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 triangle with circumcircle 
and 
a line without common points with . Denote by 
the foot of the perpendicular from the center of to . The side-lines 
intersect at the points 
different from . Prove that the circumcircles of the triangles 
, 
and 
have a common point different from or are mutually tangent at .
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
such that
is an integer.
be a triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. Let 
be the point of intersection of 
and 
.
.
be a point on segment 
. Let 
be a fixed circle passing through , and let 
be a variable point on . Let 
be the intersection of the tangent to the circumcircle of 
at 
and the tangent to the circumcircle of 
at 
. Show that as varies, lies on a fixed line.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
, let 
be the sum of the remainders when is divided by 
. For example, 
. Find all positive integers such that 
.
be the set of all positive integers. Find all functions 
such that 
and 
have the same number of 
's in their binary representations, for any 
.
is called beautiful if, for every integer 
, the base-
representation of contains the consecutive digits 
, 
, , 
(in this order, from left to right). Determine whether the set of all beautiful integers is finite.
that are relatively prime be called intertwined if among any two divisors of greater than , there exists a divisor of 
and among any two divisors of greater than , there exists a divisor of . For example, pair 
is intertwined.
for which there exists an intertwined pair such that the product 
is equal to the product of the first prime numbers. such that the product is the product of 
distinct prime numbers. such that the number of divisors of is greater than .
and 
have a common tangent that meets in and in 
. The two circles intersect at and 
where is closer to 
than . Line 
meets circle a second time in 
. Prove that 
bisects angle 
.
be real numbers satisfying 
and 
Prove that 
Let be real numbers satisfying 
and 
Prove that 
be a positive integer. Call a rearrangement 
of nice if for every 
, we have that 
is not divisible by . is odd, prove that there is no nice arrangement of . is even, find a nice arrangement of .
and a point on its side 
such that 
and 
are parallel to 
and respectively. Prove that 
denotes the area function.
and 
be points on the lines and , respectively, such that 
and 
. Assume without loss of generality that 
lies on the segment 
. Let 
be the line parallel to and through and denote 
. Then, 
. Moreover, because of the parallel lines, we have 
. Consequently, 
, as desired. Equality holds if and only if is identical to 
, i.e. iff 
.
