POTD 2006-12

Table of contents

1 December 2006

1.1 Saturday, 30th of December
1.2 Friday, 29th of December
1.3 Thursday, 28th of December
1.4 Monday, 25th of December
1.5 Sunday, 24th of December
1.6 Saturday, 23rd of December
1.7 Wednesday, 20th of December
1.8 Tuesday, 19th of December
1.9 Saturday, 16th of December
1.10 Friday, 15th of December
1.11 Friday, 8th of December

December 2006

Saturday, 30th of December

edited by landen

$f(x)\,$ and $g(x)\,$ are real valued functions defined on all of $\mathbb{R}\,.$ $f(x)\,$ is not always 0. $|f(x)|\leq1.\,$

For all $x,y\in \mathbb{R},\,$ $f(x+y)+f(x-y)=2\,f(x)\,g(y).$

Show that $|g(x)|\leq 1.$

Solution by i_c-Y

Friday, 29th of December

suggested by landen

${a k}$ is a sequence of distinct positive integers. Prove that for all positive integers $n,\ \sum_1^n{a_k\over k^2}\ge\sum_1^n{1\over k}$

Solution (http://encyclomaniacs.sound-club.org/~fs/math/POTD-2006-12-29.pdf) by flamingsp

Thursday, 28th of December

from Crito

If $a,b,c,x\,$ are real numbers such that $a,b,c>0\,$ and $\frac{xb + (1-x)c}{a} = \frac{xc + (1-x)a}{b} = \frac{xa + (1-x) b }{c}$ , then prove that $a = b = c\,$ .

Solution by int-e.

Monday, 25th of December

from Crito

Find all non-negative integers $m, n, p, q$ such that $p m * q n = (p + q) 2 + 1$ .

Sunday, 24th of December

from landen for ziggggggy and i_c-Y

1. Find the general solution:

$y''-\frac{3}{x}\,y'-\frac{5}{x^2}\,y = \log{x}$

Hint: notice $y'\,$ is divided by $x\,$ and $y\,$ is divided by $x^2\,$ so you get $0\,$ when added to $y''.\,$ Think of a super common type of function where that might happen.

2. Solve your favorite:

a. $y' + 3\,y = x^3$
b. $y'' + 5\,y' + 6\,y = e^x$
c. $y'' - 2\,y' + y = x$
d. $y''' -3\,y' +2\,y = 2\,(\sin{x}-2\,\cos{x})$

Saturday, 23rd of December

posted by landen

The positive integers $a,\; b$ are such that $15\,a+16\,b$ and $16\,a - 15\,b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

Wednesday, 20th of December

from Crito

Suppose that m and n are odd integers such that $m 2 - n 2 + 1$ divides $n 2 - 1$ . Prove that $m 2 - n 2 + 1$ is a perfect square.

Tuesday, 19th of December

from ermular

1. Find: $\sum_{k=0}^\infty \frac{4}{(4k)!}$

Solution by landen. Other solution by Kit.

Saturday, 16th of December

from Kit

1. Let $V$ be a normed space, $W \subseteq V$ a closed subspace. Fix $t > 0$ and let $A = \{ v \in V : d(v, W) < t \} \,$ .

Show there is a continuous function $f : A \to W$ with $||f(v) - v|| < t \,$

Not particularly hard, but might require some advanced knowledge.

from Crito

2. Prove that the equation $6(6 a 2 + 3 b 2 + c 2) = 5 n 2$ has no solutions in integers except a = b = c = n = 0. Solution by int-e.

3. Let a, b, c be the lengths of the sides of a triangle. Prove that

$\sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c}$

and determine when equality occurs. Solution by landen.

Friday, 15th of December

from Koro

1) Find a non-measurable subset of the plane such that all its vertical and horizontal sections are measurable.

2) Can you find it such that all vertical and horizontal sections consist of a single point?

Friday, 8th of December

from Polytope

Please do not solve in the channel before Dec 10 if you have had real analysis. This is for smart Calc students.

Let $f(x)\,$ be a function of a real variable with real values.

If $x\,$ is irrational then $f(x)=0\,$

If $x\,$ is rational $m/n\,$ in lowest terms, $f(x)=1/n\,$

$f(0)=1\,$

Show that $f(x)\,$ is nowhere differentiable.

Solution from landen
Solution from hilander

from Crito

Source: <insert-some-strange-corners-of-the-web-here>

Prove that $p(x)=(x^2+x)^{2^n}+1$ is irreducible in $\mathbb Z[x]$

Solution (http://www.efnet-math.org/math_tech/BonusProbDec0806.pdf) by hochs.

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