AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 
. Prove that exists many 
such that every equation: 
has no real roots
, Its incircle 
and 
escircle 
are tangent to 
at 
and 
respectively. 
intersects line at 
. Line 
intersects at 
, and line 
intersects at 
. Line 
intersects at 
. Prove that 
.
be a positive integer. A subset 
with four distinct elements is special if there exists a rearrangement 
of 
such that 
. Prove that the set 
cannot be partitioned into special disjoint sets.
(first player) and 
, take alternate turns in playing a game using 2016 chips as follows: the player whose turn it is, must remove 
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 
. Prove 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)
. Denote by 
the number of changes of signs in the
Then prove that
.
be an acute triangle with orthocenter , and let 
be a point on the side , lying strictly between 
and 
. The points 
and 
are the feet of the altitudes from and , respectively. Denote by 
is the circumcircle of 
, and let 
be the point on such that 
is a diameter of . Analogously, denote by 
the circumcircle of triangle 
, and let 
be the point such that 
is a diameter of . Prove that 
and are collinear.
greater than 
, such that, if 
, then 
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 are externally tangent at a point 
. Let 
be a line tangent to at 
and at 
. Let 
and 
be diameters in and respectively. Suppose points 
and 
lies on such that 
and 
are tangent to , and points 
and 
lies on such that 
and 
are tangent to ., , , lie on a circle 
., , , determine a quadrilateral with an incircle 
, and its radius is 
times the radius of .
be a positive integer. Define the function 
by 
, where 
are the decimal digits of 
(
). Prove that every sequence 
of positive integer that satisfy 
for all 
, has a finite number of distinct terms. 
be a positive integer. Define the function by 
, where are the decimal digits of (
). Prove that every sequence 
of positive integer that satisfy 
for all , has a finite number of distinct terms. 
has two positive roots 
and that 
:
: 
(definition of 
).
, then :
, then ![$ b_{k + 1} = f(b_k)\le 9^p(1 + [\log_{10}(b_k)])\le b_k$](http://latex.artofproblemsolving.com/d/2/a/d2a22f30177e475f3867157c2efcab114cf89cd6.png)
(definition of 
) and so 
then ![$ b_{k + 1} = f(b_k)\le 9^p(1 + [\log_{10}(b_k)])\le 9^p(1 + \log_{10}(x_2)) = x_2 \le M$](http://latex.artofproblemsolving.com/0/a/8/0a8c515dca7ec999790746ac4acd5a95f49c450d.png)
and 
Prove that
