,
,
,
.
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 
.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be an acute-angled triangle with 
,
be its circumcentre and orthocentre respectively. Points 
lie on segments 
respectively, such that ![\[\angle ZOB=\angle YOC = 90^{\circ}.\]](http://latex.artofproblemsolving.com/8/f/e/8fe34cc0f04ff9a48f1a7eee89fc50d57272d936.png)
The perpendicular line from 
to line 
meets lines 
and 
at 
respectively. Let the tangents to the circumcircle of 
at points 
and 
meet at point 
.
are concyclic.
Y by Adventure10
Let 
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
be a diameter of a circle 
, and let 
be the tangent at 
. Furthermore, let 
be a fixed, positive real, and consider all pairs of points 
and 
lying on , on opposite sides of , such that 
. The lines 
and 
intersect at points 
and 
, respectively. Show that all the lines 
pass through a common point.
![[asy]defaultpen(fontsize(8));
pair A=(0,0), B=(10,0), X=(0,10), Y=(0,-5), H, P, Q;
path S = Circle(midpoint(A--B),abs(A-B)/2);
P = intersectionpoints(S, X--B)[1];
Q = intersectionpoints(S, Y--B)[1];
H = extension(P,Q,A,B);
draw(S);
draw(A--Y--B--X--A--P--Q--A--B);
dot(P^^Q^^A^^B^^X^^Y^^H);
label("A",A,(-1,0));label("B",B,(1,0));label("P",P,(1,1));label("Q",Q,(1,-1));label("X",X,(-1,0));label("Y",Y,(-1,0));label("H",H,(1,-1));[/asy]](http://latex.artofproblemsolving.com/2/9/7/2970f45937b7d546c56d72b73aafaccd7195aa21.png)
Let 
.
,
remains fixed on ![$ \frac{[PAQ]}{[PBQ]}$](http://latex.artofproblemsolving.com/8/6/0/8600d3681d1728c977f8fc398417a09fb5820b98.png)
remains constant.
,
![$ \implies\frac{c}{AB^2}=\frac{AX\cdot AY}{AB^2}=\frac{AP\cdot AQ}{BP\cdot BQ}=\frac{\frac{1}{2}AP\cdot AQ\sin{PAQ}}{\frac{1}{2}BP\cdot BQ\sin{PBQ}}=\frac{[PAQ]}{[PBQ]}$](http://latex.artofproblemsolving.com/5/d/c/5dcd0940f177136adbb1c759adb80be54ffa10f3.png)
be, the points on the tangent line 
at point 
as the problem states and also such that 
and let be the point 
always passes through the point 
we have that 
we have that 
we conclude that the points 
are concyclic, with their circumcircle so be it 
which intersects the line-segment 
at points 
such that 
and 
from the right angled triangles 
we conclude that the points 
are concyclic, with circumcircle so be it 
we conclude that the point 
lies on 
because of 
as the radical center of the circles 
and the proof is completed.
with radius 
,
be a point such that 
.
be points such that 
for some positive real constant. If circles 
intersect the line through 
,
passes through a fixed point.
.
is a rectangle, so we have that 
.
,
are fixed and 
be the radi of circle.
and 
.
is constant. Let 
.
,
,
