[/quote]
,
Prove that
Let 
Prove that
and 
Prove that
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](http://latex.artofproblemsolving.com/9/3/7/9377324cb6585daaf2cbec7fed6fce2209835e22.png)
where![\[a_k =
\begin{cases}
3^k, & \text{if } k \text{ is even}, \\
0, & \text{if } k \text{ is odd}.
\end{cases}
\]](http://latex.artofproblemsolving.com/8/b/e/8be520a30bdfd9272163470ee69120c2fd3e9b59.png)
Find the remainder when 
is divided by 
,
,
,
.
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 
be a point on the same side of 
as 
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be the midpoint of 
.
are collinear.
be positive real numbers such that 
.
Where 
be the set of positive integers. Find all functions 
such that![\[f(m)+n \mid m^2f(m) - f(n) \]](http://latex.artofproblemsolving.com/8/5/5/85515a57bf60498134ceb168039fe734a5ff91a5.png)
for all positive integers 
and 
.
.
,
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.
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 inscribed circle of 
,
,
at 
,
respectively. Let 
and 
be diameters of 
and 
and 
with respect to 
cuts 
.
cuts 
.
at 
at 
.
and 
be midpoint of 
and 
,
,
and perpendicular bisector of 
are concurrent.
Y by
The vertices of octagon 
lie on the same circle. If 
and 
, what is the area of octagon ?
I approached this problem by noticing that the area of the octagon is the area of the eight isoceles triangles with lengths
, , and 
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?
lie on the same circle. If 
and 
, what is the area of octagon ?I approached this problem by noticing that the area of the octagon is the area of the eight isoceles triangles with lengths
, , and 
or 11. However, I didn't know how to find the radius. Can anyone give me a hint?This post has been edited 1 time. Last edited by Shiyul, Apr 25, 2025, 6:47 PM
Reason: latex error
Reason: latex error
denote 
.
.
,
.
and 
.
.
,
,
.
be the center of the circle. Now, let 
and 
.
which simplifies to 
.
.
and 
separately. Triangle 
is isosceles so 
.
.
.![\[
{BH}^2 = 121 + 2 -22\sqrt{2}\cos(135) = 123 + 22 = 145
\]](http://latex.artofproblemsolving.com/4/f/c/4fc501a80966268b77ed20c50cb39fec9c8fe20e.png)
Also, note that 
so 
is a right triangle so 
so 
.
so the area of the triangle becomes 
.
we find that its altitude is 
and it has an area of 6. Since our octagon is composed of four triangles congruent to 
