AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
we have 
.
.
be reals such that 
and 
Prove that
be a cyclic quadrilateral. The point 
is the reflection of 
the intersection of 
and 
, the point 
is the reflection of midpoint of 
. Also let 
be the reflection of 
midpoint of 
. Show that 
concyclic.
. Prove that: 
'is a perfect square.
be a triangle, and let 
be a point on side . A line through intersects side 
at 
and ray 
at 
. The circumcircle of triangle 
intersects the circumcircle 
of triangle again at point 
distinct from point . The lines 
and 
intersect again at 
and 
respectively.
such that 
for all real number 
and 
be a scalene triangle with point 
and 
on the plane such that 
is an equilateral . Let intersect 
and 
at and respectively and intersect 
and 
at and respectively .
, point 
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds. is an isosceles triangle.
be a positive integer such that 
.
.
m, 
m and 
m. What is its distance to the furthest corner, and what is the size of the field?
.
m
.
be the distances to the four sides. We know that 
and 
. All we need is 
which we can obtain as 
.
, and we also have 
so subtracting yields 
.
and similarly cyclicly, and solve.
, 
, 
?
.
which means 
.
because currently you only have 3 independent equations and 4 variables.
be the distances to the four sides. We know that and . All we need is which we can obtain as ., and we also have so subtracting yields . and similarly cyclicly, and solve.
Prove that
and 
Prove that
are the only options. I don't know how to prove that the former results in a rectangle.
instead of 
are ordered counterclockwise)
.
.
,
,
![$[P'APD]=[ABPD]$](http://latex.artofproblemsolving.com/9/7/3/973c478ac720156538181bbbc3020736e46ac530.png)
![$[P'AP]=\frac{9}{2}$](http://latex.artofproblemsolving.com/e/7/7/e77d6bd8153846eac8ad89c74ec1d2c47b267f90.png)
![$[PDP']=\frac{\sqrt{(9+3\sqrt{2})(9-3\sqrt{2})(1+3\sqrt{2})(-1+3\sqrt{2})}}{4}=\frac{\sqrt{(63)(17)}}{4}=\frac{3}{4}\sqrt{119}$](http://latex.artofproblemsolving.com/7/8/9/789b09c442909d86dcf8750eb5082959815af16e.png)
by heron's![$[ABPD]=[P'APD]=[P'AP]+[PDP']=\frac{18+3\sqrt{119}}{4}$](http://latex.artofproblemsolving.com/4/b/3/4b3e9d44ac09d27dd5fee8b7dfe80c99abd3a806.png)
to 
![$[BCDP]=[PP''C]+[PBP'']$](http://latex.artofproblemsolving.com/e/5/7/e57683fbe66c4cd6c56386b2805331057070e82c.png)
![$[PP''C]=16$](http://latex.artofproblemsolving.com/9/f/d/9fd2508470109c7473772b407b03f0c003bfe310.png)
![$[PBP'']=\frac{\sqrt{(17)(1)(9)(7)}}{4}=\frac{3\sqrt{119}}{4}$](http://latex.artofproblemsolving.com/1/4/4/1444bc7ecfd8cddc6c44a90d6cd409003b591e18.png)
![$[BCDP]=\frac{64+3\sqrt{119}}{4}$](http://latex.artofproblemsolving.com/8/2/a/82a30dc234da1afb0e9019c2a1ffae8ff130d375.png)
![$[ABCD]=[ABPD]+[BCDP]=\frac{41+3\sqrt{119}}{2}$](http://latex.artofproblemsolving.com/5/3/c/53c5132af09ab21e226a45ba4338f7ff106ffb37.png)
