IMO Shortlist 2010 problem G2
Dodao/la:
arhiva23. lipnja 2013. Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a point interior to triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
(with
![CA \neq CB](/media/m/5/4/0/540858eaa3372dd185bf710db16143ac.png)
). The lines
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
,
![BP](/media/m/e/e/f/eefb4fe46ab8d85b7067c29b24aa4cfc.png)
and
![CP](/media/m/6/3/0/630424587cadeb75669118dab3df6b98.png)
meet again its circumcircle
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
at
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
,
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
, respectively
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
. The tangent line at
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
to
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
meets the line
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
. Show that from
![SC = SP](/media/m/8/5/1/8517d74c844d7f6be5c1850c59b39796.png)
follows
![MK = ML](/media/m/b/7/6/b76f96a50b0c45f2c1e50f6ba7d55307.png)
.
Proposed by Marcin E. Kuczma, Poland
%V0
Let $P$ be a point interior to triangle $ABC$ (with $CA \neq CB$). The lines $AP$, $BP$ and $CP$ meet again its circumcircle $\Gamma$ at $K$, $L$, respectively $M$. The tangent line at $C$ to $\Gamma$ meets the line $AB$ at $S$. Show that from $SC = SP$ follows $MK = ML$.
Proposed by Marcin E. Kuczma, Poland
Izvor: Međunarodna matematička olimpijada, shortlist 2010