[/quote]
,
it can draw the line which forms the interior angle bisector of 
.
in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through 
and 
.
,
satisfying 
,
?
and 
such that![\[
1^n + 2^n + 3^n + \cdots + n^n = x!
\]](http://latex.artofproblemsolving.com/6/4/b/64b5f5919fd3710cc15e96724918eae216c8e4f0.png)
be a cyclic convex quadrilateral and 
be its circumcircle. Let 
be the intersection of the diagonals of 
and 
.
be the center of the circle tangent to sides 
,
,
,
be the midpoint of the arc 
and 
.
opposite 
.
that has 8097 distinct points with area of a triangle that has 3 points belong to 
.
that has the area 
contains at least 2025 points that belong to 
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.
be the incentre of acute triangle 
with 
.
of 
,
and 
,
meets 
.
meets 
.
and 
meet again at 
.
and 
meet on the line through 
.
and 
Prove that
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)
and 
Prove that
integers are such that 
is prime and 
.
such that 
.
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
.
.
Y by Adventure10, Mango247
This problem came up during a discussion session at a conference I attended. It looked easy and olympiad-like. But I cannot come up with a solution. I feel like I am missing something trivial.
Let
be a positive integer and 
be a subset of 
such that the sum of the elements of is not divisible by 
. Prove that the elements of can be ordered in a way such that no consecutive block of the ordered set has sum divisible by .
For example, if
and 
, then we can order as 
, and no consecutive block of integers has sum divisible by 
.
Let
be a positive integer and 
be a subset of 
such that the sum of the elements of is not divisible by 
. Prove that the elements of can be ordered in a way such that no consecutive block of the ordered set has sum divisible by .For example, if
and 
, then we can order as 
, and no consecutive block of integers has sum divisible by 
.
