[/quote]
,
such that 
is a perfect square but none of the integers 
are perfect squares.
and ![$P_1(x),P_2(x), \ldots P_n(x) \in \mathbb{Z}[x]$](http://latex.artofproblemsolving.com/c/6/4/c64176e52d94032e571b4579e35f9e6adfde675a.png)
such that 
.
such that every equation: 
has no real roots
,
and 
e
are tangent to 
at 
and 
respectively. 
intersects line 
.
intersects 
,
intersects 
.
intersects 
.
.
be a positive integer. A subset 
with four distinct elements is special if there exists a rearrangement 
of 
such that 
.
cannot be partitioned into 
(first player) and 
,
chips from the remaining pile of chips, where 
. No one can skip a turn. The player who at some point is unable to make a move (cannot remove chips from the pile) loses the game. Who among the two players can force a win on this game?
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
.
the number of changes of signs in the
Then prove that
.
be an acute triangle with orthocenter 
be a point on the side 
and 
.
and 
are the feet of the altitudes from 
is the circumcircle of 
,
be the point on 
is a diameter of 
the circumcircle of triangle 
,
be the point such that 
is a diameter of 
and 
greater than 
,
,
satisfying 
and their sum is an even number, consider the quadratic polynomial:![\[
P(x) = x^2 - (m^2 - m + 1)x + (m^2 - n^2 - m)(n^2 + 1).
\]](http://latex.artofproblemsolving.com/d/5/1/d51aaddfddfe06cc91abfdd3476a155770a8a5fd.png)
are positive integers but are not perfect squares.
and 
Prove that
.
be a line tangent to 
and 
.
and 
be diameters in 
and 
lies on 
and 
are tangent to 
and 
lies on 
and 
are tangent to 
.
,
times the radius of 
Y by
The number 
has the following nice properties:
(a) It is the sum of the fourth powers of fuve distinct positive integers.
(b) It is the sum of six consecutive positive integers.
In fact,
(1)
(2)
Prove that is the smallest number that satises both (a) and (b).
(You may assume that (1) and (2) are correct!)
has the following nice properties:(a) It is the sum of the fourth powers of fuve distinct positive integers.
(b) It is the sum of six consecutive positive integers.
In fact,
(1)
(2)Prove that
is the smallest number that satises both (a) and (b).(You may assume that (1) and (2) are correct!)
consecutive integers, it must be 
.
t
and 
(
)
and the second one is 
.
