AoPS Academy
![\[
f(x)=
\begin{cases}
\displaystyle\int_{x}^{\infty}\! u^{-1}e^{-u}\,du, & x>0,\\[6pt]
0, & x\le 0
\end{cases}
\]](http://latex.artofproblemsolving.com/e/a/7/ea723fe708579bd6e3e140f3ad705f7207bd2097.png)
is a probability density. Determine its characteristic function and give its power–series expansion.
is a random variable that takes on only nonnegative integer values, with ![$E[X]=1,$](http://latex.artofproblemsolving.com/e/8/2/e826f5b79a2adeacdaccd3884658e94a3b1f8e22.png)
![$E[X^2]=2,$](http://latex.artofproblemsolving.com/5/9/1/5915ee4a36f8615ddc8075a197f3b44a902936c6.png)
and ![$E[X^3]=5.$](http://latex.artofproblemsolving.com/d/1/1/d11c925af9f188f0140a89a0fbd6c83e1dd4fc12.png)
(Here ![$E[Y]$](http://latex.artofproblemsolving.com/a/2/9/a29ea940a5d54ad2a5d668a596f9db0c88fa1123.png)
denotes the expectation of the random variable 
) Determine the smallest possible value of the probability of the event 
variables 
, 
, 
, 
be chosen uniformly at random in the range ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
. Show that the value of ![\[\mathbb{E}(\min(X)\cdot \max(X)) = \dfrac{1}{n+2}.\]](http://latex.artofproblemsolving.com/b/8/4/b84d5581f617beda3c068283978a8c49e93613e0.png)
I'm pretty sure this requires calculus (although would love a non-calculus based proof), hence why I posted in College Math.
be a finite group, 
a subgroup of and 
a character of a -representation such that 
. Prove that the order of divides 
. I don't know many things about group representations. So any idea is useful.
. [/center]
, and 
, 
. This is proven in solution 1a-1. We will refer to this as Trick 1.
has the property for every positive integer 
, its 
term is greater than the mean of the first 
terms, and the sum of its first terms is a multiple of . Let 
be the number of such sequences satisfying 
. Compute the remainder when is divided by 
.
has side length 
. Point lies on segment 
such that line 
is perpendicular to line 
. Given that 
, the area of can be expressed as 
, where 
and are relatively prime positive integers. Compute 
.
and 
satisfy 
, 
, and 
. If the number of possible ordered pairs 
is equal to 
, compute the remainder when is divided by .
be a triangle. Point 
lies on side 
, point 
lies on side 
, and point 
lies on side 
such that 
, 
, and 
. Let be the foot of the altitude from 
to . Given that 
, 
, and ![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
, the value of 
can be expressed as , where and are relatively prime positive integers. Compute .
and tells Bob that the three resulting remainders are 
, 
, and , in some order. For how many values of is it possible for Bob to uniquely determine ?
satisfying the system of equations 
The value of 
can be expressed as 
, where 
and 
are positive integers such that is square-free. Compute 
. be the set of all monotonically increasing six-term sequences whose terms are all integers between 
and 
inclusive. We say a sequence 
in is symmetric if for every integer 
, the number of terms of 
that are at least 
is 
. The probability that a randomly chosen element of is symmetric is 
, where and are relatively prime positive integers. Compute ., let 
denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer 
such that the equation 
has at least ten positive integer solutions . be a quadratic polynomial with a positive leading coefficient. There exists a positive real number 
such that 
and 
for 
. Compute 
.
such that 
and 
is a multiple of 
.
be positive integers, is even such that 
. Prove that there exist positive integers 
satisfying 
and
are reducible polynomials in 
However, 
is an irreducible polynomial in So what are all the such that the polynomial:
is reducible in 
?
be a linear transformation on linear space 
satisfying:
but 
, and 
is the eigensubspace of eigenvalue . Prove that can be decomposed to 
-cyclic subspace's direct sum.
and 
.
be a basis of a complementary of 
in .
be a basis of a complementary of 
in .
be a basis of .
, where 
. associated to this basis is 
, where 
is the nilpotent Jordan block of dimension .
.
Prove that
Prove that
be reals such that 
and 
.
.
