#11) 2012 IMO P5 Barybash
by Kayak, Feb 13, 2018, 1:59 PM
Since everyone's copy-pasting previously done bashes, why should I not ?
2012 G5 Let
be a triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. Let 
be the point of intersection of 
and 
.
Show that
.
Proposed by Josef Tkadlec, Czech Republic[
2012 G5 Let
be a triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. Let 
be the point of intersection of 
and 
.Show that
.Proposed by Josef Tkadlec, Czech Republic[
.
, and as 
. Let 
. We get the displacement vector 
. The condition that length of 
implies 
. Now observe that only the positive root satisfies 
,
, where we defined 
.
, and analgously, 
, hence 
. Now, the displacement vector (after normalization) equates to 
. Now, we need to show that the lenght of this displacemnt vector is symmetric in 
and 
,![$$ Length[MN]^2 = \frac{b^2d^2}{(c^2+d(a+b)+x^2)^2}[\frac{Terms}{ (c^2+x^2+ad)^2 } ]$$](http://latex.artofproblemsolving.com/9/9/3/993362f4272a430655e3c7698a5b8b6eae7ef7c0.png)
, where 
![$$ = a^2 [ (a^2+b^2+x^2+ad)^2 - b^4-b^2a^2-2b^2x^3+b^2d^2] = a^2[(c^2+x^2+ad)^2 - b^2(b^2+a^2+2x^2-d^2)] = a^2[(c^2+x^2+ad)^2 - b^2(d^2-d^2)] = a^2(c^2+x^2+ad)^2$$](http://latex.artofproblemsolving.com/b/5/3/b5321d5e1e51f787c8c6bf10745668b9680e2b07.png)
, so we finally get ![$$ Length [MN]^2 = \frac{b^2d^2}{(c^2+d(a+b)+x^2)^2}[\frac{Terms}{ (c^2+x^2+ad)^2 } ] = \frac{b^2d^2}{(c^2+d(a+b)+x^2)^2}[\frac{a^2(c^2+x^2+ad)^2}{ (c^2+x^2+ad)^2 } ] = \frac{a^2b^2d^2}{(c^2+d(a+b)+x^2)^2}$$](http://latex.artofproblemsolving.com/d/6/6/d66a4469629d93af5941d5b1eec6585529fe73ff.png)
, and as of course it's symmetric in This post has been edited 2 times. Last edited by pro_4_ever, Mar 6, 2018, 9:05 AM
March 2018
February 2018