[/quote]
,
be positive real numbers such that 
.
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 
,
,
,
,
respectively. Let 
and 
be diameters of 
and 
be the pole of 
and 
with respect to 
cuts 
.
cuts 
.
at 
.
at 
.
and 
be midpoint of 
and 
,
,
and perpendicular bisector of 
are concurrent.
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](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 
is divided by 
is 
What is the remainder when 
is divided by 
Prove that
Let 
Prove that
![$A \in \mathbb Q[x]$](http://latex.artofproblemsolving.com/d/1/5/d158f344cf30f8beb5f2199da53272bd8e304704.png)
with ![$\exists Q \in \mathbb Q[x], Q(A(x))= x^{2025!+2}+x^2+x+1$](http://latex.artofproblemsolving.com/1/1/9/119a7b617e09ba16bb266b44ab3c85acc529f88f.png)
and 
.
,
and positive real number 
,
,
,
,
holds.
.
are the orthocenter and excenter. Let 
as 
,
is equilateral. Let 
be a point on the same side of 
as 
is equilateral. Let 
be a point on the same side of 
is equilateral. Let 
.
are collinear.
be real numbers satisfying 
and 
Prove that 
Let 
and 
Prove that 
in the plane.Let 
be reflections of 
across the line 
be the midpoints of 
,
be the reflections of 
and 
are natural numbers such that 
is divisible by 
and 
is divisible by 
,
.
Y by
Let 
be four distinct points in the plane.
Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, then
the oriented angle
is equal to the oriented angle DCB.
2.If
lies on the segment 
, then
the oriented angle
plus the oriented angle 
equals 
.
3.If the oriented angle
plus the oriented angle 
equals 0°, then
lines
and are parallel.
4.If the oriented angle plus the oriented angle equals , then
lines and are parallel.
be four distinct points in the plane.Which of the following statements, expressed using oriented angles, are always true?
1.If lines
and 
are distinct and parallel, thenthe oriented angle
is equal to the oriented angle DCB.2.If
lies on the segment 
, thenthe oriented angle
plus the oriented angle 
equals 
.3.If the oriented angle
plus the oriented angle 
equals 0°, thenlines
and are parallel.4.If the oriented angle
plus the oriented angle equals , thenlines
and are parallel.
