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 a circumcircle of triangle 
. Also, let 
be an angle bisector of angle 
, 
be a midpoint of arc 
of circle containing the point 
, and let 
be an incenter of a triangle . Circle cuts line 
at point 
and circle with diameter 
at 
. If the circumcircle of triangle 
intersects again at 
, prove that , and are collinear.
such that 
. Find the minimum value of:
be an acute triangle incribed in a circle 
.Let be the midpoint of 
.Let 
and 
be altitudes from 
and of triangle , respectively, and let them intersect at .Let be the intersection point of tangents to the circle at points 
.Prove that 
and 
are concurrent.
+1 w
be an odd prime and 
is a bijection and 
is an integer such that 
Show that 
is a multiple of .
be a triangle. is the midpoint of segment , and points 
, 
are selected on sides , 
respectively such that , , are collinear. The circumcircles 
and 
intersect at a point 
. The circumcircle 
intersects line again at a point 
.
, and the tangent to at point 
concur. with 
and 
, let be the midpoint of 
. Choose point on the extension of 
past 
and point 
on segment 
such that lies on 
. Let 
be on the opposite side of 
from such that 
and 
. Let 
intersect the circumcircle of 
again at 
, and let 
intersect the circumcircle of 
again at 
. Prove that , 
, and 
are collinear.
people are playing a game at Cheesy's math casino, where is a prime number. Let 
be a positive integer. A subset of length 
from the set of integers from 
to inclusive is randomly chosen, with an equal probability (
and is fixed). The winner of Cheesy's game is person 
, if the sum of the chosen numbers are congruent to 
for 
., find all values of such that no person will sue Cheesy for creating unfair games (i.e. all the winning outcomes are equally likely)., let 
be the set of positive integers that do not exceed and are coprime to . Define 
as the smallest positive integer that allows to be partitioned into disjoint subsets, each forming an arithmetic progression.
satisfying 
, 
, and 
.
be a quadrilateral and let 
be a line. Let intersect the lines 
at points 
respectively. Given that these six points on are in the order 
, show that the circles with diameter 
are coaxal.
be a positive integer greater or equal to 
and let 
, 
, ..., 
be a sequence of non-negative real numbers. Find the maximum value of 
in terms of .
positive real numbers such that 
Prove that![\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+(k+5)(a+b+c) \geqslant 3(k+6),\]](http://latex.artofproblemsolving.com/3/d/a/3da7c98ff14a70b4bbbb6504bfb2c397fd148e45.png)
for all ![$0 \leqslant k \leqslant k_0 = \frac{3\big(\sqrt[3]{2}+\sqrt[3]{4}\big)-7}{2}.$](http://latex.artofproblemsolving.com/7/7/5/77526925e38a958cb4167555e8fbc1056506c522.png)
real number on the board such that each number is more than 
.
on the board and write 
.
real number on the board such that each number is more than . on the board and write .
or![$$a=b=\frac{\sqrt[3]{4}+2\sqrt[3]{2}-2}{3},\,c=\frac{2\big(1+\sqrt[3]{2}\big)}{3},$$](http://latex.artofproblemsolving.com/c/b/d/cbd0471249c6ac97348f9748ee55aa5e56af03ca.png)
or any cyclic permutation.
.
.