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.
integers are such that 
is prime and 
. Prove that there exists a positive integer 
such that 
.
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
. Prove that there exists a positive integer 
satisfying 
.
+1 w
is described on a circle 
, and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
define 
be the product of first 
primes. Determine if there exists positive integers 
with the following property :![\[ m?=n(n+1)(n+2)(n+3) \]](http://latex.artofproblemsolving.com/4/5/c/45cb4a00f9395b8c6edb08a343c3bbd664916a4d.png)
be reals such that 
and 
Prove that
Let be reals such that and 
Prove that
Let be reals such that and 
Prove that
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.
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.
?
be real numbers such that![\[a+b+c+d=-2\]](http://latex.artofproblemsolving.com/c/7/3/c73dff68dcdfbe4f6b6b2f52b378548e47193a16.png)
![\[ab+ac+ad+bc+bd+cd=0\]](http://latex.artofproblemsolving.com/9/f/4/9f450dcb150229ee76e3debdaceadddb4e451458.png)
is not greater than 
.
and distinct positive integers 
are given. Does there exist a polynomial ![$p(x)\in\Bbb{Z}[x]$](http://latex.artofproblemsolving.com/5/1/a/51a23efd70e495d2fe516b642f0a6e71e5875ba2.png)
of degree 
that satisfies the following conditions?
of positive integers such that
be a triangle with 
. The points 
and 
lie on the side such that is perpendicular to and 
. The line that bisects 
meets 
and 
at 
and 
, respectively. Prove that the quadrilateral 
is cyclic.
be a triangle with . The points and lie on the side such that is perpendicular to and . The line that bisects meets and at and , respectively. Prove that the quadrilateral is cyclic.
, done.
and 
Prove that
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
.
being cyclic. Which is obvious since 
.
.