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.
with the following properties 
If each positive integer is arbitrarily colored red or blue
distinct red positive integers 
satisfying
or there are 
distinct blue positive integers 
satisfiying
be reals such that 
.Prove that
Where 
such that 
Prove that 
2. Let 
be sidelengths of a triangle such that 
Prove that 
be three collinear points and 
three other collinear points. Let 
be the intersection of the lines 
and 
,respectively. If 
and 
.Prove that 
. Let 
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
be a fixed acute-angled triangle. Consider some points 
and 
lying on the sides 
and 
, respectively, and let 
be the midpoint of 
. Let the perpendicular bisector of intersect the line 
at 
, and let the perpendicular bisector of 
intersect the lines and at 
and 
, respectively. We call the pair 
, if the quadrilateral 
is cyclic.
and 
are interesting. Prove that 
is given. Over the side draw an equilateral triangle 
on the outside. The midpoint of the segment 
is 
and the midpoint of the side 
is 
. Prove that 
. be an acute triangle. Let 
be the reflection of point over the line . Let 
and be the circumcenter and the orthocenter of triangle , respectively, and let be the midpoint of segment 
. Let 
and 
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle , where point lies on the arc not containing . If 
is a point on the line 
such that 
, prove that 
.
be a regular hexagon with sidelength s. The points 
and 
are on the diagonals 
and 
, respectively, such that 
. Prove that the three points 
, and are on a line.
, and 
be the angles of a triangle, and let 
, and 
. If 
, show that![\[s(s - a)(s - b)(s -c) \geq 0.\]](http://latex.artofproblemsolving.com/1/3/4/1344a0cbbd4e1aa4bcf6a1e777501c62daff32b3.png)
has 
. Point is the foot of the perpendicular from to . Points and lie on sides and 
, respectively, such that lies inside triangle 
and ![\[
\angle CHS - \angle CSB = 90^{\circ}, \quad \angle THC - \angle DTC = 90^{\circ}. \]](http://latex.artofproblemsolving.com/f/8/c/f8c35912c531eb2dae8f62b74666b7757d2b9ba3.png)
Prove that line is tangent to the circumcircle of triangle 
.
cannot cover entirely a square surface of side 
, but they can cover more than 
of it.
is chosen inside a convex quadrilateral . Could it happen that
respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.
respectively, each sphere externally touches the other spheres. Suppose that there is another sphere that is externally tangent to all those four spheres, determine the radius of this sphere.
, see two pictures.
green, 
red, 
blue, 
purple.
.
.
and radius , satisfying:![\[\left\{\begin{array}{l} MM_{2}+2=r \\ MM_{3}+3=r \\ MM_{4}+3=r \\ \end{array} \right.\]](http://latex.artofproblemsolving.com/3/8/9/3892236aaf28f2b35e8c3e787906817f6aefa78d.png)
![\[\left\{\begin{array}{l} \sqrt{c^{2}+4+d^{2}}+2=r \\ \sqrt{(c-\sqrt{21})^{2}+d^{2}}+3=r \\ \sqrt{(c-\frac{\sqrt{21}}{7})^{2}+(d-\frac{12\sqrt{7}}{7})^{2}}+3=r \\ \end{array} \right.\]](http://latex.artofproblemsolving.com/4/a/a/4aa1941e16092a6e720214a1225dc5b30501ab95.png)
and 
.
of the sphere, externally tangent to all those four spheres..
, see two pictures. green, red, blue, purple.
.. and radius , satisfying: and .
should be 
; and since the fifth sphere is externally (not internally) tangent to the other spheres, hence 
... Consequently, 
is not the right answer.
Prove that
Where 
.
.