Kružnica
![K_0](/media/m/4/4/9/4495396974fa91e54fbf1f9965940fc2.png)
ima polumjer
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
i
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
je točka na toj kružnici. Kružnica
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
ima polumjer
![r<1](/media/m/1/7/d/17de95b36dfdb2f212f160b9fde71516.png)
i s unutrašnje strane dira
![K_0](/media/m/4/4/9/4495396974fa91e54fbf1f9965940fc2.png)
u točki
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
. Točka
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
leži na kružnici
![90^{\circ}](/media/m/2/9/4/29404077a84f1539d9b7d5dcccb02023.png)
u smjeru obrnutom od smjera kazaljke na satu s obzirom na
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
. Kružnica
![K_2](/media/m/3/1/d/31d4e457d9fd74913d14bd19a565ce00.png)
ima polumjer
![r^2](/media/m/3/f/9/3f9dd2e5318a5ff9912cd70a759fa6cf.png)
i s unutrašnje strane dira
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
u točki
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
. Na navedeni način konstruiran je niz kružnica
![K_1,K_2,K_3,...](/media/m/5/8/6/5862448bf3f0b773364f22bd72b28613.png)
i točaka
![A_1,A_2,A_3,...](/media/m/4/6/0/4607eecd5c10e1c946657ce3070e8912.png)
na navedenim kružnicama, pri čemu kružnica
![K_n](/media/m/7/a/b/7ab875a68e7151ae2a463a233f3695dd.png)
ima radijus
![r^n](/media/m/4/3/4/43497581556b4d8b1ca3afae0631f556.png)
i iznutra dodiruje kružnicu
![K_{n-1}](/media/m/3/7/8/378a4d4d298d028e0a2270e81c529676.png)
u točki
![A_{n-1}](/media/m/1/1/e/11efb314e8b09672fd7279b4721c612a.png)
. Također, točka
![A_n](/media/m/5/2/c/52cc7b12306c4c6a541b1b5322ccf2d6.png)
nalazi se na
![90^{\circ}](/media/m/2/9/4/29404077a84f1539d9b7d5dcccb02023.png)
obrnuto od smjera kazaljke na satu od točke
![A_{n-1}](/media/m/1/1/e/11efb314e8b09672fd7279b4721c612a.png)
, kao što je prikazano na slici.
Postoji točka
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
koja se nalazi unutar svih kružnica. Kada
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
iznosi
![\frac{11}{60}](/media/m/c/1/6/c1618d785d278424d176bc3e7020bcb5.png)
, udaljenost središta
![K_0](/media/m/4/4/9/4495396974fa91e54fbf1f9965940fc2.png)
od
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
je
![\frac{m}{n}](/media/m/1/e/c/1eccd5cda4efb1b60e9314a587458f47.png)
, gdje su
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
i
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
relativno prosti prirodni brojevi. Koliko iznosi
![m+n](/media/m/1/2/c/12cfeebf074af065b2efa21bce4eb0fe.png)
?
Radius of a circle
![K_0](/media/m/4/4/9/4495396974fa91e54fbf1f9965940fc2.png)
is
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
.
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
is a point on that circle. Radius of a circle
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
is
![r<1](/media/m/1/7/d/17de95b36dfdb2f212f160b9fde71516.png)
.
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
touches
![K_0](/media/m/4/4/9/4495396974fa91e54fbf1f9965940fc2.png)
internally at
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
. We choose a point
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
so that it lies
![90^{\circ}](/media/m/2/9/4/29404077a84f1539d9b7d5dcccb02023.png)
counterclockwise from
![A_0](/media/m/2/f/f/2ff029e43a310c5b6d8137a3edb7609c.png)
on
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
. Circle
![K_2](/media/m/3/1/d/31d4e457d9fd74913d14bd19a565ce00.png)
has a radius
![r^2](/media/m/3/f/9/3f9dd2e5318a5ff9912cd70a759fa6cf.png)
and touches
![K_1](/media/m/4/8/7/4879c18473f8b1c4ba76dd53c7cbd958.png)
internally at point
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
. This way we have constructed a sequence of circles
![K_1,K_2,K_3,...](/media/m/5/8/6/5862448bf3f0b773364f22bd72b28613.png)
and a sequence of points
![A_1,A_2,A_3,...](/media/m/4/6/0/4607eecd5c10e1c946657ce3070e8912.png)
on said circles, where circle
![K_n](/media/m/7/a/b/7ab875a68e7151ae2a463a233f3695dd.png)
has a radius
![r^n](/media/m/4/3/4/43497581556b4d8b1ca3afae0631f556.png)
and touches the circle
![K_{n-1}](/media/m/3/7/8/378a4d4d298d028e0a2270e81c529676.png)
internally at
![A_{n-1}](/media/m/1/1/e/11efb314e8b09672fd7279b4721c612a.png)
. Also,
![A_n](/media/m/5/2/c/52cc7b12306c4c6a541b1b5322ccf2d6.png)
lies
![90^{\circ}](/media/m/2/9/4/29404077a84f1539d9b7d5dcccb02023.png)
counterclockwise from
![A_{n-1}](/media/m/1/1/e/11efb314e8b09672fd7279b4721c612a.png)
on the circle
![K_n](/media/m/7/a/b/7ab875a68e7151ae2a463a233f3695dd.png)
, as shown in the figure below.
There exists a point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
inside all these circles. When
![r=\frac{11}{60}](/media/m/d/b/a/dba970be0fc9acdf92962b21e87d333d.png)
, the distance between the center of
![K_0](/media/m/4/4/9/4495396974fa91e54fbf1f9965940fc2.png)
and
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
is
![\frac{m}{n}](/media/m/1/e/c/1eccd5cda4efb1b60e9314a587458f47.png)
, where
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
are relatively prime. Find
![m+n](/media/m/1/2/c/12cfeebf074af065b2efa21bce4eb0fe.png)
.
[lang=hr]
Kružnica $K_0$ ima polumjer $1$ i $A_0$ je točka na toj kružnici. Kružnica $K_1$ ima polumjer $r<1$ i s unutrašnje strane dira $K_0$ u točki $A_0$. Točka $A_1$ leži na kružnici $K_1$ $90^{\circ}$ u smjeru obrnutom od smjera kazaljke na satu s obzirom na $A_0$. Kružnica $K_2$ ima polumjer $r^2$ i s unutrašnje strane dira $K_1$ u točki $A_1$. Na navedeni način konstruiran je niz kružnica $K_1,K_2,K_3,...$ i točaka $A_1,A_2,A_3,...$ na navedenim kružnicama, pri čemu kružnica $K_n$ ima radijus $r^n$ i iznutra dodiruje kružnicu $K_{n-1}$ u točki $A_{n-1}$. Također, točka $A_n$ nalazi se na $K_n$ $90^{\circ}$ obrnuto od smjera kazaljke na satu od točke $A_{n-1}$, kao što je prikazano na slici.
\\
Postoji točka $P$ koja se nalazi unutar svih kružnica. Kada $r$ iznosi $\frac{11}{60}$, udaljenost središta $K_0$ od $P$ je $\frac{m}{n}$, gdje su $m$ i $n$ relativno prosti prirodni brojevi. Koliko iznosi $m+n$?
\begin{center}
\begin{figure} \includegraphics{kruznice.png} \end{figure} \end{center}
[/lang]
[lang=en]
Radius of a circle $K_0$ is $1$. $A_0$ is a point on that circle. Radius of a circle $K_1$ is $r<1$. $K_1$ touches $K_0$ internally at $A_0$. We choose a point $A_1$ so that it lies $90^{\circ}$ counterclockwise from $A_0$ on $K_1$. Circle $K_2$ has a radius $r^2$ and touches $K_1$ internally at point $A_1$. This way we have constructed a sequence of circles $K_1,K_2,K_3,...$ and a sequence of points $A_1,A_2,A_3,...$ on said circles, where circle $K_n$ has a radius $r^n$ and touches the circle $K_{n-1}$ internally at $A_{n-1}$. Also, $A_n$ lies $90^{\circ}$ counterclockwise from $A_{n-1}$ on the circle $K_n$, as shown in the figure below.
\\
There exists a point $P$ inside all these circles. When $r=\frac{11}{60}$, the distance between the center of $K_0$ and $P$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime. Find $m+n$.
\begin{center}
\begin{figure} \includegraphics{kruznice.png} \end{figure} \end{center}
[/lang]