IMO Shortlist 1966 problem 63
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Let
be a triangle, and let
,
,
be three points in the interiors of the sides
,
,
of this triangle. Prove that the area of at least one of the three triangles
,
,
is less than or equal to one quarter of the area of triangle
.
Alternative formulation: Let
be a triangle, and let
,
,
be three points on the segments
,
,
, respectively. Prove that
,
where the abbreviation
denotes the (non-directed) area of an arbitrary triangle
.
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.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)
![AQR](/media/m/0/b/9/0b9d88c62dec426ecd50b78d94deb9ba.png)
![BRP](/media/m/8/d/9/8d909025dae92a0fdef40654f99611b6.png)
![CPQ](/media/m/3/f/a/3fa3bde551a500e7f67ac1706c8a3256.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
Alternative formulation: Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
![Q](/media/m/4/5/c/45ce8d14aa1eb54f755fd8e332280abd.png)
![R](/media/m/4/d/7/4d76ce566584cfe8ff88e5f3e8b8e823.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)
![\min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|](/media/m/f/d/4/fd4cd74b538ec28e6fe33c7128fc8cfc.png)
where the abbreviation
![\left|P_1P_2P_3\right|](/media/m/7/5/e/75eb2a10a52f41ed67fe3fb49fc9629e.png)
![P_1P_2P_3](/media/m/3/5/5/3556b2178ef8de9f97d48162d503e6c2.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1966