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.
denote the set of all positive integers less than 
that are relatively prime to . Let 
. Is the polynomial 
reducible over the rational numbers, given that it has integer coefficients?
is defined by 
and for all positive integers 
, 
. Show that for every positive integer 
, there exists a suitable positive integer 
such that 
is a multiple of .
. Let 
be a subset of such that no sum of three distinct elements of is divisible by 
. What is the maximum value of 
.
has circumcircle 
and incenter 
. The incircle of triangle touches side 
at 
respectively. Circumcircle of triangle 
intersects again at 
distinct from 
, circumcircle of triangle 
intersects again at 
distinct from 
. Prove that the 
points 
are concyclic.
lies inside triangle so that 
. The point is such that 
is a parallelogram. Prove that 
.
, point 
on 
satisfies 
and 
.
be the point where the perpendicular from 
to line 
meets the angle bisector of 
.
holds. is an isosceles triangle.
be a positive integer, and let its positive divisors be![\[
d_1 < d_2 < \cdots < d_k.
\]](http://latex.artofproblemsolving.com/6/3/e/63ec82aba0d6c4eece89af6a1247a89498616281.png)
Define 
to be the number of ordered pairs 
with 
such that 
.
. when![\[
n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k},
\]](http://latex.artofproblemsolving.com/2/7/2/272dc2cc33ebbf695311cb61567fa0297e1299b6.png)
where the 
are distinct primes and the 
are positive integers.
such that 
.
that verify the following equation for all 
:
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
that satisfy the equation 
.
is chosen at random on a circular disc. What is the chance that the chord with as mid-point is longer than the radius?
of a circle are chosen at random. What is the chance that both 
are longer than the radius?
is chosen at random on a circular disc. What is the chance that the chord with as mid-point is longer than the radius? of a circle are chosen at random. What is the chance that both are longer than the radius?
as a midpoint so probability is 
?
then if the minor angle formed by 
is less than 
then one of the chords is less than the radius. So the diameter 
(unit circle) so we have 
.
The distance of the chord that contains 
.
,
The probability of this occurring is
and 
.
,
.