IMO Shortlist 2003 problem G7
Kvaliteta:
Avg: 0,0Težina:
Avg: 9,0 Let
be a triangle with semiperimeter
and inradius
. The semicircles with diameters
,
,
are drawn on the outside of the triangle
. The circle tangent to all of these three semicircles has radius
. Prove that
Alternative formulation. In a triangle
, construct circles with diameters
,
, and
, respectively. Construct a circle
externally tangent to these three circles. Let the radius of this circle
be
.
Prove:
, where
is the inradius and
is the semiperimeter of triangle
.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
![\frac{s}{2}<t\le\frac{s}{2}+\left(1-\frac{\sqrt{3}}{2}\right)r.](/media/m/0/d/7/0d7d34fa067c4e42cd569aa9c814ba9e.png)
Alternative formulation. In a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![w](/media/m/a/7/a/a7abf250ebf14efa424fde966849d5f9.png)
![t](/media/m/7/f/6/7f630d3904cfcd77d22bd7938423df6c.png)
Prove:
![\frac{s}{2}<t\le\frac{s}{2}+\frac12\left(2-\sqrt3\right)r](/media/m/3/3/e/33ea79cb660b9ba799a6d7d41b84798c.png)
![r](/media/m/3/d/f/3df7cc5bbfb7b3948b16db0d40571068.png)
![s](/media/m/9/0/8/908014cbadb69e42261a56b450a375b9.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
Izvor: Međunarodna matematička olimpijada, shortlist 2003