AoPS Academy
![\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\]](http://latex.artofproblemsolving.com/f/d/0/fd0ab11635ee44ad272a54c0f20a2bc47a3596de.png)
cannot all be real if 
.
with integer coefficients with the following property: for every positive integer 
, the numbers ![\[ Q(0), \; Q(1), Q(2), \; \dots, \; Q(n-1) \]](http://latex.artofproblemsolving.com/2/d/e/2de4f57066a97f863558aa819659216b9c6a13c1.png)
produce at most 
distinct residues when taken modulo 
.
+1 w
be a cyclic convex quadrilateral, and let 
, 
. 
has center 
and diagonals 
and 
cut at 
. Prove the followings :
intersect at a common point called the Miquel point of .
and 
. Precisely, 
is the inverse of with respect to 
.
. swapping the pairs 
and 
.
be the midpoints of 
respectively. Show they are collinear.
be a positive integer. Let 
be a sequence of real numbers such that![\[\max_{1 \leq k \leq n} \frac{x_1 + \dots + x_k}{k} = 1 \quad \text{and} \quad \min_{1 \leq k \leq n} \frac{x_1 + \dots + x_k}{k} = 0.\]](http://latex.artofproblemsolving.com/f/1/a/f1afd7d53391f845f9cb0fdedb5b814527164cc8.png)
Find the minimum and maximum possible values of 
.
. I note the middlepoint 
and the point 
which belongs to the crease. Thus,
, i.e. 
in the point , then 
. If I note the intersections 
, 
, then 
, 
) a.s.o.
is so folded that 
falls on the mid point of of . Prove that the crease will divide 
in the ration 
.
, B
, C
, D
. Then, CD's middle point is M
,
. BM's middle point is N
.
.
, which divides BC in the ratio 
.
is isosceles 
Therefore, 
, where 
lies on ,
to be 
~ 
~ 
be the crease, where 
is the point on 
be the midpoint of 
and 
.
.
,
be the square. Let its sides be = 
units.
is the midpoint of 
.
is the crease formed.
and 
.
.
.
.
,
.
