IMO Shortlist 1996 problem C7
Dodao/la:
arhiva2. travnja 2012. let
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
be a finitive set and
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
and
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
be two injective surjective functions from
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
to
![V](/media/m/5/d/1/5d1544cc9c474ed7006c60d2c6dfebf6.png)
.let
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
and
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be two sets such that they are defined as following
we know that
![S \cup T = V](/media/m/b/b/d/bbd1a2ad997e1053d93d4cf7b3382e89.png)
, prove:
for each
![w \in V : f(w) \in S](/media/m/3/9/0/3909b5c199ff37beb048f282374d12fa.png)
if and only if
%V0
let $V$ be a finitive set and $g$ and $f$ be two injective surjective functions from $V$to$V$.let $T$ and $S$ be two sets such that they are defined as following
$S = \{w \in V: f(f(w)) = g(g(w))\}$
$T = \{w \in V: f(g(w)) = g(f(w))\}$
we know that $S \cup T = V$, prove:
for each $w \in V : f(w) \in S$ if and only if $g(w) \in S$
Izvor: Međunarodna matematička olimpijada, shortlist 1996