POTD 2007-02

Table of contents

February 2007

Monday, 26th of February

from dedekind

for a\in (0,1)\, and s>0\, find:

\int_0^{\infty} \frac{s\,x^{-a}}{1+s\,x}\,dx

in terms of a famous "higher" function

Saturday, 24th of February

from Polytope

Define two sequences \{a_n\}\, and \{b_n\}\, by

a_0 = 2\,, b_0 = 1\,,

a_{n+1} = \frac{2a_n\,b_n}{(a_n+b_n)}, and b_{n+1} = \sqrt{a_{n+1}\,b_n}.

Show that \lim a_n = \lim b_n = \frac{2\pi}{\sqrt{27}}

Here is a small hint.

Thursday, 22nd of February

from Crito

Find all positive integer pairs (a,n) such that \frac{(a+1)^n-a^n}{n} is an integer.

Solution by int-e

Wednesday, 21st of February

from Crito

Show that there exist infinitely many square-free positive integers n that divide 2005n − 1.

Solution by int-e

Tuesday, 20th of February

from Crito

Let a,b,c,d\, be positive real numbers such that a+b+c+d=1\,. Prove that

6(a^3+b^3+c^3+d^3)\ge(a^2+b^2+c^2+d^2)+\frac{1}{8}\,

Solution from landen

Tuesday, 13th of February

from math channel discussion

For k a positive integer and p a prime, what are the possible values of:

s(k)=\sum_{n=1}^{p-1}n^k \mod{p}

Solutions from int-e (first solver) and landen (different method).

Saturday, 10th of February

Mathica

Suppose that 0 < x_1 < x_2 < \ldots < x_n; define the function f(t) on [0,xn] as the proportion of those xi that are \le t. Let g on [0,xn] be the least concave majorant of f, meaning the unique concave function everywhere \ge f that is pointwise \le any other concave function that is pointwise \ge f. Evaluate \int_0^{x_n} g(t) dtas a function of \lbrace x_1,  x_2, \ldots, x_n\rbrace.

Tuesday, 6th of February

Crito

n\, is a natural number. d\, is the least natural number that for each a\, with \gcd(a,n)=1\,, we have that a^{d}\equiv1\pmod{n}\,. Prove that there exists a natural number b\, with \mbox{ord}_{n}b=d \,.