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.
, draw exterior equilateral triangles on sides 
and 
to obtain 
and 
, respectively. Let 
be the intersection of the altitude through 
and the median through 
. Let 
be the intersection of the altitude through 
and line 
. Let 
be the intersection of the median through and the line 
. Prove that , , and lie on a common line.
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points and of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
so that the sum of its edges' lengths is minumum.
with equilateral. Let be the area of the triangle of sides 
, 
and 
. Show that 
where 
is the circumradius and 
is the volume of the tetrahedron. be circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at . Let 
be the midpoint of 
, and let the line 
meet again at . The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
, the following inequality holds:![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
we consider a variable point 
which moves on line 
towards . Halfway there, it stops and starts moving in a straight line line towards . Halfway there, it stops and starts moving in a straight line towards , and halfway there it stops and starts moving in a straight line towards , and so on. Show that will get as close as we want to the vertices of a fixed triangle with area 
.
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
be a triangle with incenter and -excenter 
, whose incircle touches 
at 
. The line 
meets at and is the midpoint of 
. Show that 
.
be a triangle, and let be an arbitrary point on line 
, where 
is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of . be an equilateral triangle, be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of , 
, , show that there is a triangle with sides 
, 
and 
.
.
.
.
intersects 
in the point 
. intersrects the circumcircle 
in the point 
.
.
satisfy that: 
,
.
at 
,
.