IMO Shortlist 1996 problem G4
Dodao/la:
arhiva2. travnja 2012. Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be an equilateral triangle and let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a point in its interior. Let the lines
![AP](/media/m/7/b/0/7b05fe3b464ec24a15fa5701f4d14b61.png)
,
![BP](/media/m/e/e/f/eefb4fe46ab8d85b7067c29b24aa4cfc.png)
,
![CP](/media/m/6/3/0/630424587cadeb75669118dab3df6b98.png)
meet the sides
![BC](/media/m/5/0/0/5005d4d5eac1b420fbabb76c83fc63ad.png)
,
![CA](/media/m/a/a/e/aaec86bc003cfdb64d54116a4cabd387.png)
,
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
at the points
![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)
, respectively. Prove that
![A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A](/media/m/5/1/9/519b1f34186a8dd6ab6934ae9cf19c82.png)
.
%V0
Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that
$A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.
Izvor: Međunarodna matematička olimpijada, shortlist 1996