[/quote]
,
be a cyclic quadrilateral. A circle centered at 
passes through 
and 
and meets lines 
and 
again at points 
and 
(distinct from 
)
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle 
and 
are similar.
satisfying 
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
crosses are placed. A set of 2025 cells we will call balanced if no two of them are in the same row or column. It is known that any balanced set has at least 
crosses.
,
onto 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
and 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
such that 
![\[
C(p-1, k-1) \equiv (-1)^{k-1} \pmod{p}
\]](http://latex.artofproblemsolving.com/4/0/7/4072235afbaa1f2d8d4c9ca044615a113a4306ce.png)
for 
,
is the binomial coefficient 
choose 
.
satisfy that: 
and 
.
intersect at points 
and 
and 
.
be positive integers and 
be an odd prime such that:![\[
a^p \equiv 1 \pmod{p^n}.
\]](http://latex.artofproblemsolving.com/3/4/8/34815b8211ded2f37b0653525dc4b436356e6eee.png)
Prove that:![\[
a \equiv 1 \pmod{p^{n-1}}.
\]](http://latex.artofproblemsolving.com/0/7/a/07a4ed2ab3c94f41028080f7789cfb6e2973955c.png)
be a function such that for every regular 
-
we have 
.
for all reals 
.
Y by Adventure10, Mango247
Let us consider a convex hexagon ABCDEF. Let 
be midpoints of the sides 
respectively. Denote by 
and 
, respectively, the perimeter of the hexagon 
and hexagon 
. Suppose that all inner angles of hexagon are equal. Prove that ![\[ p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .\]](//latex.artofproblemsolving.com/c/8/1/c813cb30724ca31ce41ca84a5d74221d9cf25b82.png)
When does equality hold ?
be midpoints of the sides 
respectively. Denote by 
and 
, respectively, the perimeter of the hexagon 
and hexagon 
. Suppose that all inner angles of hexagon are equal. Prove that ![\[ p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 .\]](http://latex.artofproblemsolving.com/c/8/1/c813cb30724ca31ce41ca84a5d74221d9cf25b82.png)
When does equality hold ?
.
on 
+ projection of 
on 
.
.
,
,
,
,
,
,
,
,
,
,
,
,
,
then: 
so: 
is equilateral triangle
,
,
,
,
or 
is regular hexagon
