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.
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
. The line 
and the circle of radius centered at 
cross at 
and 
, respectively. Prove that , the circle on diameter 
and circle 
are concurrent.
for which the following statement holds: among any five positive real numbers 
(not necessarily distinct), one can always choose distinct subscripts 
such that![\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]](http://latex.artofproblemsolving.com/a/e/e/aeeab8e846634c7922e9312ef139739ad3fbe6eb.png)
be a triangle with circumcircle 
, incenter 
, and -excenter 
. Let the incircle and the -excircle hit 
at 
and 
, respectively, and let 
be the midpoint of arc without . Consider the circle tangent to at and arc 
at 
. If 
intersects again at 
, prove that 
and 
meet on .
be an acute triangle with 
. be the circumcircle circumscribed to the triangle and the midpoint of the smallest arc of this circle. Let and points of the segments 
and 
respectively such that 
. Let 
be the second intersection point of the circumcircle circumscribed to 
with . Let 
and be the intersections of lines 
and 
with other than 
, respectively. Let 
and 
be the intersection points of lines 
and 
with lines and respectively. Show that the 
line passes through the midpoint of
be an acute-angled triangle with 
A circle passes through and 
and crosses the segments and again at 
and 
respectively. The circumcircles of 
and meet each other at points and 
The segments 
and 
intersect at 
Let 
and 
be the reflections of in the lines 
and 
respectively. Prove that the points 
and 
are concyclic.
be a center of circle which passes through vertices of quadrilateral 
, which has perpendicular diagonals. Prove that sum of distances of point to sides of quadrilateral is equal to half of perimeter of .
satisfy the equality, 
prove that the triangle must have a right angle.
be a cyclic quadrilateral and let the intersection point of lines and 
be . Let the points and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.
![\[\frac{\sin^2A+\sin^2B+\sin^2C}{\cos^2A+\cos^2B+\cos^2C} = 2\]](http://latex.artofproblemsolving.com/9/2/a/92a524bbdca92e9dd4091275714bf346bad96cd7.png)
is a pentagon whose vertices lie on circle where 
. Let 
and intersect at , 
meet 
at . is the midpoint of arc on , not containing . If 
holds, then what is the value of 
?
be the sequence 
so that 
for each 
and so that the decimal number 
is divisible by 
for each 
. Show that the decimal 
is irrational.
, then![\[2^{m+kn}\mid\overline{\underbrace{b_n\cdots b_1\ldots b_n\cdots b_1}_{k\text{ groups}}a_m\cdots a_1}.\]](http://latex.artofproblemsolving.com/2/4/d/24da518ef9cbdd1a586ddcbf96ed11c624b787a4.png)
Let 
, 
, then for all natural numbers 
, we have![\[2^{m+kn}\mid 2^mt+\sum_{i=0}^{k-1}10^{m+in}s,\]](http://latex.artofproblemsolving.com/6/5/b/65b8cf593896270e409398548b18b3a970da9e5e.png)
so![\[2^{kn}\mid t+5^m\sum_{i=0}^{k-1}10^{in}s=t+5^ms\cdot\frac{10^{kn}-1}{10^n-1}.\]](http://latex.artofproblemsolving.com/7/5/6/756e3b847083b40e993eb16623ba45459cceba86.png)
Thus, 
, so![\[2^{kn}\mid\left(10^n-1\right)t-5^{ms}.\]](http://latex.artofproblemsolving.com/9/2/a/92afc9a5d20487e2daf14014560ab9e56db1f56d.png)
This is not true for sufficiently large , since the right side is a fixed non-zero integer. Contradiction.
