[/quote]
,
such that 
and 
with another positive integer 
,
is a perfect square. For example, if the number on the blackboard is 
,
,
,
,
.
,
with circumcenter 
and orthocenter 
and that line 
is not parallel to line 
.
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
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 cyclic quadrilateral. A circle centered at 
and 
)
Prove that if lines 
are concurrent, then triangle 
are similar.
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.
,
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 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
.
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 
.
Y by Adventure10, Rounak_iitr
The circle 
and the line 
have no common points. Let 
be the diameter of perpendicular to , with 
closer to than 
. An arbitrary point 
, is chosen on . The line 
intersects at 
. The line 
is tangent to at 
, with and on the same side of . Let 
intersect at 
, and let 
intersect at 
. Let 
be the reflection of 
in . Show that 
, and are collinear.
and the line 
have no common points. Let 
be the diameter of perpendicular to , with 
closer to than 
. An arbitrary point 
, is chosen on . The line 
intersects at 
. The line 
is tangent to at 
, with and on the same side of . Let 
intersect at 
, and let 
intersect at 
. Let 
be the reflection of 
in . Show that 
, and are collinear.
and 
different fom 
By Pascal theorem for the degenerate cyclic hexagon 
the intersections 
and 
are collinear 
hence 
Quadrilateral 
is cyclic on account of 
Therefore 
is the reflection of 
i.e. 
,
,
we have 
cyclic, which implies 
.
,
we have 
.![\[DC\cdot DA=DE^2=DF^2\implies \dfrac{DF}{DC}=\dfrac{DA}{DF}.\]](http://latex.artofproblemsolving.com/c/7/7/c77bdc4498794a70bd0cf3640dafa639bf5e2f92.png)
Therefore 
.
,
,
,
are collinear.
