IMO Shortlist 2000 problem G5
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Let
be an acute-angled triangle, and let
be the circumcircle of triangle
.
The tangent to the circle
at the point
meets the tangent to the circle
at
at the point
. The line
intersects the line
at
, and
is the midpoint of the segment
.
Similarly, the tangent to the circle
at the point
meets the tangent to the circle
at the point
at the point
. The line
intersects the line
at
, and
is the midpoint of the segment
.
a) Show that
.
b) If
, determine the values of
and
for the triangles
which maximise
.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
The tangent to the circle
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![B^{\prime}](/media/m/f/2/2/f224d52d8ce00cb36346ab94021a247d.png)
![BB^{\prime}](/media/m/0/2/4/02413e7de3a7aca4fca295dd24e2af45.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![E](/media/m/8/b/0/8b01e755d2253cb9a52f9e451d89ec11.png)
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
![BE](/media/m/e/e/2/ee25cd134664bc0c8d7fdbba81e54f90.png)
Similarly, the tangent to the circle
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
![A^{\prime}](/media/m/a/8/4/a84516137f20981f16b27c93610a7150.png)
![AA^{\prime}](/media/m/8/8/8/88851c33932839337050781bd30af3d4.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
a) Show that
![\measuredangle ABM = \measuredangle BAN](/media/m/5/8/9/589593507aa45494adbb8a0333535f42.png)
b) If
![AB = 1](/media/m/3/8/a/38adba4756a30af2973c64bb397c0261.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![AC](/media/m/6/4/7/647ef3a5d68f07d59d84afe03a9dc655.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\measuredangle ABM](/media/m/2/f/4/2f47570c747342495988a4b8af48422d.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2000