to
, where
and
, so that
. Since
was arbitrary, we have that
is constant and equal to
.
We conclude that the solutions are , where
.( By Jorge Miranda [3]
)
Solution 4
Clearly , so
for all
.
If for all
, then by taking
we get
, so
is identically null (which checks).
If, contrariwise, for some
, it follows
for all
.
Now it immediately follows , hence
.
For this implies
.
Assume
; then
, absurd.
Therefore , and now
in the given functional equation yields
for all
, therefore
constant, with
, i.e.
(which obviously checks).( By mavropnevma [4])
Video Solution
https://youtu.be/rY5YgNnVn7g [Video Solution by little fermat]
See Also
2010 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |