[/quote]
,
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
for which 
is the square of an integer
of natural numbers with 
of 
if the product of the elements of 
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
be an integer greater than zero. Then, the value of the sum of the reciprocals of the cubes of the roots of the equation![\[
mx^4 + 8x^3 - 139x^2 - 18x + 9 = 0
\]](http://latex.artofproblemsolving.com/a/3/a/a3a9bca412e927632e4921c51f2a6593409031d9.png)
is equal to:
for which there exists an even natural number 
such that the number![\[
(a - 1)(a^2 - 1)\cdots(a^n - 1)
\]](http://latex.artofproblemsolving.com/9/c/a/9caf4eeb82ff46b5ba55ab4b6bc28f0cace586ec.png)
is a perfect square.
and 
be on segment 
of an acute triangle 
such that 
and 
.
be the points on 
and 
,
and 
.
and 
is on the circumference of triangle 
,![\[
\sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x
\]](http://latex.artofproblemsolving.com/9/5/5/955863a0cf8747ae45b736b0631f243d3908eb84.png)
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
.
)
such that the equality 
holds for all pairs of real numbers 
.
is an obvious solution. Now, let 
.
or 
.
.
,
irreducible over 
?
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
matrix 
whose entries are all in 
,
is the matrix of all ones?
Y by TJK7, ahmedosama, Adventure10, Mango247
This is an integral marathon! Standard marathon rules, hide problems and solutions. I would appreciate it if you would add hint(s) to the problem as to help "guide" the reader to the correct solution (as I currently suck at integrating... rip). The first problem will be easy, and each problem after it will increase in difficulty minimally.
Remember: If you forget the +C term, you're going to have a bad time.
Now:
Remember: If you forget the +C term, you're going to have a bad time.
Now:
This post has been edited 4 times. Last edited by WireCat, Aug 3, 2016, 2:15 PM
,
,
And the area bounded by it is given by 
in terms of 
so the sides are each 
.
or 
.
.
.
by Taylor series? Can we evaluate 
,
.
.
around 
.![\[
\int_0^1 \frac{1-x}{(1+x)\ln x} dx. \]](http://latex.artofproblemsolving.com/6/b/d/6bd10a888da34f1b80fe292c964c11ec6da96c2a.png)
