Difference between revisions of "1994 IMO Problems/Problem 5"

Revision as of 23:16, 6 October 2023

Problem

Let $S$ be the set of real numbers strictly greater than $-1$ . Find all functions $f:S \to S$ satisfying the two conditions:

1. $f(x+f(y)+xf(y)) = y+f(x)+yf(x)$ for all $x$ and $y$ in $S$ ;

2. $\frac{f(x)}{x}$ is strictly increasing on each of the intervals $-1<x<0$ and $0<x$ .

Solution

The only solution is $f(x) = \frac{-x}{x+1}.$

Setting $x=y,$ we get

$f(x+f(x)+xf(x)) = x+f(x)+xf(x).$

Therefore, $f(s) = s$ for $s = x+f(x)+xf(x).$

Note: If we can show that $s$ is always $0,$ we will get that $x+f(x)+xf(x) = 0$ for all $x$ in $S$ and therefore, $f(x) = \frac{-x}{x+1}.$

Let $f(s)=s.$ Setting $x=y=s,$ we get

$f(s + f(s) + sf(s)) = f(2s + s^2) = 2s+s^2.$

If $t=2s+s^2,$ we have $f(t)=t$ as well.

Consider $s > 0.$ We get $t = 2s + s^2 > s.$ Since $\frac{f(x)}{x}$ is strictly increasing for $x>0$ and $t > s$ in this domain, we must have $\frac{f(t)}{t} > \frac{f(s)}{s}$ but since $f(s) = s$ and $f(t) = t,$ we also have that $\frac{f(s)}{s} = 1 = \frac{f(t)}{t}$ which is a contradiction. Therefore $s \leq 0$

Consider $-1 < s < 0.$ Using a similar argument, we will get that $t = 2s + s^2 < s$ but also $\frac{f(s)}{s} = 1 = \frac{f(t)}{t},$ which is a contradiction.

Hence, $s$ must be $0.$ Since $s=0,$ we can conclude that $x+f(x)+xf(x) = 0$ and therefore, $f(x) = \frac{-x}{x+1}$ for all $x$ .

@@ Line 8: / Line 8: @@
 ==Solution==
-{{solution}}
+The only solution is <math>f(x) = \frac{-x}{x+1}.</math>
+Setting <math>x=y,</math> we get
+<cmath>f(x+f(x)+xf(x)) = x+f(x)+xf(x).</cmath>
+Therefore, <math>f(s) = s</math> for <math>s = x+f(x)+xf(x).</math>
+Note: If we can show that <math>s</math> is always <math>0,</math> we will get that <math>x+f(x)+xf(x) = 0</math> for all <math>x</math> in <math>S</math> and therefore, <math>f(x) = \frac{-x}{x+1}.</math>
+Let <math>f(s)=s.</math> Setting <math>x=y=s,</math> we get
+<cmath>f(s + f(s) + sf(s)) = f(2s + s^2) = 2s+s^2.</cmath>
+If <math>t=2s+s^2,</math> we have <math>f(t)=t</math> as well.
+Consider <math>s > 0.</math> We get <math>t = 2s + s^2 > s.</math> Since <math>\frac{f(x)}{x}</math> is strictly increasing for <math>x>0</math> and <math>t > s</math> in this domain, we must have <math>\frac{f(t)}{t} > \frac{f(s)}{s}</math> but since <math>f(s) = s</math> and <math>f(t) = t,</math> we also have that <math>\frac{f(s)}{s} = 1 = \frac{f(t)}{t}</math> which is a contradiction. Therefore <math>s \leq 0</math>
+Consider <math>-1 < s < 0.</math> Using a similar argument, we will get that <math>t = 2s + s^2 < s</math> but also <math>\frac{f(s)}{s} = 1 = \frac{f(t)}{t},</math> which is a contradiction.
+Hence, <math>s</math> must be <math>0.</math> Since <math>s=0,</math> we can conclude that <math>x+f(x)+xf(x) = 0</math> and therefore, <math>f(x) = \frac{-x}{x+1}</math> for all <math>x</math>.

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Difference between revisions of "1994 IMO Problems/Problem 5"

Revision as of 23:16, 6 October 2023

Problem

Solution