Difference between revisions of "2005 IMO Shortlist Problems/A2"

Latest revision as of 19:59, 28 March 2010

Problem

(Bulgaria) Let $\mathbb{R}^+$ denote the set of all postive real numbers. Determine all functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that

$\displaystyle f(x)f(y) = 2f(x+yf(x))$

for all positive real numbers $\displaystyle x$ and $\displaystyle y$ .

Solutions

Solution 1

Lemma 1. $f$ is non-decreasing.

Proof. Suppose, on the contrary, that there exist $x> z$ such that $f(x) < f(z)$ . We set $y = \frac{x-z}{f(z)-f(x)} > 0$ , so $x + yf(x) = z + yf(z)$ , to obtain

$f(x)f(y) = 2f(x+yf(x)) = 2f(z+yf(z)) = f(z)f(y)$ ,

or $f(x) = f(z)$ , a contradiction. ∎

Lemma 2. $f$ is not strictly increasing.

Proof. Assume the contrary. Then for all $x ,y$ we have

$f(x)f(y) = 2f(x + yf(x)) > 2f(x)$ ,

so $f(y) > 2$ . Furthermore, since $f$ is injective, we have $f(x+1 \cdot f(x)) = f(x)f(1) = f(1+ x\cdot f(1))$ , which implies $x+ f(x) = 1+ x \cdot f(1))$ , or

$f(x) = (f(1)-1)x + 1$ .

But then for $x$ arbitrarily close to 0, $f(x)$ becomes less than 2, a contradiction. Thus $f$ is not strictly increasing. ∎

Now, let $x>z$ , $f(x) = f(z)$ . If $y$ is in the interval $(0, (x-z)/f(x)$ , then $z< z+yf(z) \le x$ , so

$f(z) \le f(z+yf(z)) \le f(x) = f(z)$ ,

so $f(x+yf(z)) = f(x)$ . It follows that

$f(z)f(y) = 2f(z + yf(z)) = 2f(x) = 2f(z)$ ,

so $f(y) = 2$ , for all $y$ in that interval.

But if ${} f(y_0) = 2$ , then setting $x=y=y_0$ in the original equation gives $f(3^n y_0) = 2$ , by induction. It follows that for any $x$ , there exist $a<x$ and $c>x$ such that $f(a) = f(c) = 2$ , and since $f$ is non-decreasing, we must have $f(x) = 2$ for all $x$ . It is easy to see that this satisfies the given equation. Q.E.D.

Solution 2

Lemma 1. There exist distinct positive $\displaystyle a,b$ such that $\displaystyle f(a) = f(b)$ .

Proof. Suppose the contrary, i.e., suppose $\displaystyle f$ is injective. Then for any $\displaystyle x$ , we have

$f(x+1\cdot f(x)) = f(x)f(1) = f(1 + x\cdot f(1))$ ,

which implies

$\displaystyle x + f(x) = 1 + f(1) \cdot x$
$\displaystyle f(x) = (f(1)-1)x +1$ .

This means that either $\displaystyle f(1)=1$ and $\displaystyle f(x) = 1$ for all $\displaystyle x$ (a contradiction, since that is not injective), or $\displaystyle f(x) = ax+1$ , for some real $\displaystyle a$ . But setting $\displaystyle y=x$ then gives us a quadratic in $\displaystyle x$ with a nonzero leading coefficient, which has at most two real roots, implying that $\displaystyle x$ can only assume two different values, a contradiction. ∎

Lemma 2. There exist $\displaystyle c_0$ and infinitely many $\displaystyle c_n = 3^n \cdot c_0$ such that $\displaystyle f(c_n) = 2$ , for all nonnegative integers $\displaystyle n$ .

Proof. Let $\displaystyle a,b$ be the distinct positive reals of Lemma 1 such that $\displaystyle f(a)=f(b)$ ; without loss of generality, let $\displaystyle a>b$ . Letting $\displaystyle c_0 = \frac{a-b}{f(b)}$ yields

$f(b)f(c_0) = 2f\left( b + \frac{a-b}{f(b)} \right) = 2f(a) = 2f(b)$ .

Since $f(b) \in \mathbb{R}^+$ , this implies $\displaystyle f(c_0) = 2$ . We now prove that $\displaystyle f(c_n) = 2$ for all nonnegative $\displaystyle n$ , by induction. We have just proven our base case. Now, assume $\displaystyle f(c_n) = 2$ . Setting $\displaystyle x=y=c_n$ gives us

$2\cdot 2 = f(c_n)f(c_n) = 2 f(c_n + c_n f(c_n)) = 2f(3c_n) = 2f(c_{n+1})$ ,

so $\displaystyle f(c_{n+1}) = 2$ , as desired. ∎

Lemma 3. For all $x\in \mathbb{R}^+$ , $f(x) \ge 1$ .

Proof. We note that $\displaystyle y = \frac{-x}{f(x)-1}$ is the solution to the equation $\displaystyle y = x + y\cdot f(x)$ , which is positive when $\displaystyle f(x)<1$ , so if this is the case, setting $\displaystyle y$ to this value gives us

$\displaystyle f(x)f(y) = 2f(y)$ ,

and since $\displaystyle f(y)>0$ , this implies, $\displaystyle f(x) = 2 \not< 1$ , a contradiction. ∎

Lemma 4. $\displaystyle f(x) \ge 2$ .

Proof. We will prove by induction that $f(x) \ge 2^{1- \frac{1}{2^n}}$ , for all $\displaystyle n$ . Our base case comes from Lemma 3. Now, if for all $\displaystyle x$ , $\displaystyle f(x) \ge 2^{1-\frac{1}{2^n}}$ , then for some $\displaystyle z$ ,

$[f(x)]^2 = f(x)f(x) = 2f(z) \ge 2 \cdot 2^{1-\frac{1}{2^n}} = 2^{2\left(1 - \frac{1}{2^{n+1}} \right) }$ ,

so $f(x) \ge 2^{\frac{1}{2^{n+1}}}$ , as desired.

Now, suppose there exists some $\displaystyle f(x) < 2$ . Then there exists some $\displaystyle n$ such that $2^{1-\frac{1}{2^n}} > f(x)$ . But this gives us $f(x) \ge 2^{1-\frac{1}{2^n}} > f(x)$ , a contradiction. Thus for all $\displaystyle x$ , $\displaystyle f(x) \ge 2$ . ∎

We will now prove that $\displaystyle f(x) = 2$ is the only solution to the functional equation. Consider any value of $\displaystyle x$ . By Lemma 2, there exists some $\displaystyle {} c_n > x$ such that $\displaystyle f(c_n) = 2$ . Setting $\displaystyle y= \frac{c_n -x}{f(x)}$ then gives us

$f(x) f\left( \frac{c_n -x}{f(x)} \right) = 2f(c_n) = 4$ .

But from Lemma 4, we know $f(x), f\left( \frac{c_n-x}{f(x)} \right) \ge 2$ , so we must have $\displaystyle f(x) = f\left( \frac{c_n -x}{f(x)} \right) = 2$ . This constant function clearly satisfies the given equation. Q.E.D.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

Resources

@@ Line 12: / Line 12: @@
 === Solution 1 ===
-'''Lemma 1.''' <math> \displaystyle f </math> is non-decreasing.
+'''Lemma 1.''' <math>f </math> is non-decreasing.
-''Proof.''  Suppose, on the contrary, that there exist <math> \displaystyle x> z</math> such that <math> \displaystyle f(x) > f(z) </math>.  We set <math> \displaystyle y = \frac{x-z}{f(z)-f(x)} > 0 </math>, so <math> \displaystyle x + yf(x) = z + yf(z) </math>, to obtain
+''Proof.''  Suppose, on the contrary, that there exist <math>x> z</math> such that <math>f(x) < f(z) </math>.  We set <math>y = \frac{x-z}{f(z)-f(x)} > 0 </math>, so <math>x + yf(x) = z + yf(z) </math>, to obtain
 <center>
-<math> \displaystyle f(x)f(y) = 2f(x+yf(x)) = 2f(z+yf(z)) = f(z)f(y) </math>,
+<math>f(x)f(y) = 2f(x+yf(x)) = 2f(z+yf(z)) = f(z)f(y) </math>,
 </center>
-or <math> \displaystyle f(x) = f(z) </math>,  a contradiction.  {{Halmos}}
+or <math>f(x) = f(z) </math>,  a contradiction.  {{Halmos}}
-'''Lemma 2.''' <math> \displaystyle f </math> is not strictly increasing.
+'''Lemma 2.''' <math>f </math> is not strictly increasing.
-''Proof.'' Assume the contrary.  Then for all <math> \displaystyle x ,y </math> we have
+''Proof.'' Assume the contrary.  Then for all <math>x ,y </math> we have
-<center> <math> \displaystyle f(x)f(y) = 2f(x + yf(x)) > 2f(x) </math>, </center>
+<center> <math>f(x)f(y) = 2f(x + yf(x)) > 2f(x) </math>, </center>
-so <math> \displaystyle f(y) > 2 </math>.  Furthermore, since <math> \displaystyle f </math> is [[injective]], we have <math> \displaystyle f(x+1 \cdot f(x)) = f(x)f(1) = f(1+ x\cdot f(1)) </math>, which implies <math> \displaystyle x+ f(x) = 1+ x \cdot f(1)) </math>, or
+so <math>f(y) > 2 </math>.  Furthermore, since <math>f </math> is [[injective]], we have <math>f(x+1 \cdot f(x)) = f(x)f(1) = f(1+ x\cdot f(1)) </math>, which implies <math>x+ f(x) = 1+ x \cdot f(1)) </math>, or
-<center> <math> \displaystyle f(x) = (f(1)-1)x + 1 </math>. </center>
+<center> <math>f(x) = (f(1)-1)x + 1 </math>. </center>
-But then for <math> \displaystyle x </math> arbitrarily close to 0, <math> \displaystyle f(x) </math> becomes less than 2, a contradiction.  Thus <math> \displaystyle f </math> is not strictly inreasing.  {{Halmos}}
+But then for <math>x </math> arbitrarily close to 0, <math>f(x) </math> becomes less than 2, a contradiction.  Thus <math>f </math> is not strictly increasing.  {{Halmos}}
-Now, let <math> \displaystyle x>z </math>, <math> \displaystyle f(x) = f(z) </math>.  If <math> \displaystyle y </math> is in the interval <math> \displaystyle (0, (x-z)/f(x) </math>, then <math> z< z+yf(z) \le x </math>, so
+Now, let <math>x>z </math>, <math>f(x) = f(z) </math>.  If <math>y </math> is in the interval <math>(0, (x-z)/f(x) </math>, then <math> z< z+yf(z) \le x </math>, so
 <center>
 <math> f(z) \le f(z+yf(z)) \le f(x) = f(z) </math>, </center>
-so <math> \displaystyle f(x+yf(z)) = f(x) </math>.  It follows that
+so <math>f(x+yf(z)) = f(x) </math>.  It follows that
 <center>
-<math> \displaystyle f(z)f(y) = 2f(z + yf(z)) = 2f(x) = 2f(z) </math>,
+<math>f(z)f(y) = 2f(z + yf(z)) = 2f(x) = 2f(z) </math>,
 </center>
-so <math> \displaystyle f(y) = 2 </math>, for all <math> \displaystyle y </math> in that interval.
+so <math>f(y) = 2 </math>, for all <math>y </math> in that interval.
-But if <math> \displaystyle  {} f(y_0) = 2 </math>, then setting <math> \displaystyle x=y=y_0 </math> in the original equation gives <math> \displaystyle f(3^n y_0) = 2 </math>, by induction.  It follows that for any <math> \displaystyle x </math>, there exist <math> \displaystyle a<x </math>  and <math> \displaystyle c>x </math> such that <math> \displaystyle f(a) = f(c) = 2 </math>, and since <math> \displaystyle f </math> is non-decreasing, we must have <math> \displaystyle f(x) = 2 </math> for all <math> \displaystyle x </math>.  It is easy to see that this satisfies the given equation.  Q.E.D.
+But if <math> {} f(y_0) = 2 </math>, then setting <math>x=y=y_0 </math> in the original equation gives <math>f(3^n y_0) = 2 </math>, by induction.  It follows that for any <math>x </math>, there exist <math>a<x </math>  and <math>c>x </math> such that <math>f(a) = f(c) = 2 </math>, and since <math>f </math> is non-decreasing, we must have <math>f(x) = 2 </math> for all <math>x </math>.  It is easy to see that this satisfies the given equation.  Q.E.D.
 === Solution 2 ===

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