(a) Be M an internal point of the convex quadrilateral ABCD. Prove that $|MA|+|MB| < |AD|+|DC|+|CB|$. (b) Be M an internal point of the triangle ABC. Note $k=\min(|MA|,|MB|,|MC|)$. Prove $k+|MA|+|MB|+|MC|<|AB|+|BC|+|CA|$.
Problem
Source: flanders junior olympiad '05
Tags: IMO Shortlist
themonster
25.04.2005 19:45
Extend segment $AM$ until it touches $P$ on $DC$ . Extend $BM$ to $O$ on $DC$. Since the shortest distance between two points is a line, $PD+AD>AM$. Likewise $OC+BC>BM$. Thus $AM+BM<PD+AD+OC+BC<AD+DC+CB$.
darij grinberg
25.04.2005 19:53
Flanders Junior 05 = IMO Shortlist 99 problem 1 ... http://www.mathlinks.ro/Forum/viewtopic.php?t=19769 gives proofs of part (b). Part (a) occurs as Lemma in post #6. Darij
Peter
25.04.2005 20:00
Haha... Flanders is even more crazy than I had ever imagined An IMO shortlist problem for untrained 14yo pupils!! waaaay to go!!
Psycho
25.04.2005 21:52
Peter VDD wrote: Haha... Flanders is even more crazy than I had ever imagined An IMO shortlist problem for untrained 14yo pupils!! waaaay to go!! 14 year old? most are 15 i guess, and some are already 16 (although SOME are 14 indeed )
Peter
26.04.2005 01:03
I was 14 when I entered 3rd year... and so were most of the people I know. Add to that that many grade-skippers reach the final round aswell, some are even just 13 [ youngest participant is 12 I believe ]
Psycho
27.04.2005 16:39
aren't most of the FJO from fourth year? which would make them 15 or 16...
Peter
27.04.2005 20:39
most? I don't know, but still I believe the "average" participant is at most 15, not 16