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.
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
be a triangle whose incircle is tangent to 
, 
, 
at 
, respectively. Let 
and 
be the midpoints of 
and 
, respectively. Suppose that points 
and 
lie on 
so that 
and 
. Prove that lines 
and 
intersect on the circumcircle of 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
be a triangle, and let be on so that 
. Let 
intersect the circumcircle of again at . Let 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
. If is the circumcenter of 
, prove that 
.
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form 
, where 
and 
are not necessarily distinct. Determine 
.
be a cyclic quadrilateral with 
. Let be the midpoint of the arc 
not containing . Suppose there is a point inside such that 
and 
.
, and are concurrent.
. By one question, we may take three different indices 
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
be an equilateral triangle with side length 
and be a point inside the triangle such that 
. What is the area of a triangle with side lengths 
?
be the number of positive integers that divide the integer 
. For all positive integral divisors of 
, what is the sum of numbers 
?
such that the value of the
is a square of an integer.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
, , 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
be a triangle with 
let 
be the circumcircle of 
and let 
be its radius. Point is chosen on 
such taht 
and point 
is the foot of the perpendicular from to . Ray 
mets again at . Point is chosen on line 
such that 
and 
lie on a line in that order. Finally, let be a point satisfying 
and 
. Prove that lies on .
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.
.
.