[/quote]
,
be integers such that there are consecutive integers 
satisfy 
.
is a perfect square.
a
Prove that
on one side and a 
on the other. Harry has 
of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing 
t
the process starting with the configuration 
would be 
,
,
be the number of operations before Harry stops. For example, 
and 
.
possible initial configurations 
,
,
of the
,
,
, 
.
such that
non-negative real numbers,but 
0![\[1\leq \frac{a+b}{a+4b+c}+\frac{b+c}{b+4c+a}+\frac{c+a}{c+4a+b}\leq \frac{4}{3}\]](http://latex.artofproblemsolving.com/8/b/7/8b7d7b8cbcab8f2eed31e17aac717fa3c70c92f1.png)
with 
.
and 
midpoint 
.
perpendicular to 
.
be a point on the segment 
such that 
and point 
be the intersection of 
and 
.
concyclic
be an integer. Consider an 
chessboard consisting of 
unit squares. A configuration of 
square which does not contain a rook on any of its 
unit squares.
students. Let, for 
,
student have 
many friends. For 
let us define 
to be the number of students who have strictly more than 
friends. Show that 
such that:
with all 
is real number
be a line. Let 
at points 
respectively. Given that these six points on 
,
are coaxal.
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
Prove that
Y by Adventure10, Mango247
Four points 
are chosen on a circle in such a way that arcs 
and 
are of the same length and the 
is longer than these three. Line 
and the line tangent to the circle at 
intersect at 
. Let 
be the other endpoint of the diameter starting at 
of the circle. Prove that triangle 
is equilateral.
are chosen on a circle in such a way that arcs 
and 
are of the same length and the 
is longer than these three. Line 
and the line tangent to the circle at 
intersect at 
. Let 
be the other endpoint of the diameter starting at 
of the circle. Prove that triangle 
is equilateral.
(that is simple to prove)
