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.
be real numbers such that 
. Prove that
Let be real numbers such that 
. Prove that
and 
, we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.
be a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points 
and 
. Let 
and 
. Prove that the points 
, 
, and are concyclic. and are not the diagnols)
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
. Prove that there exists a positive integer satisfying 
.
?
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 
.
+1 w
be a triangle with right 
. Consider points 
and 
such that 
. Prove that 
![\[
\begin{gathered}
\left\{ \begin{gathered}
{\rm A}{\rm Z} = {\rm A}D \hfill \\
EK//AD \hfill \\
\end{gathered} \right.\mathop {}\nolimits_{}^{} \Rightarrow \mathop {}\nolimits_{}^{} \left\{ \begin{gathered}
\measuredangle Z{\rm B}C = \sphericalangle ZCB = 90^O - \varphi \hfill \\
BZ = BD \hfill \\
\end{gathered} \right.\mathop {}\nolimits_{}^{} \Rightarrow \hfill \\
\hfill \\
\left\{ \begin{gathered}
BZ = ZC \hfill \\
ZK = DE = DC \hfill \\
BE = BK = BZ - ZK = ZC - DC = ZD = 2AD \hfill \\
\end{gathered} \right. \hfill \\
\end{gathered}
\]](http://latex.artofproblemsolving.com/5/2/9/5291f211f7b47c15e749b8a7325177ea149f028e.png)
be reals such that 
and 
.
