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 
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.
be a point in the exterior of the circle 
of center 
and radius 
, and let 
, 
be the tangent segments from to the circle. On the segment consider the point 
such that 
.Let the line from parallel to 
intersect the segment at . If is a point on the segment other than so that 
, and if the incircle of the triangle has radius 
, then find the area of 
in terms of 
.
be the set of positive integers. Determine all functions 
such that 
is divisible by 
for all positive integers 
.
be a set of integers containing the numbers 
and 
. Suppose further that any integer root of any non-zero polynomial with coefficients in also belongs to . Prove that 
belongs to .
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is . Prove that there exists a vertex that is present in every clique of size 
if and only if 
and 
are squares of rational numbers and are in Arithmetic Progression is an integer.
such that, for all positive integers 
and 
, if divides 
, then divides 
.
be a cyclic quadrilateral whose diagonals 
and 
meet at 
. The extensions of the sides 
and 
beyond and meet at 
. Let 
be the point such that 
is a parallelogram, and let 
be the image of under reflection in . Prove that 
are concyclic.
be a scalene triangle with point and on the plane such that 
is an equilateral . Let 
intersect 
and 
at and respectively and intersect 
and 
at and 
respectively .
, let 
. Point 
lies on segment 
such that 
. Let 
and 
be the circumcenter and incenter of triangle 
, respectively. Let 
and 
be the circumcenter and incenter of triangle 
, respectively. Suppose that lines 
and 
intersect at point 
. Find the length of 
.
satisfying
Prove
and 
are positive real numbers, prove that
be triangle with circumcenter 
and orthocenter . 
intersect again at 
respectively. Lines through parallel to 
intersects 
at 
respectively. Point 
such that 
is a parallelogram. Prove that lines 
and are concurrent at a point on 
.
be an acute triangle with orthocenter . Let be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that bisects 
. Suppose that the line intersects the circumcircle of the triangle 
at and 
. Prove that 
.
be a positive integer. Prove that there exists a positive integer 
and distinct primes 
such that if 
denotes the number of positive integers less than or equal to and not divisible by any of , then
be a positive integer. Prove that there exists a positive integer and distinct primes such that if denotes the number of positive integers less than or equal to and not divisible by any of , then
