AoPS Academy
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 . 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 triangle, inscribed in parabola. Tangents in points 
forms triangle . Prove that 
. (
is area of triangle 
).
be triangle, inscribed in parabola. Tangents in points forms triangle . Prove that . ( is area of triangle ).
be any point on the Steiner circumellipse of 
and let 
be the cevian triangle of WRT 
Then ![$[DEF]=2 \cdot [ABC].$](http://latex.artofproblemsolving.com/3/3/e/33ec9a123e2b3b63c128a4fe7fe4aa12551261a1.png)
be the antimedial triangle of ( against 
). The Steiner circumellipse 
of is obviously the Steiner inellipse of tangent to its sides at 
are collinear on the polar of WRT and similarly 
and 
are respectively collinear. Thus by Pappus theorem for the hexagon 
it follows that 
is at infinity, i.e. 
and similarly we get 
![$[DEF]=[DYZ]=[CYZ]=2\cdot [ABC]. \ \blacksquare$](http://latex.artofproblemsolving.com/b/4/0/b4028b012dc90ae29c0bc707cbf447409f6ab058.png)
be the perspector of the parabola. Since the center of any inconic is the isotomcomplement of its perspector (in the parabola case at infinity), then it follows then that the isotomic conjugate of WRT must be on the line at infinity is on the Steiner circumellipse of and the result follows from the lemma.
