POTD 2007-02

Table of contents

1 February 2007

1.1 Monday, 26th of February
1.2 Saturday, 24th of February
1.3 Thursday, 22nd of February
1.4 Wednesday, 21st of February
1.5 Tuesday, 20th of February
1.6 Tuesday, 13th of February
1.7 Saturday, 10th of February
1.8 Tuesday, 6th of February

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 $2005 n - 1$ .

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 $[0, x n]$ as the proportion of those $x i$ that are $\le t$ . Let $g$ on $[0, x n]$ 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 $\int_0^{x_n} g(t) dt$ as 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 $\mbox{ord}_{n}b=d \,.$

Retrieved from "http://efnet-math.org/w/POTD_2007-02"