satisfying
Prove
,
,
,
.
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 
represents 
.
and a line 
in the plane.Let 
be reflections of 
across the line 
be the midpoints of 
,
be the reflections of 
.
are the orthocenter and excenter. Let 
be a point on the same side of 
as 
is equilateral. Let 
as 
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be the midpoint of 
.
are collinear.
for which there exists positive integers 
and 
satisfying ![\[ \gcd(a,b) + \operatorname{lcm}(b,c) = \operatorname{lcm}(c,a)^3. \]](http://latex.artofproblemsolving.com/0/6/0/0602a5b960fc784175f1208cf3db1d244c067377.png)
for which the roots of the equation 
are integers.
for some base 
,
.
,
,
lie on the graph of 
.
,
.
and 
n
.
representation of 
?
has side length 
.
be 
and the circle centered at 
occurs when 
for the function 
.
Configuration 
,
:
.
,
,
,
.
islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands 
and 
.
.
,
for prime 
and 
for 
.
such that 
is a perfect power of 
.
be a triangle whose 
-
,
,
,
and the perimeter of 
,
of functions 
satisfying:
,
,
.
can be written as 
where 
are distinct primes, compute 
.
and that the first (leftmost) two digits of 
are 10. Compute the number of integers 
such that 
starts with either the digit 
or 
(in base 
)
be the circumcenter of 
intersects 
(on the major arc 
,
such that 
is perpendicular to 
.
and 
,
.
be the isosceles triangle with side lengths 
.
within this triangle, and compute 
,
denote the regular polygon with 
sides such that the vertices of 
denote the polygon that results after applying f a total of k times. The area of 
where 
for some positive integers 
where 
is not divisible by the square of any prime and 
.
. This function can be expressed in the form 
for sequences of integers 
,
.
.
,
,
be 
,
and the midpoint of 
at 
.
,
is 
,
?
be a polynomial with nonnegative integer coefficients. If 
and 
,
is divided by 
Prove that
Let 
Prove that
. Prove that
Y by microsoft_office_word, Adventure10, Mango247
Let 
be the circumcircle of the acute nonisosceles triangle 
. Point 
lies on the altitude from 
. Let 
and 
be the feet of the altitudes from P to 
, 
respectively. Circumcircle of triangle 
intersects the circle in 
, different from . Prove that the lines 
, 
and 
are concurrent.
be the circumcircle of the acute nonisosceles triangle 
. Point 
lies on the altitude from 
. Let 
and 
be the feet of the altitudes from P to 
, 
respectively. Circumcircle of triangle 
intersects the circle in 
, different from . Prove that the lines 
, 
and 
are concurrent.This post has been edited 1 time. Last edited by Snakes, Mar 6, 2017, 7:50 PM
be the antipode of 
,
are colinear, so it is enough to show that that 
passes through 
and 
,
is concyclic. Now we have that 
.
.
and 
,
we get the desired result.
,
and 
is cyclic we get 
and 
we get the perpendicular bisectors of 
and 
intersect in 
be that point 
are collinear, it so easy note that 
.
,
,
,
,
,
hence 
impies 
and 
cyclic where 
.
colinear. So, we reduce the problem into showing 
colinear.
.
meets 
on the circle. Let this point be 
meets 
at 
which is also on the circle.
.
from where we obtain that 
is cyclic. Let 
be feet of altitude from 
and 
we obtain that 
are cyclic too.
With this we get that 
is cyclic. Now let 
.
then points 
lie on line perpendicular to 
.
is the line perpendicular to 
is cyclic.So from radical center 
concur at 
are perpedicular to 
so 
collinear so we are done
