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.
, the point 
is the midpoint of 
and 
is a point on the side such that 
. The cotangents of the angles 
, 
, and 
are positive integers 
.
is also an integer and equals 
.
, some of which consisting of Fibonacci numbers.
be an acute triangle with incentre 
and 
. Let lines 
and 
intersect the circumcircle of at 
and 
, respectively. Consider points 
and 
such that 
and 
are parallelograms (with 
, and 
). Let 
be the point of intersection of lines 
and 
. Prove that points 
, and are concyclic.
and a positive integer such that for every integer 
, the number 
is a perfect square.
of primes such that for all positive integers 
, there exists a positive integer 
and prime 
such that 
but 
.
chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)
intersect at the point 
. On the segment 
, where 
is the altitude of the triangle, the point 
is marked. Points 
and 
are the feet of perpendiculars from point on straight lines 
and 
, respectively. The lines 
and 
intersect at 
. Prove that 
.
be real numbers such that 
. Find the value of 
be real numbers such that 
and 
. Find the value of 
, let 
be a point on arc of circle which doesn't contain 
. 
and intersect at 
. Let 
and 
be the reflection of about to 
and 
, respectively. 
intersects at , and 
intersects at . Prove that circumcenter of triangle 
lies on .
Prove that
Let 
Prove that
Let 
Prove that![$$ a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$$](http://latex.artofproblemsolving.com/8/6/6/866623c65f54d6147a43b1b241c2cba96863b624.png)
such that 
for all 
such that 
.
be a positive integer such that the sum of its positive divisors is at least 
. Prove that has at least 
distinct prime factors.
and 
intersect each other at points and 
. Their external common tangent (closer to ) touches at and at . Let 
be the reflection of in line 
. Prove that 
.
.
concyclic and thus 
.
concyclic, we get , we are done. concyclic, we get , we are done.
and intersect each other at points and . Their external common tangent (closer to ) touches at and at . Let be the reflection of in line . Prove that .
is the 
point in 
, Anyways, here's a solution with just angle chasing,
is cyclic, 
, and 
.
and 
.
.
. is tangent to 
and 
at and respectively, we have 
and 
.
.
are concyclic.
, and too.
such that 
coprime, 
and 
.
Prove that
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
and we know 
is cyclic
hence done