be the circumcenter, symmedian point of 
.
at 
at 
,
at 
are concurrent.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
be an integer. Prove that 
can be written as a polynomial with integer coefficients in the variables 
,
and 
.
,
are real numbers such that 
is an integral multiple of 
.
,
for all positive integers 
.
be the orthocenter of an acute-angled triangle 
.
centered at the midpoint of 
and passing through 
and 
.
,
,
and 
.
Y by
Let 
be an isosceles trapezoid inscribed in circle 
, such that 
. Point 
is chosen on the arc 
of not containing 
. Let and 
intersect at 
. Show that if is chosen such that 
, the area of 
is maximized.
be an isosceles trapezoid inscribed in circle 
, such that 
. Point 
is chosen on the arc 
of not containing 
. Let and 
intersect at 
. Show that if is chosen such that 
, the area of 
is maximized.This post has been edited 1 time. Last edited by mira74, Mar 20, 2020, 9:59 PM
Reason: prob #
Reason: prob #
![$[AEF]=[AED]-[AFD]$](http://latex.artofproblemsolving.com/1/c/8/1c86696bceb31ca43ab3513cc76305bbefa0cd93.png)
,![$[AFD]$](http://latex.artofproblemsolving.com/8/a/1/8a1a8e000537e273c7df2c009e5ff08992bead39.png)
is fixed, to maximize ![$[AEF]$](http://latex.artofproblemsolving.com/1/3/c/13c70c42a7bfa54d17fc9289b2134a7a1fd4abcf.png)
we need to maximize ![$[AED]$](http://latex.artofproblemsolving.com/f/3/0/f30eaecc95015ef575f560aae40c6bccfbcc1a86.png)
,
.
![$[AFD] $](http://latex.artofproblemsolving.com/2/0/5/2053266984e429673ef5f60af88ab91e9723da16.png)
is fixed since 
is fixed and the locus of 
is a line parallel to 
so the altitude is fixed.
varies over arc 
of 
(means arc 
)
is fixed, so we need to minimize the sum of areas 
and 
.
be the circumcenter of 
the radii. To minimize 
we need to maximize ![$[EOA]+[EOD]=\frac {1}{2}(r^2 (\sin(\angle EOA)+\sin (\angle EOD))) $](http://latex.artofproblemsolving.com/9/8/1/981aadf3c9c5cb5cf16772ef5d897e5e2ac4f558.png)
.
moreover when 
.![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
be the area of 
)
The area of 
is a constant.
The area of 
However, note that the altitude from 
a constant. Thus, ![$[ADF]$](http://latex.artofproblemsolving.com/8/9/f/89f26005cdaa38dff904ae294a8067607d663a16.png)
is solely determined by the trapezoid 
![$[AEF]=[AED]-[AFD].$](http://latex.artofproblemsolving.com/f/6/f/f6f75d3f4cfceba0bb723d10425fae14e1c49941.png)
Therefore, maximizing ![$[AED]=\frac{AD\cdot h}{2},$](http://latex.artofproblemsolving.com/5/0/1/501d0d3d76876693460ab1096933e268552b786f.png)
where 
is the length of the altitude from 
However, this is trivial by the definition of a circle, and we are finished.
