[/quote]
,
satisfying the following condition: for every monic polynomial 
of degree at most 
and 
distinct integers 
such that ![\[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]](http://latex.artofproblemsolving.com/e/e/8/ee8adc79dc258b021529245fec16e081da2d1b0a.png)
.
At each step, Sheldon must choose either two rows or two columns.
,
are entries in 
and 
are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.
?
to satisfy that the equation 
has at least 1 roots 
and for 
,
Prove that every natural number occurs exactly once in this sequence.
inscribed in 
,
is the midpoint of arc 
of 
intersects 
at 
.
intersects 
at 
different from 
is the orthocenter of triangle 
= 
for which the number![\[x^4 + x^3 + x^2 + x + 1\]](http://latex.artofproblemsolving.com/a/f/f/affe758628a67ad6f72e82decfe71356c24daa6e.png)
?
such for any 
.
.
and 
intersect the circumcircle of 
and 
,
and 
and 
are parallelograms (with 
,
)
be the point of intersection of lines 
and 
.
,
Find the range of values that can be taken by :
are positive reals.
there are elements 
such that 
.
would be divisible by 
for any integer values of 
is not a factor of 
Y by Adventure10, Mango247
Two circles 
and 
intersect at two points 
and 
. Some line through the point meets the circle at a point 
(apart from ), and the circle at a point 
(apart from ). Another line through the point meets the circle at a point 
(apart from ), and the circle at a point 
(apart from ). Assume that the point lies between the points and and between the points and .
Finally, let
and 
be the midpoints of the segments 
and 
.
Prove that the triangles
, 
and 
are similar to each other.
and 
intersect at two points 
and 
. Some line through the point meets the circle at a point 
(apart from ), and the circle at a point 
(apart from ). Another line through the point meets the circle at a point 
(apart from ), and the circle at a point 
(apart from ). Assume that the point lies between the points and and between the points and .Finally, let
and 
be the midpoints of the segments 
and 
.Prove that the triangles
, 
and 
are similar to each other.
follows from
because
and we have
is taken to 
by a spiral similarity about 
,
so 
,
