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        riddles >> putnam exam (pure math) >> Triangles
        
(Message started by: THUDandBLUNDER on Dec 5th, 2004, 11:48pm)
Title: Triangles
Post by THUDandBLUNDER on Dec 5th, 2004, 11:48pm
For i = 1,2 let Ti be a triangle with side lengths ai, bi, ci, and area Ai.
Suppose that
a1 [smiley=eqslantless.gif] a2
b1 [smiley=eqslantless.gif] b2
c1 [smiley=eqslantless.gif] c2
T2 is an acute triangle.

Does it follow that A1 [smiley=eqslantless.gif] A2 ?
Title: Re: Triangles
Post by towr on Dec 6th, 2004, 12:39am
::[hide]
a2+b2=c2+e
for e -> 0, A2 -> 0
So for any positive A1, A2 may be smaller
[/hide]::
Title: Re: Triangles
Post by Icarus on Dec 6th, 2004, 2:00am
But for small e, T2 is no longer acute, so the problem is not quite that simple.
Title: Re: Triangles
Post by towr on Dec 6th, 2004, 2:28am
Sorry, I read 'has an acute triangle' as 'has an acute angle', rather than as 'is an acute triangle'.
(The former is of course trivially true for any triangle, so I suppose the latter was intended)
Title: Re: Triangles
Post by THUDandBLUNDER on Dec 6th, 2004, 2:58am

on 12/06/04 at 02:28:47, towr wrote:
Sorry, I read 'has an acute triangle' as 'has an acute angle', rather than as 'is an acute triangle'.
(The former is of course trivially true for any triangle, so I suppose the latter was intended)

Sorry for the typo. I have amended it.
Title: Re: Triangles
Post by Icarus on Dec 6th, 2004, 5:47pm
I guess it is that simple after all:





At least one of the three angle conditions
[alpha]1 [le] [alpha]2
[beta]1 [le] [beta]2
[gamma]1 [le] [gamma]2

must hold, since both sets of angles sum up to [pi]. Assume wlog that [gamma]1 [le] [gamma]2.
Since [gamma]2 < [pi]/2, sin [gamma]1 [le] sin [gamma]2.
So A1 = (1/2)a1b1sin [gamma]1 [le] (1/2)a2b2sin [gamma]2 = A2.
Title: Re: Triangles
Post by Barukh on Dec 7th, 2004, 6:47am
Very elegant, Icarus!

A friend of mine gave the following nice argument: consider 2 triangles with the same base and same altitude to the base (= same area). Look at the 4 sides remained. Because the angles at the base are acute, it is easy to see that the biggest and the smallest side belong to the same triangle.

BUT: does it really belong to the Putnam exam section? IMHO, putting it here, drives away the attention of many potential puzzlers.

It seems like THUD&BLUNDER discovered recently an unknown Putnam exam treasure  ;D
Title: Re: Triangles
Post by THUDandBLUNDER on Dec 7th, 2004, 8:15am

Quote:
BUT: does it really belong to the Putnam exam section?

It seems like THUD&BLUNDER discovered recently an unknown Putnam exam treasure  

Aren't those two points of view inconsistent?  :)


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