IMO Shortlist 1974 problem 10
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Let
be a triangle. Prove that there exists a point
on the side
of the triangle
, such that
is the geometric mean of
and
, iff the triangle
satisfies the inequality
.
CommentAlternative formulation, from IMO ShortList 1974, Finland 2: We consider a triangle
. Prove that:
is a necessary and sufficient condition for the existence of a point
on the segment
so that
is the geometrical mean of
and
.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![DB](/media/m/1/a/c/1ac48340338d92304ef2ae402df64471.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\sin A\sin B\le\sin^2\frac{C}{2}](/media/m/6/2/a/62a2670917b0b8b664cefa2baae42e0d.png)
CommentAlternative formulation, from IMO ShortList 1974, Finland 2: We consider a triangle
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)](/media/m/6/7/4/6748958b583c4a47671a23f33834f110.png)
![D](/media/m/7/0/0/7006c4b57335ab717f8f20960577a9ef.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![AD](/media/m/6/9/6/69672822808d046d0e94ab2fa7f2dc80.png)
![BD](/media/m/1/1/f/11f65a804e5c922ee28a53b1df04d138.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1974