IMO 2015 zadatak 4
Kvaliteta:
Avg: 3,0Težina:
Avg: 6,0 Neka je
opisana kružnica trokuta
i
njeno središte. Kružnica
sa središtem
siječe dužinu
u točkama
i
, tako da su točke
,
,
i
u parovima različite i leže na pravcu
tim redom. Neka su
i
sjecišta kružnica
i
takva da točke
,
,
,
i
leže na kružnici
tim redom. Neka je
drugo sjecište opisane kružnice trokuta
i dužine
. Neka je
drugo sjecište opisane kružnice trokuta
i dužine
.
Pretpostavlja se da su pravci
i
različiti i da se sijeku u točki
. Dokaži da točka
leži na pravcu
.
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![\overline{BC}](/media/m/8/8/1/8818caad7d36e134c54122cbf46f1cd9.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![\Gamma](/media/m/4/e/0/4e08987e1d0700578a2eb5c2fc65dc3b.png)
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![F](/media/m/3/e/8/3e8bad5df716d332365fca76f53c1743.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
![\Omega](/media/m/b/5/7/b57bf55357e41163629a7a5e4a145f2b.png)
![K](/media/m/e/1/e/e1ed1943d69f4d6a840e99c7bd199930.png)
![BDF](/media/m/f/5/0/f5003fffaa52534b09542e985cfb81b0.png)
![\overline{AB}](/media/m/a/1/a/a1a42310b1a849922197735f632d57ec.png)
![L](/media/m/f/c/1/fc1ae4eb78da7d1352cbf1f8217ab286.png)
![CGE](/media/m/8/9/3/89317da92a915b649690d3e022e4b3b0.png)
![\overline{CA}](/media/m/c/e/9/ce9fb8497710464615e1d00d148c5663.png)
Pretpostavlja se da su pravci
![FK](/media/m/1/8/5/185a554a2095520f3032542c84f6aa69.png)
![GL](/media/m/4/d/0/4d0810e955d25de2b306692cbde18c2b.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![AO](/media/m/d/9/3/d93e4e1fde6437bd5210d0a50abb3ca8.png)
Izvor: Međunarodna matematička olimpijada 2015, drugi dan