POTD 2006-09

September 2006

Saturday, 30th of September

Today's POTDs courtesy of xkcd (http://xkcd.com/c135.html).

Friday, 29th of September

from Magnus-

Find a positive constant $k\,$ , such that $xy^2 - y^2 - x +y = k\,$ has precisely 3 solutions in positive integers.

Solution by flamingsp: Click here (http://www.infimum.org/metadata/POTD-2006-09-29.pdf)

Wednesday, 27th of September

(1) Prove that no order can be defined in the complex field that turns it into an ordered field.

(2) If $d(x,y)\,$ is a metric on set $X\,$ show that $delta(x,y) = \frac{d(x,y)}{1+d(x,y)}$ is also a metric on $X\,$ and that $X\,$ is bounded in terms of this metric. Do the two metrics $d(x,y)\,$ and $delta(x,y)\,$ determine the same open subsets of $X\,$ ? [Or Give a counter example]

Sunday, 24th of September

from Magnus-

Given $x,y\,$ in positive integers. Determine the least value of $|11\,x^5 - 7\,y^3|$

Saturday, 23rd of September

from Johnster via Kit

Let $X\,$ be a compact Hausdorff space and $\sigma : X \to X\,$ be a homeomorphism with no fixed points.

Let $I \subseteq C(X)\,$ be the ideal in generated by functions of the form $f - f \sigma\,$ . Show that $I\,$ is dense in (and hence equal to) $C(X)\,$ .

Not good enough at Topology? Here is a problem to attack with your computer algebra system skills.

from neobuddha via landen

Metal tanks are made by cutting rectangular squares out of the corners of rectangular sheets of metal. Then the sides are folded up and welded to form an open aquarium-shaped tank. The tank is to hold 10 cubic meters and the sheets can be gotten any size but they are $3X\,$ long and $X\,$ wide. What size sheets should be used so that the finished weight of the tanks is minimized.

A variation on this problem is very common in Calc I. In the common variation the area of the uncut metal is to be minimized for a given volume. The problem above is only slightly different but is harder. landen got an exact symbolic answer but not by hand. Get out your favorite CAS.

Friday, 22nd of September

from A 1987 STEP paper via Kit

I'm not sure if this is actually a good question, but it's so utterly random that I had to post it.

My two friends, who shall remain nameless, but whom I shall refer to as P and Q, both told me this afternoon that there is a body in my fridge. I'm not sure what to make of this, because P tells the truth with a probability of only p, while Q (independently) tells the truth with probability q. I haven't looked in the fridge for some time, so if you had asked me this morning, I would have said that there was just as likely to be a body in it as not. Clearly, in view of what P and Q told me, I must revise this estimate. Explain carefully why my new estimate of the probability of there being a body in the fridge should be

$\frac{pq}{1 - p - q + 2pq}$

I have now been to look in the fridge, and there is indeed a body in it; perhaps more than one. It seems to me that only my enemy A, or my enemy B, or (with a bit of luck) both A and B could be in my fridge, and this evening I would have judged these three possibilities to be equally likely. But tonight I asked P and Q separately whether or not A was in the fridge, and they each said that he was. What should be my new estimate of the probability that both A and B are in my fridge?

Of course, I always tell the truth.

Tuesday, 19th of September

from fido7.ru.math via Inept

Let $S\,$ be a finite set with $|S|\,$ being a prime number. Let $f:S \times S\rightarrow S\,$ be a binary operation such that $f(a,a)=a\,$ and $f(f(a,b),f(c,d))=f(a,d)\,$ for any $a,b,c,d\,$ from $S\,$ . Prove that either $f(a,b)=a\,$ for all $a,b \in S\,$ , or $f(a,b)=b\,$ for all $a,b \in S\,$ .

Solution (http://efnet-math.org/~david/POTD.pdf) by Kit. Solution by HiLander. Solution by koro

Sunday, 17th of September

From atomic by way of R^^n Find:

$\sum_{k=-\infty}^{\infty}\frac{1}{(3\,k+1)\,(4\,k+2)}$

Solution (http://efnet-math.org/~david/POTDSep17.pdf) by Kit

Saturday, 16th of September

from stakked, by way of HiLander:

Show that for every $c \in [-1,1]$ , there is an $a \in \mathbb{R}$ with $\lim_{n \to \infty} \sin(2\pi a n!) = c.$

Hint: Consider the POTD from 9/4.

Friday, 15th of September

from cheater

Prove:

$\sum_{k=1}^{\infty} \frac{1}{\left(k\left( 4k^2-1\right)\right)} = 2\ln 2-1$

Solution (http://encyclomaniacs.sound-club.org/~fs/temp/POTD-2006-09-15.pdf) from flamingspinach

Wednesday, 13th of September

from Karlsen

Show that $a_{n+1} = \frac{a_n }{ 2} + 1, a_0 = 0$ , converges, and find (and prove) a formula for $a_n.\,$

Solution (http://encyclomaniacs.sound-club.org/~fs/temp/POTD-2006-09-13.pdf) from flamingspinach

Tuesday, 12th of September

from scoobydew

Calc I students

$\mbox{Find: }\lim_{x\rightarrow \infty} \frac{x^2(1+\sin^2 x)}{(x+\sin x)^2}$

Monday, 11th of September

from cheater

Rated fairly hard. Students without some analysis will probably have trouble.

Let $a_n\,$ be a sequence of non-negative real numbers which satisfy

$a_{m+n} \leq a_m + a_n$

Show that

$\lim \limits_{n \to \infty} \frac{a_n}{n} = \inf\limits_{n > 0}\frac{a_n}{n}$

Solution by HiLander

Friday, 8th of September

Constructed by landen for Calc I Students. Please don't spoil in #math 'til Saturday UTC

$\lim_{x\rightarrow 0}{{{\sqrt{x^2+1}-\sqrt{1-x^2}}\over{1-\cos x}}}$

Solution by binrapt. Multiple Solutions (http://kilian.byethost5.com/mytemp/potd_9_11_2006.pdf) by _kmh_

Thursday, 7th of September

From a HS Contest. landen solved it with common theorems so it isn't too hard. Show that for positive real numbers $x,y,z\,$ :

$\left(y\,z+x\,z+x\,y\right)\,\left({{1}\over{\left(z+y\right)^2}}+ {{1}\over{\left(z+x\right)^2}}+{{1}\over{\left(y+x\right)^2}}\right) \geq {{9}\over{4}}$

Tuesday, 5th of September

From Polytope

Let $a_n = \int_0^{\pi/4} \tan^n x\, dx$ .

Show that $a_n + a_{n+2} = \frac{1}{n+1}$ for all non-negative integers $n$ .
Using this result, show that $\frac11 - \frac13 + \frac15 - \frac17 + \cdots = \frac{\pi}{4}$ .

Monday, 4th of September

From Kit

Show that

$\lim\limits_{n \to \infty} n \sin (2 \pi e n!) = 2 \pi$

Saturday, 2nd of September

From maks

Prove that every real number is the sum of two Liouville numbers. (A Liouville number is an irrational number x such that, for every positive integer n, there exists a rational number p/q so that |x - p/q| < 1/q^n)

Friday, 1st of September

From evilgeek

$\mbox{Find: }\sum_{n=1}^{\infty}\left( \frac{\lfloor \sqrt{n}\rfloor -\lfloor \sqrt{n-1} \rfloor}{n} \right)$

Solution from Karlsen.

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