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.
and colors are given. We will say that a set of 
points on a plane is 
, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of the least integer 
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
points of every color there exists a subset.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
, let 
be the set of all positive integers not exceeding that are relatively prime to . Consider the polynomial![\[
P_n(x) = \sum_{k \in S_n} x^{k - 1}
\]](http://latex.artofproblemsolving.com/3/a/c/3ac60cb70baf4dfcf297e9af87480a8ee34ba70c.png)
and a polynomial 
with integer coefficients such that![\[
P_n(x) = (x^{r_n} + 1) Q_n(x).
\]](http://latex.artofproblemsolving.com/7/0/e/70e19a9d3e0a7e97abc7aa08d3f23126c14ea7d0.png)
such that 
is irreducible in ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
integers are such that 
is prime and 
. Prove that there exists a positive integer 
such that 
.
board, each square is colored either white or black, with all the squares on the border of the board being black. Additionally, no 
square within the board has all four squares of the same color. Prove that the board contains a square colored like a chessboard.
, where 
, then the number of integers in the range of 
is __
is inscribed in circle 
. Let 
denote its incenter and 
its 
-excenter. Let 
denote the midpoint of arc 
. Line 
meets a second time at 
. The perpendicular to 
at meets sides 
and 
at 
and 
respectively. The circumcircle of 
meets 
a second time at 
, and the circumcircle of 
meets 
a second time at 
. Prove that 
passes through the antipodal to on .
+1 w
and 
be two similar triangles (with the same orientation) and let 
be a parallelogram. Show that![\[\vert OF\vert \cdot \vert ON\vert=\vert OA\vert \cdot \vert OT\vert.\]](http://latex.artofproblemsolving.com/e/6/e/e6e4efefba491414308a738d6efb05bcb2347609.png)
. Angle chasing gives that 
and 
are cyclic and 
. Thus 
.
so 
and 
.
inversion on triangle 
.
is tangent to 
and 
is tangent to 
, goes to a point lying on the lines parallel to and passing through and respectively. But this point is , so in fact and swap. The desired equality follows.
mod 
. We have 
and by the inverse of the secant-tangent theorem, 
is a tangent to the circumcircle of 
and thus 
. But since is a parallelogram, we find 
. Now, let 
. We have 
. Therefore 
lies on the circumcircle of 
. Since 
, 
is tangent to 
. But is a point over arc , so 
, so 
is also cyclic. Hence, 
, meaning that 
. We deduce
and rearranging gives the result.
be reals such that 
and 
.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
satisfying 
,
,
,
