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.
satisfy the following two conditions:
exist such that 
.
, exist such that 
.
of the set 
, let 
. Find the maximum number of elements in the set 
that satisfies all of the three following conditions: are subsets of 
.
of , 
is not empty.
of , 
.
, then
. Half chord 
be a cyclic 
-gon and let 
for all 
. Define 
as the intersection of diagonals 
and 
for all integers .
satisfying 
for all integers . Prove that the points 
are concyclic.
be a triangle and let 
and 
be variable points on sides 
and 
, respectively, such that 
. Let 
denote the point on 
such that 
is parallel to 
, and let 
denote the point on 
such that 
is parallel to 
. Prove that as 
vary, the circumcircle of 
passes through a fixed point, other than .
and 
. satisfies the following three conditions:
All elements of are a subset of .
.
satisfies 
. Let the incenter of triangle be 
, which touches 
at 
, respectively. Let 
be the midpoint of 
. Let the circle centered at passing through 
intersect 
at 
, respecively. Let line 
meet at 
, line 
meet 
at 
. Prove that the three lines 
are concurrent.
and 
Prove that
Let and 
Prove that
is given, find the largest real number 
such that there exists a geometric sequence 
with common ratio satisfying
for all positive integers 
. Here, 
denotes the distance from the real number 
to the nearest integer.
, where the common ratio 
. Given that
.
is your favourite function, 
is your mother's favourite function, and 
is your father's favourite function. It was discovered that for any reals 
, 
Prove that you are boring. is 
, then is called a perfect number. For example, the sum of the positive divisors of 6 is 
, hence 6 is a perfect number.
, where 
are positive integers, and 
are odd primes.
is a triangle with circumcenter 
and 
is a circle tangent to and internally to 
The tangent to intersects again at 
If 
are the contact points of with 
respectively, prove that 
and 
Prove that
The correct is above. I fixed it.
be intersection of 2 circles and 
be intersection of 
and 
bisects 
.
and by ptolemy
.
be intersection of 
an
and 
are similar. By tangency on P, we get 
.