[/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 Adventure10, Mango247, Mango247, Mango247
In a 
board, a piece called dragon moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board.
The draconian distance between two squares is defined as the least number of moves a dragon needs to move from one square to the other.
Let
be a corner square, and 
the square neighbor of that has only a point in common with . Show that there exists a square 
of the board, such that the draconian distance between and is greater than the draconian distance between and .
board, a piece called dragon moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board.The draconian distance between two squares is defined as the least number of moves a dragon needs to move from one square to the other.
Let
be a corner square, and 
the square neighbor of that has only a point in common with . Show that there exists a square 
of the board, such that the draconian distance between and is greater than the draconian distance between and .
,
grid is 
,
and wlog because of symmetry the first coordinate is horizontal and the second coordinate is vertical. We will call "horizontal" a move that changes the horizontal coordinate by 4 and the vertical coordinate by 1; "vertical" moves are defined reciprocally.
,
.
.
horizontal moves and 
vertical moves, it must hold that 
and 
,
,
.
Overall very simple (to IMO standards) problems, but good fun nonetheless 
