Challenge Intermediate Math Challenge - May 2018

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QuantumQuest

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It's time for an intermediate math challenge! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread!

RULES:
1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored.
2) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common knowledge to all mathematicians". Whether the latter is satisfied will be decided on a case-by-case basis.
3) If you have seen the problem before and remember the solution, you cannot participate in the solution to that problem.
4) You are allowed to use google, wolframalpha or any other resource. However, you are not allowed to search the question directly. So if the question was to solve an integral, you are allowed to obtain numerical answers from software, you are allowed to search for useful integration techniques, but you cannot type in the integral in wolframalpha to see its solution.
5) Mentors, advisors and homework helpers are kindly requested not to post solutions, not even in spoiler tags, for the challenge problems, until 16th of each month. This gives the opportunity to other people including but not limited to students to feel more comfortable in dealing with / solving the challenge problems. In case of an inadvertent posting of a solution the post will be deleted by @fresh_42

##1.## (solved by @julian ) Let ##A## be an ##\text{n x n}## matrix that is real skew symmetric, and with positive numbers ## \{x_1, ..., x_k\}##
where ##k \geq 2##. Prove that:

##\prod_{i = 1}^k \det\Big(A +x_i I\Big) \geq \det\Big(A +\big( \prod_{i = 1}^k x_i I \big)^\frac{1}{k}\Big)^k## ##\space## (by @StoneTemplePython)

##2.## (solved by @Biker ) Solve ##\mathcal{I}=\int_{-1}^0 \,x \cdot \sqrt{x^2+x+1} \,dx## . Hint: Use ##\cosh^2x - \sinh^2 x=1## . ##\space## (by @fresh_42)

##3.## (solved by @nuuskur ) Show that there are infinitely many prime numbers of the form ##4k + 3##, ##k \in \mathbb{N} - \{0\}## ##\space## (by @QuantumQuest)

##4.## (solved by @lpetrich ) What's the 100th digit after the decimal point of ##(1 + \sqrt{3})^{5000}## ##\space## (by @StoneTemplePython)

##5.## (solved by @lpetrich ) Given the differential operators ## D_n := x^n \cdot \dfrac{d}{dx}\,\,(n \in \mathbb{Z})## on smooth real valued functions ##\mathcal{C}^\infty(\mathbb{R})## .
Determine for which subsets ##L \subseteq \mathbb{Z}## the set ##\{D_n \,\vert \,n \in L\}## is a basis for a finite dimensional Lie algebra and which Lie algebra is it. ##\space## (by @fresh_42)

##6.## (solved by @lpetrich, @julian ) Calculate the integral ## I = \int_{0}^{\pi} \sin^{2n} x dx## , ##n \in \mathbb{N}## ##\space## (by @QuantumQuest)

##7.## (solved by @julian ) Solve ##\sum_{k=1}^\infty \dfrac{1}{k \binom{2k}{k}}## . ##\space## (by @fresh_42)

##8.## (solved by @julian ) Find the coordinates of the center of gravity for the arc of the curve ##y = a \cosh(\frac{x}{a})## for ##-a \leq x \leq a## ##\space## (by @QuantumQuest)

##9.## (resolved in post #62) For a given real Lie algebra ##\mathfrak{g}## , we define

##\mathfrak{A(g)} = \{\,\alpha \, : \,\mathfrak{g}\longrightarrow \mathfrak{g}\,\,: \,\,[\alpha(X),Y]=-[X,\alpha(Y)]\text{ for all }X,Y\in \mathfrak{g}\,\}##


the set of antisymmetric transformations of ##\mathfrak{g}##. Remember that a real Lie algebra is a real vector space equipped with a multiplication for which holds

  • anti-commutativity: ##[X,X]=0##
  • Jacobi-identity: ##[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0##

a) Show that ##\mathfrak{A(g)}\subseteq \mathfrak{gl}(g)## is a Lie subalgebra in the Lie algebra of all linear transformations of ##\mathfrak{g}## with the commutator as Lie product: ##[\alpha, \beta]= \alpha \beta -\beta \alpha## .
b) Show that ##\mathfrak{g} \ltimes \mathfrak{A(g)}## is a semidirect product (##\,\mathfrak{A(g)}## is the ideal ) given by

##[X,\alpha]:=[\operatorname{ad}X,\alpha]=\operatorname{ad}X\,\alpha - \alpha\,\operatorname{ad}X##


c) Show that for all ##\alpha \in \mathfrak{A(g)}## and ##X,Y,Z \in \mathfrak{g}##

##[α(X),[Y,Z]]+[α(Y),[Z,X]]+[α(Z),[X,Y]]=0## ##\space## (by @fresh_42)

##10.## (solved by @julian ) Calculate the integral ##\int_{c}^{} \frac{e^{kz}}{z}dz## where ##c## is the circle ## z = e^{i\theta}## with ##-\pi \leq \theta \leq \pi## and then prove that ##\int_{-\pi}^{\pi} e^{k\cos\theta} \cos(k\sin\theta)d\theta = 2\pi## ##\space## (by @QuantumQuest)
 
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I'll try problem 3. I have seen the original proof of there being infinitely many primes. I haven't done this particular exercise before, but a similar approach works. I employ only some introductory results about divisibility and the fundamental theorem of arithmetic.
Denote by ##\mathbb N ## the positive integers.
We assume towards a contradiction there are finitely many primes of the form ##4k+3## with ##k\in\mathbb N##. Let these be ##p_1,\ldots , p_t##. Consider
[tex]
M := 4p_1\ldots p_t
[/tex]
By hypothesis, ##M+3 ## is composite, therefore divisible by a prime. Note that ##\neg (3\mid M+3)##, otherwise ##3\mid M ##, then, by Euclid's lemma, ##3\mid p_j## which is possible only if ##k=0 ##. Since ##M+3 ## is odd, its prime factors must be of the form ##4k+1 ## or ##4k+3##. However, if any ##p_j\mid M+3 ##, then ##p_j\mid 3##, which is impossible. Now, all the prime factors of ##M+3 ## are of the form ##4k+1 ##. However
[tex]
(4k+1)(4l+1) = 4(4kl + k + l) +1.\tag{1}
[/tex]
Coupled with the fundamental theorem of arithmetic, (1) implies the number ##M+3## must be of the form ##4k+1 ##, a contradiction. Therefore, there must be infinitely many primes of the form ##4k+3 ##.
 
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3 is one of your ##p_i##. You constructed a number that is divisible by 3, and then claimed that it cannot be divisible by any prime of the form 4k+3. But it can, and we know an explicit example - it is divisible by 3.
 
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3 is one of your ##p_i##. You constructed a number that is divisible by 3, and then claimed that it cannot be divisible by any prime of the form 4k+3. But it can, and we know an explicit example - it is divisible by 3.
I have assumed ##p_j = 4k+3, k\in\mathbb N ## (positive integers) as stated in problem 3. The number ##M## doesn't contain ##3##.
 
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Ah, I didn't see the "positive integers" part. Then it works (a comment that 3 is not a ##p_i## would have helped), although you should mention that M+3 is not a multiple of 3. Without that observation you don't get a contradiction.
 
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Ah, I didn't see the "positive integers" part. Then it works (a comment that 3 is not a ##p_i## would have helped), although you should mention that M+3 is not a multiple of 3. Without that observation you don't get a contradiction.
I had assumed the observation is trivial, but you are right. Furthermore, I had invoked Euclid's lemma without mentioning it in my first draft.
 

fresh_42

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I had assumed the observation is trivial, but you are right. Furthermore, I had invoked Euclid's lemma without mentioning it in my first draft.
I think you could have made it easier to read. I fought my way through, but with a little help from your side it would have been much easier to read (##\,p_i \neq 3## per assumption, a few steps in the ##3|M## part and finally why ##M+3=4N+1## is impossible).

You reminded me of what I once told a student on how to write a thesis:
  1. Write it down so that you can understand each step, even after a year.
  2. Remove all but every third line (but keep a copy).
  3. Undelete four consecutive lines every two pages, so that the professor has something to criticize.
 

Math_QED

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I think you could have made it easier to read. I fought my way through, but with a little help from your side it would have been much easier to read (##\,p_i \neq 3## per assumption, a few steps in the ##3|M## part and finally why ##M+3=4N+1## is impossible).

You reminded me of what I once told a student on how to write a thesis:
  1. Write it down so that you can understand each step, even after a year.
  2. Remove all but every third line (but keep a copy).
  3. Undelete four consecutive lines every two pages, so that the professor has something to criticize.
Can't see why the third one is necessary.
 

QuantumQuest

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I'll try problem 3. I have seen the original proof of there being infinitely many primes. I haven't done this particular exercise before, but a similar approach works. I employ only some introductory results about divisibility and the fundamental theorem of arithmetic.
Checking your solution under spoiler I think that it is right but as already noted, it could be somewhat more well structured i.e. I needed quite a number of hops to understand some things. In all other respects it is good.
 
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##2.## Solve ##\mathcal{I}=\int_{-1}^0 \,x \cdot \sqrt{x^2+x+1} \,dx## . Hint: Use ##\cosh^2x - \sinh^2 x=1## . ##\space## (by @fresh_42)
You could use hyperbolic function and you could use normal trig function. I will go with trig functions instead. I hope it is correct
##\mathcal{I}=\int_{-1}^0 \,x \cdot \sqrt{(x+1/2)^2+3/4)} \,dx##
Use this substitution:
## \frac{\sqrt(3)}{2} tan(\theta) = x +1/2 ##
##\mathcal{I}=\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \, (\frac{ \sqrt{3}}{2} tan(\theta) - 1/2) \cdot \frac{3}{4} sec^3(\theta) \,d\theta##
##\mathcal{I}=\frac{3}{4} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \, (\frac{ \sqrt{3}}{2} tan(\theta) sec^3(\theta) - 1/2 sec^3(\theta)) \,d\theta##
Notice that the ## tan(\theta) sec^3(\theta)## is an odd function, no need to calculate the integral and the other part is an even function
##\mathcal{I}=\frac{-3}{4} \int_{0}^{\frac{\pi}{6}} \, sec^3(\theta) \,d\theta##
Using integration by parts for ##sec^3(\theta)## which is easy. You get:
##\mathcal{I}=\frac{-3}{8} ( \left. tan(\theta) sec(\theta) + ln(sec(\theta) + tan(\theta)) \right|_0^{\frac{\pi}{6}} ##
you get the answer -0.455989
 
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@QuantumQuest Is the solution to 6 is
##\frac{(2n-1)^2 \cdot (2n-3)^2 .... 1^2}{(2n)!} \pi## ?
 

fresh_42

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You could use hyperbolic function and you could use normal trig function. I will go with trig functions instead. I hope it is correct
##\mathcal{I}=\int_{-1}^0 \,x \cdot \sqrt{(x+1/2)^2+3/4)} \,dx##
Use this substitution:
## \frac{\sqrt(3)}{2} tan(\theta) = x +1/2 ##
##\mathcal{I}=\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \, (\frac{ \sqrt{3}}{2} tan(\theta) - 1/2) \cdot \frac{3}{4} sec^3(\theta) \,d\theta##
##\mathcal{I}=\frac{3}{4} \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \, (\frac{ \sqrt{3}}{2} tan(\theta) sec^3(\theta) - 1/2 sec^3(\theta)) \,d\theta##
Notice that the ## tan(\theta) sec^3(\theta)## is an odd function, no need to calculate the integral and the other part is an even function
##\mathcal{I}=\frac{-3}{4} \int_{0}^{\frac{\pi}{6}} \, sec^3(\theta) \,d\theta##
Using integration by parts for ##sec^3(\theta)## which is easy. You get:
##\mathcal{I}=\frac{-3}{8} ( \left. tan(\theta) sec(\theta) + ln(sec(\theta) + tan(\theta)) \right|_0^{\frac{\pi}{6}} ##
you get the answer -0.455989
That's correct, although you could have made life much easier for an old man. I don't know all the trig formulas in mind anymore. And to demonstrate the integration by parts would have been a nice service for the younger among us. However, the argument with the odd function was nice, so I'll not complain about the wrong round-off of the result, although an exact solution would have been better:
$$
\int_{-1}^0 \,x \cdot \sqrt{x^2+x+1} \,dx = -\frac{1}{4} - \frac{3}{16} \log 3 \approx -0.45598980\ldots \approx -0.456
$$
 
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That's correct, although you could have made life much easier for an old man. I don't know all the trig formulas in mind anymore. And to demonstrate the integration by parts would have been a nice service for the younger among us. However, the argument with the odd function was nice, so I'll not complain about the wrong round-off of the result, although an exact solution would have been better:
$$
\int_{-1}^0 \,x \cdot \sqrt{x^2+x+1} \,dx = -\frac{1}{4} - \frac{3}{16} \log 3 \approx -0.45598980\ldots \approx -0.456
$$
Yeah, I am sorry. When I saw the question, I imagined having an exam with this question. So I tried the first thing that came to my mind, Trig subs. When I reached the answer. I tried the hint, Which is much simpler and you can still do the odd-even trick which will simplify it even more. So I just posted the answer that first came to my mind without the hint. Because that is a big hint.
 

fresh_42

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So I just posted the answer that first came to my mind without the hint. Because that is a big hint.
No problem, keeps my mind busy :wink:. There are a few questions on this level which could have well been on "B" level, and it was only the machinery that made them "I". And the metric problem on the "B" list is actually the easiest of all, but still untouched. Seems we can't accurately predict the level, yet. As I said before, it's not as easy as we thought to find good problems ... and to assess their difficulty.
 

QuantumQuest

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Is the solution to 6 is
No. I would really like to give the correct answer but I can't as this is a challenge. But what I can ask you is to think about the problem again. If you come up with a way which is in the right direction, I maybe able to give you some hint along the way.
 
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No. I would really like to give the correct answer but I can't as this is a challenge. But what I can ask you is to think about the problem again. If you come up with a way which is in the right direction, I maybe able to give you some hint along the way.
Weird, I tried a few examples with my definite integral calculator and it seems to be correct. Isn't the integral (sin x)^2n?

I used integration by part to get a recursive formula for (sinx)^n then noticed that the integration is from 0 to pi which simplified the integral to the last term
 

QuantumQuest

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@Biker

I see that you put good efforts so I'll ask can you change the limits of integration to a circle and maybe think outside real numbers?
 

julian

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Is the solution to problem 8 this:

The center of gravity is ##(0,\overline{y})## where

##
\overline{y} = \lim_{\delta s \rightarrow 0} \frac{\sum y \; \delta s}{\sum \delta s} = \frac{1}{s} \int y ds
##

where ##\delta s## is a small arc length. The ##x##-component of the center of gravity is zero because ##a \cosh (x/a)## is symmetric in ##x##. Now

##
d s = \sqrt{(d x)^2 +(dy)^2} = \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^{1/2} dx
##

So that

##
s = \int_{-a}^a \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^{1/2} dx \quad \mathrm{and} \quad \overline{y} s = \int_{-a}^a y \left( 1 + \left( \frac{dy}{dx} \right)^2 \right)^{1/2} dx .
##

We first calculate the arc length ##s##:

\begin{align}
s & = \int_{-a}^a \left[ 1 + \left( \sinh \frac{x}{a} \right)^2 \right]^{1/2} dx
\nonumber \\
& = \int_{-a}^a \left[ \cosh^2 \frac{x}{a} \right]^{1/2} dx
\nonumber \\
& = \int_{-a}^a \cosh \frac{x}{a} dx
\nonumber \\
& = \left[ a \sinh \frac{x}{a} \right]_{-a}^a
\nonumber \\
& = 2a \sinh (1) .
\nonumber
\end{align}

And now

\begin{align}
\overline{y} s & = \int_{-a}^a a \cosh \frac{x}{a} \times \cosh \frac{x}{a} dx
\nonumber \\
& = \int_{-a}^a \frac{a}{2} (\cosh \frac{2x}{a} + 1) dx
\nonumber \\
& = \frac{a}{2} \left[ \frac{a}{2} \sinh \frac{2x}{a} + x \right]_{-a}^a
\nonumber \\
& = \frac{a^2}{4} 2 \sinh (2) +a^2= \frac{a^2}{2} \sinh (2) + a^2 .
\nonumber
\end{align}

So that

##
\overline{y} = \frac{a^2}{2} (\sinh (2) +2) / 2a \sinh (1) = \frac{a}{4} \frac{\sinh (2) + 2}{\sinh (1)} = \frac{a}{4} \frac{e^2 - e^{-2} + 2}{e - e^{-1}} .
##
 
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dRic2

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I'm a bit embarrassed to ask because I'm not skilled enough to solve any of this problems, but, out of curiosity,

if ##F(z) = \int \frac {e^{kz}} z dz##

is it ok to assume:

##I = \int_c \frac {e^{kz}} z dz = F(z_a) - F(z_b) = F(e^{i\pi}) - F(e^{-i\pi})## ?

or did I misunderstood the problem?

PS: Don't eat me alive if I wrote a blasphemy :biggrin::biggrin:
 
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I'm a bit embarrassed to ask because I'm not skilled enough to solve any of this problems, but, out of curiosity,

if ##F(z) = \int \frac {e^{kz}} z dz##

is it ok to assume:

##I = \int_c \frac {e^{kz}} z dz = F(z_a) - F(z_b) = F(e^{i\pi}) - F(e^{-i\pi})## ?

or did I misunderstood the problem?

PS: Don't eat me alive if I wrote a blasphemy :biggrin::biggrin:
Generally an integral is not determined by the end points of the path of integration.
 

QuantumQuest

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I'm a bit embarrassed to ask because I'm not skilled enough to solve any of this problems, but, out of curiosity,

if ##F(z) = \int \frac {e^{kz}} z dz##

is it ok to assume:

##I = \int_c \frac {e^{kz}} z dz = F(z_a) - F(z_b) = F(e^{i\pi}) - F(e^{-i\pi})## ?

or did I misunderstood the problem?

PS: Don't eat me alive if I wrote a blasphemy :biggrin::biggrin:
How did you get that? Note that there is a known formula that can make things a lot easier for this particular problem. Can you recognize which is it?
 

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