IMO Shortlist 2007 problem G6
Kvaliteta:
Avg: 0,0Težina:
Avg: 8,0 Determine the smallest positive real number
with the following property. Let
be a convex quadrilateral, and let points
,
,
, and
lie on sides
,
,
, and
, respectively. Consider the areas of triangles
,
,
and
; let
be the sum of the two smallest ones, and let
be the area of quadrilateral
. Then we always have
.
Author: unknown author, USA
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![A_1](/media/m/5/a/6/5a6ce1347567551c02239ff8d4ebee67.png)
![B_1](/media/m/5/d/9/5d9518a7c0ead344571aac61b51bb25c.png)
![C_1](/media/m/b/0/b/b0b10dc32c3e01824e0f0b6753ac2537.png)
![D_1](/media/m/f/e/6/fe67388584f844e56a8db45e4e8768ca.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![DA](/media/m/a/0/8/a081cf3dbb7eaedd62ea487a4cd46956.png)
![AA_1D_1](/media/m/5/a/8/5a8648f829b1ae9de4d4b07f616a5101.png)
![BB_1A_1](/media/m/1/e/2/1e2c48e7431cf29b103d526e2637591a.png)
![CC_1B_1](/media/m/a/b/7/ab7031bd8c18c2120c966dd6b2e0a3f4.png)
![DD_1C_1](/media/m/f/1/1/f11b998b6f53db93d18ae09c54c9d40e.png)
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
![S_1](/media/m/7/7/e/77ed808eaa71be903a10ce754f90a904.png)
![A_1B_1C_1D_1](/media/m/8/e/a/8ea8991888072519f65b0a7e2f45de2d.png)
![kS_1\ge S](/media/m/3/4/1/341083ebb13784cb931279519286da48.png)
Author: unknown author, USA
Izvor: Međunarodna matematička olimpijada, shortlist 2007