[/quote]
,
on one variables with real coefficients such that the equation 
has exactly 
ordered pairs 
as solutions?
.
be positive reals such that 
.
(Shining Sun, USA)
be an acute triangle with 
,
of side 
,
)
.
and 
,
,
.
intersects line 
at a second point 
.
and 
are perpendicular.
such that exactly 
of the integers 
are squarefree.
be a non-constant monic polynomial with integer coefficients and let 
be an infinite sequence. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
.
cells of an 
square grid are colored black, and the remaining cells are white. The cost of such a coloring is the minimum number of white cells that need to be recolored black so that from any black cell 
,
through a sequence 
of black cells where each consecutive pair 
are adjacent (sharing a common side) for every 
.
denote the maximum possible cost over all initial colorings with exactly 
such that![\[
\frac{1}{3}n^{\alpha} \leq f(n) \leq 3n^{\alpha}
\]](http://latex.artofproblemsolving.com/0/a/0/0a06d4ff52479c8cb785c6450cea8748bf7769b6.png)
for any 
.
be the orthocenter of triangle 
.
be the circumradius of 
be a circle, 
be a diameter of the circle and for each point 
let 
be the foot of the perpendicular from 
and 
the base of the perpendicular from 
on 
.
be the geometric locus of all points 
grid, two cells are considered neighboring if they share a common edge. Kritesh performs the following actions:
in any cell of the grid.
.
, s.t :
![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
of positive integers satisfying the following two conditions:
,
is in 
,
and 
;
such that 
is not divisible by 
Y by
The number 
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
so not exactly once
?
such that 
,
is a perfect square, and no two 
are the same.
