POTD 2006-11

Table of contents

1 November 2006

1.1 Thursday, 30th of November
1.2 Wednesday, 22nd of November
1.3 Thursday, 16th of November
1.4 Saturday, 11th of November
1.5 Friday, 10th of November
1.6 Tuesday, 7th of November
1.7 Sunday, 5th of November
1.8 Saturday, 4th of November
1.9 Friday, 3rd of November

November 2006

Thursday, 30th of November

from Crito

Let $x\,$ and $y\,$ be positive integers such that $xy|x^2 +y^2 +1\,$ . Show that $\frac{x^2+y^2+1}{xy}=3$

Solution by int-e

Wednesday, 22nd of November

from dedekind

Let $1 = a_1 \leq a_2 \leq \ldots$ and $a m a n = a m n$ . Prove that either $a n = 1$ for all $n$ , or $a n = n 1 / p$ for some $p > 0$ .

Solution by Kit

Thursday, 16th of November

Show that $\sum^{\infty}_{n=1}\frac{(-1)^{n-1}\sin(\log n)}{n^\alpha}$ converges iff $\alpha>0\,$ .

Solution by landen by famous secret method.

Saturday, 11th of November

from Crito

1. $a_{n}\,$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3\,$ , and

$a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}$ Prove that for each natural $n\,$ , $a_{n}\,$ is integer.

Solution

2. Let $m\,$ and $n\,$ be positive integers such that $n \leq m$ . Prove that

$2^n n! \leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n$

3. For positive $a,b,c,\,$

$a^{2}+b^{2}+c^{2}+abc=4\,$ Prove $a+b+c \leq3$

4. $a_{0}=2,\ a_{1}=1\,$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1},\$ $m\,$ is an even number, and $p\,$ is prime number such that $p\,$ divides $a_{m}-2\,$ . Prove that $p\,$ divides $a_{m+1}-1\,$ .

5. $π(n)$ is the number of primes that are not bigger than n. For $n=2,3,4,6,8,27,30,33,\dots$ we have $π(n) | n$ . Do there exist infinitely many integers n that $π(n) | n$ ?

Friday, 10th of November

from Kit

Let $X$ be a separable topological space, and $C (X)$ the space of continuous functions from $X$ to $\mathbb{R}$ with the product topology. Show that compact subsets of $C (X)$ are metrizable.