IMO Shortlist 1973 problem 17
Dodao/la:
arhiva2. travnja 2012. ![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
is a set of non-constant functions
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
. Each
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
is defined on the real line and has the form
![f(x)=ax+b](/media/m/d/c/c/dccd03e7b3535f337f65717677d0e8fa.png)
for some real
![a,b](/media/m/7/d/8/7d8bdace47e602448e6040957d8cf923.png)
. If
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
and
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
are in
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
, then so is
![fg](/media/m/0/8/2/082efab275e5a39db7c685a6d72ca607.png)
, where
![fg](/media/m/0/8/2/082efab275e5a39db7c685a6d72ca607.png)
is defined by
![fg(x)=f(g(x))](/media/m/9/5/b/95bd73520debc646354e2cb792ade05b.png)
. If
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
is in
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
, then so is the inverse
![f^{-1}](/media/m/5/9/6/596944925463a79c6b6bf127303d03de.png)
. If
![f(x)=ax+b](/media/m/d/c/c/dccd03e7b3535f337f65717677d0e8fa.png)
, then
![f^{-1}(x)= \frac{x-b}{a}](/media/m/d/a/8/da8004b202b45992dc26f8037eb0bf84.png)
. Every
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
in
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
has a fixed point (in other words we can find
![x_f](/media/m/e/8/f/e8f71d47db6784f89d645a2d5b73d196.png)
such that
![f(x_f)=x_f](/media/m/9/7/7/977b2d2c39ca49a4236cdecb5d905e2b.png)
. Prove that all the functions in
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
have a common fixed point.
%V0
$G$ is a set of non-constant functions $f$. Each $f$ is defined on the real line and has the form $f(x)=ax+b$ for some real $a,b$. If $f$ and $g$ are in $G$, then so is $fg$, where $fg$ is defined by $fg(x)=f(g(x))$. If $f$ is in $G$, then so is the inverse $f^{-1}$. If $f(x)=ax+b$, then $f^{-1}(x)= \frac{x-b}{a}$. Every $f$ in $G$ has a fixed point (in other words we can find $x_f$ such that $f(x_f)=x_f$. Prove that all the functions in $G$ have a common fixed point.
Izvor: Međunarodna matematička olimpijada, shortlist 1973