[/quote]
,
red cards, 
white cards, and 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
lies on side 
and point 
lies on side 
.
and 
be points on segments 
and 
,
is parallel to 
.
be a point on line 
,
.
be the point on line 
,
.
,
.
meet at point 
.
on line 
be 
.
meet 
respectively. Let 
be the midpoint of 
lie on a circle.
such that 
satisfying 
where 
.
cannot be a perfect cube of a positive integer.
has an incenter 
Suppose 
.
.
meets 
again at point 
.
and 
.
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
and 
.
.
and 
with another positive integer 
,
is a perfect square. For example, if the number on the blackboard is 
,
,
,
,
.
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
is perpendicular to 
.
is a diameter, and 
and 
are parallel. Let 
.
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)
and 
)
Prove that if lines 
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)
Y by Mango247
The centers of three spheres with radii 
form an isosceles triangle with side 
. Find the locus of the intersection points of the medians of triangles 
such that point 
lies on the first sphere, point 
lies on the second sphere, and point 
lies on the third sphere.
form an isosceles triangle with side 
. Find the locus of the intersection points of the medians of triangles 
such that point 
lies on the first sphere, point 
lies on the second sphere, and point 
lies on the third sphere.This post has been edited 1 time. Last edited by parmenides51, Aug 4, 2020, 6:09 AM
