Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2007 IMO Shortlist Problems/A4"

(arrgh, didn't have time to finish writing this solution)
 
 
Line 8: Line 8:
 
== Solution ==
 
== Solution ==
  
−
We will show that <math>f(x) = 2x</math> is the unique solution to this equation.  To this end, let <math>g(x) = f(x) - x</math>.
+
We will show that <math>f(x) = 2x</math> is the unique solution to this equation.  To this end, let <math>g(x) = f(x) - x</math>. The given condition then translates to
 +
<cmath> g(x+y+ g(y)) + x+y+g(y) = g(x+y) + x+y + g(y) + y , </cmath>
 +
or
 +
<cmath> g(x+y+g(y)) = g(x+y) + y .</cmath>
  
 
'''Lemma 1.''' The function <math>g</math> is injective.
 
'''Lemma 1.''' The function <math>g</math> is injective.
Line 21: Line 24:
  
 
'''Lemma 3.''' For all <math>a,b</math>, <math>g(a+b) = g(a) + g(b)</math>.
 
'''Lemma 3.''' For all <math>a,b</math>, <math>g(a+b) = g(a) + g(b)</math>.
 +
 +
''Proof.''  Pick an arbitrary positive real <math>c > \max(g(a)+g(b), g(a))</math>.  Then by Lemma 2,
 +
<cmath> g(c+g(a)+g(b)) = g(c+g(a))+b = g(c) + a+b = g(c+g(a+b)) . </cmath>
 +
Since <math>g</math> is injective, it follows that <math>c+g(a)+g(b) = c + g(a+b)</math>.  The lemma then follows.  <math>\blacksquare</math>
 +
 +
Now, let <math>x</math> be any positive real; pick some <math>y>x</math>.  Then by Lemmata 3 and 2,
 +
<cmath> g(x) + g(y) = g(x+y) = g(y) + x . </cmath>
 +
Hence <math>g(x)=  x</math> and <math>f(x) = g(x) + x = 2x</math>.  Therefore the function <math>x \mapsto 2x</math> is the only possible solution to the problem.  Since this function evidently satisfies the problem's condition, it is the unique solution, as desired.  <math>\blacksquare</math>
 +
 +
== Resources ==
 +
 +
* [[2007 IMO Shortlist Problems]]
 +
* <url>viewtopic.php?p=1165901#1165901 Discussion on AoPS/MathLinks</url>
 +
 +
[[Category:Olympiad Algebra Problems]]

Latest revision as of 16:44, 4 September 2008

Problem

(Thailand) Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that \[f(x+f(y)) = f(x+y) + f(y)\] for all $x,y \in \mathbb{R}^+$. (Symbol $\mathbb{R}^+$ denotes the set of all positive real numbers [sic].)

Solution

We will show that $f(x) = 2x$ is the unique solution to this equation. To this end, let $g(x) = f(x) - x$. The given condition then translates to \[g(x+y+ g(y)) + x+y+g(y) = g(x+y) + x+y + g(y) + y ,\] or \[g(x+y+g(y)) = g(x+y) + y .\]

Lemma 1. The function $g$ is injective.

Proof. Suppose $g(a) = g(b)$. Then \[a = g(b+a+g(a)) - g(b+a) = g(a+b+g(b)) - g(a+b) = b,\] as desired. $\blacksquare$

Lemma 2. If $a>b$, then $g(a+g(b)) = g(a) + b$.

Proof. Set $y= b$, $x=a-b$. $\blacksquare$

Lemma 3. For all $a,b$, $g(a+b) = g(a) + g(b)$.

Proof. Pick an arbitrary positive real $c > \max(g(a)+g(b), g(a))$. Then by Lemma 2, \[g(c+g(a)+g(b)) = g(c+g(a))+b = g(c) + a+b = g(c+g(a+b)) .\] Since $g$ is injective, it follows that $c+g(a)+g(b) = c + g(a+b)$. The lemma then follows. $\blacksquare$

Now, let $x$ be any positive real; pick some $y>x$. Then by Lemmata 3 and 2, \[g(x) + g(y) = g(x+y) = g(y) + x .\] Hence $g(x)= x$ and $f(x) = g(x) + x = 2x$. Therefore the function $x \mapsto 2x$ is the only possible solution to the problem. Since this function evidently satisfies the problem's condition, it is the unique solution, as desired. $\blacksquare$

Resources

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_IMO_Shortlist_Problems/A4&oldid=27761"