# Math Challenge - November 2020

• Challenge
• Featured
There should be some intermediate questions for early undergraduates who haven't taken any advanced math yet.

member 587159
@etotheipi
Moreover, Taylor and MVT are not really school level. Induction is.
I saw both Taylor and MVT in high school, but only one induction proof.

Mentor
There should be some intermediate questions for early undergraduates who haven't taken any advanced math yet.
We tried, but it didn't work. What is intermediate? Taylor and MVT in high schools (see post #24), or are even induction and limits too difficult? Especially school systems vary significantly all over the world.

And on the other end: Problems #4 and #5 are not very sophisticated. I would have called them "intermediate", and even the algebraic questions are not so difficult. We even had questions which could have been solved by looking up some definitions on Wikipedia, and even this was too much to do. People get scared by technical terms, but our questions are already heavily limited to only easy concepts. I tried to find some more interesting questions in my books ... no chance! I would need to hold a lecture beforehand, regardless whether a question would had been easy or difficult, simply to set up the language.

We have a thread with solutions to many questions:
where I think can be found problems at any level.

StoneTemplePython
And on the other end: Problems #4 and #5 are not very sophisticated. I would have called them "intermediate", and even the algebraic questions are not so difficult. We even had questions which could have been solved by looking up some definitions on Wikipedia, and even this was too much to do.
I agree. I found problem #5 could be solved easily in a few minutes by referring to https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections. I like this problem because it showed me a beautiful connection between linear algebra and geometry. Thank you for making the effort to post the monthly challenges, I learn a lot from them

etotheipi and fresh_42
etotheipi
What is a Euclidean normal form, and what do we mean by ##(x,y)^{\tau}##?

member 587159
What is a Euclidean normal form, and what do we mean by ##(x,y)^{\tau}##?

I believe it means the vector ##(x,y)## transposed, so written as column vector ##\begin{pmatrix}x \\ y\end{pmatrix}##.

Mentor
What is a Euclidean normal form, and what do we mean by ##(x,y)^{\tau}##?
Yes, it is only a habit in this case to write vectors or coordinates as column. The Euclidean normal form is, when the axis of the conic sections are the coordinate axis, and its center the origin of the Cartesian coordinate system: rotations, translations; basically such that all interesting quantities can directly be read.

etotheipi
So if I understand correctly,
$$A_Q = \begin{pmatrix} 34 & 12 & 10 \\ 12 & 41 & 55 \\ 10 & 55 & 50\end{pmatrix}$$and ##f(x,y) = \mathbf{x}^{\tau}A_Q \mathbf{x}##, ##\text{det}(A_Q) = -31250##, and ##\text{det}(A_{33}) = 1250##, and ##\frac{\text{det}(A_Q)}{\text{det}(A_{33})} = -25##, so I guess when you say you want it in normal form, since ##A_{33}## has eigenvalues ##\lambda = 25, 50##, it will be$$50u^2 + 25v^2 = 25 \iff 2u^2 + v^2 = 1 \iff \left(\frac{u}{1/\sqrt{2}}\right)^2 + \left(\frac{v}{1}\right)^2 = 1$$so vertices at ##(u,v) = \left( \pm 1/\sqrt{2}, 0\right), \left( 0, \pm 1\right)## and foci at ##(0, \pm \frac{1}{2})##? Is that what you had in mind, @Fred Wright?

Mentor
So if I understand correctly,
$$A_Q = \begin{pmatrix} 34 & 12 & 10 \\ 12 & 41 & 55 \\ 10 & 55 & 50\end{pmatrix}$$and ##f(x,y) = \mathbf{x}^{\tau}A_Q \mathbf{x}##, ##\text{det}(A_Q) = -31250##, and ##\text{det}(A_{33}) = 1250##, and ##\frac{\text{det}(A_Q)}{\text{det}(A_{33})} = -25##, so I guess when you say you want it in normal form, since ##A_{33}## has eigenvalues ##\lambda = 25, 50##, it will be$$50u^2 + 25v^2 = 25 \iff 2u^2 + v^2 = 1 \iff \left(\frac{u}{1/\sqrt{2}}\right)^2 + \left(\frac{v}{1}\right)^2 = 1$$so vertices at ##(u,v) = \left( \pm 1/\sqrt{2}, 0\right), \left( 0, \pm 1\right)## and foci at ##(0, \pm \frac{1}{2})##? Is that what you had in mind, @Fred Wright?
Well, you could have shown how you did it. It's a bit difficult to learn anything from your solution. And the foci are ##(0,\pm 1/\sqrt{2})## if I wasn't mistaken.

Office_Shredder
Staff Emeritus
Gold Member
As far as making questions more accessible, you could have just asked something like construct a sequence of functions that are differentiable whose limit is not differentiable, and not scared off people that haven't heard of a Banach space.

Mentor
As far as making questions more accessible, you could have just asked something like construct a sequence of functions that are differentiable whose limit is not differentiable, and not scared off people that haven't heard of a Banach space.
That isn't what the problem asks for.

member 587159
etotheipi
Well, you could have shown how you did it. It's a bit difficult to learn anything from your solution. And the foci are ##(0,\pm 1/\sqrt{2})## if I wasn't mistaken.
Yes, sorry, I forgot to square root. Anyway, I'll give a more complete method here for anyone that wants it. First we need to find the centre of $$f(x,y) = 34x^2 + 24xy +41y^2 + 20x + 110y + 50=0$$Set ##f_y = 0## and ##f_x = 0## to get\begin{align*}68x + 24y + 20 &= 0 \\ 82y + 24x + 110&=0 \end{align*}this is solved by ##\mathbf{x}_c = (x_c,y_c)^{\tau} = (\frac{1}{5}, \frac{-7}{5})^{\tau}##, i.e. this is the centre of the ellipse. Now we perform a coordinate transformation ##\mathbf{x}' = \mathbf{x} -\mathbf{x}_c## so that the centre is at our new origin ##\mathcal{O}'##, i.e.$$34(x' + \frac{1}{5}) + 24(x' + \frac{1}{5})(y' - \frac{7}{5}) + 41(y' - \frac{7}{5})^2 + 20(x' + \frac{1}{5}) + 110(y' - \frac{7}{5}) + 50 = 0$$ $$34x'^2 + 24x'y' + 41y'^2 = 25$$Now we perform a further coordinate transformation, this time a rotation (preserving the origin, ##\mathcal{O}'' = \mathcal{O}'##), so that the new axis intersect the ellipse at the vertices. This will be ##\mathbf{x}'' = R\mathbf{x}'##, where ##R = \begin{pmatrix} c & s \\ -s & c\end{pmatrix}##, and ##c=\cos{\theta}##, ##s = \sin{\theta}## for some angle ##\theta## to be determined. Running through the algebra with ##x' = cx'' - sy''## and ##y' = sx'' + cy''## yields$$34(cx'' -sy'')^2 + 24(cx'' - sy'')(sx'' + cy'') + 41(sx'' + cy'') = 25$$ $$x''^2 [ 34 c^2 + 41s^2 + 24cs] + y''^2 [41c^2 + 34s^2 - 24cs] + x''y''[24c^2 - 24s^2 + 14cs]$$We require that the coefficient of ##x''y''## is brought to be zero, i.e. that ##s = 4/5## and ##c=3/5##. Now it's just a case of substituting those back into the equation,$$50x''^2 + 25y''^2 = 25 \iff 2x''^2 + y''^2 = 1$$and for brevity I'll re-label ##x'' \equiv u## and ##y'' \equiv v##. And that completes the transformation!

fresh_42
Yes, sorry, I forgot to square root. Anyway, I'll give a more complete method here for anyone that wants it. First we need to find the centre of $$f(x,y) = 34x^2 + 24xy +41y^2 + 20x + 110y + 50=0$$Set ##f_y = 0## and ##f_x = 0## to get\begin{align*}68x + 24y + 20 &= 0 \\ 82y + 24x + 110&=0 \end{align*}this is solved by ##\mathbf{x}_c = (x_c,y_c)^{\tau} = (\frac{1}{5}, \frac{-7}{5})^{\tau}##, i.e. this is the centre of the ellipse. Now we perform a coordinate transformation ##\mathbf{x}' = \mathbf{x} -\mathbf{x}_c## so that the centre is at our new origin ##\mathcal{O}'##, i.e.$$34(x' + \frac{1}{5}) + 24(x' + \frac{1}{5})(y' - \frac{7}{5}) + 41(y' - \frac{7}{5})^2 + 20(x' + \frac{1}{5}) + 110(y' - \frac{7}{5}) + 50 = 0$$ $$34x'^2 + 24x'y' + 41y'^2 = 25$$Now we perform a further coordinate transformation, this time a rotation (preserving the origin, ##\mathcal{O}'' = \mathcal{O}'##), so that the new axis intersect the ellipse at the vertices. This will be ##\mathbf{x}'' = R\mathbf{x}'##, where ##R = \begin{pmatrix} c & s \\ -s & c\end{pmatrix}##, and ##c=\cos{\theta}##, ##s = \sin{\theta}## for some angle ##\theta## to be determined. Running through the algebra with ##x' = cx'' - sy''## and ##y' = sx'' + cy''## yields$$34(cx'' -sy'')^2 + 24(cx'' - sy'')(sx'' + cy'') + 41(sx'' + cy'') = 25$$ $$x''^2 [ 34 c^2 + 41s^2 + 24cs] + y''^2 [41c^2 + 34s^2 - 24cs] + x''y''[24c^2 - 24s^2 + 14cs]$$We require that the coefficient of ##x''y''## is brought to be zero, i.e. that ##s = 4/5## and ##c=3/5##. Now it's just a case of substituting those back into the equation,$$50x''^2 + 25y''^2 = 25 \iff 2x''^2 + y''^2 = 1$$and for brevity I'll re-label ##x'' \equiv u## and ##y'' \equiv v##. And that completes the transformation!
How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.

etotheipi
How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.

Coordinate transformations I learned from Douglas Gregory, Classical Mechanics! It's not so bad, you just need to be careful about when you're leaving the vector alone but changing its representation i.e. w.r.t. a new coordinate system (passive), or if you're actually transforming the vector (active), and not getting the signs mixed up!

Mayhem
Mentor
How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.
It is not too difficult:

Given the red one, asked for the blue one. You have to rotate the curve, stretch it, and shift the center into the origin. You can either handle this with linear algebra (hint in post #29, or wait for the long version in my solution manual), or simply perform coordinate transformations (post #44) until it fits. We want to have an ellipse at the end: ##\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1##.

Imagine the ellipse is a crystal instead, and the red one is what you see under the microscope, whereas you need the blue version to identify it by the pictures in the book.

Mayhem
LCSphysicist
2020 Award
How did you learn this stuff when you're a first year uni student? I'm at the top of my first year math course for chemistry, but I haven't even seen half of this stuff before.
In physics i risk to say there is change of coordinates in all place, mainly in mechanic, since it is important to all the laws be the same in all coordinates, i think the college starts to talk about it very early when the course is physics. For example, in my first semester of physics i saw it, probably the same with ethoteipi

Mentor
In physics i risk to say there is change of coordinates in all place, mainly in mechanic, since it is important to all the laws be the same in all coordinates, i think the college starts to talk about it very early when the course is physics. For example, in my first semester i saw it.
I would even go as far as to say: physics is all about coordinates. Without coordinates, how do you measure? And it is the reason why general relativity is hard to understand: all of a sudden, coordinates become dependent on the observer! Mathematically you just want to know whether it is an ellipse, a parabola or an hyperbola.

etotheipi
Problem 5, I can do the first two parts today, too tired to attempt the harder final parts,\begin{align*} \text{det}\begin{bmatrix}5 - \lambda &0&1&6\\3&3-\lambda&5&2\\0&0&3-\lambda&0\\6&0&3&-\lambda\end{bmatrix} &= \text{det}\begin{bmatrix}0&0&3-\lambda&0\\6&0&3&-\lambda\\5 - \lambda &0&1&6\\3&3-\lambda&5&2\end{bmatrix} \\ \\ &= (3-\lambda) \text{det} \begin{bmatrix} 6 & 0 & -\lambda \\ 3-\lambda & 0 & 6 \\ 3 & 3-\lambda & 2 \end{bmatrix} \\ \\ &= (3-\lambda)\left( 36(\lambda -3) -\lambda(\lambda-3)^2 \right) \\ &= (\lambda - 3)^2 (\lambda + 4)(\lambda -9) = 0 \end{align*}so ##\lambda_1 = 3##, ##\lambda_2 = -4##, ##\lambda_3 = 9## are the eigenvalues of ##A##. Now for the eigenvectors, ##\vec{u}, \vec{v}## and ##\vec{w}##,\begin{align*}\begin{bmatrix} -4 & 0 & 1 & 6 \\ 3 & -6 & 5 &2 \\ 0 & 0 & -6 & 0 \\ 6 & 0 & 3 & 9\end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff u_1 = \frac{3}{2} u_4, u_2 = \frac{13}{12}u_4, u_3 = 0 \\ \\\begin{bmatrix} 9 & 0 & 1 & 6 \\ 3 & 7 & 5 &2 \\ 0 & 0 & 7 & 0 \\ 6 & 0 & 3 & 4\end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff v_1 = -\frac{2}{3}v_4, v_2 =0, v_3 = 0 \\ \\ \begin{bmatrix} 2 & 0 & 1 & 6 \\ 3 & 0 & 5 &2 \\ 0 & 0 & 0 & 0 \\ 6 & 0 & 3 & -3\end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff w_1 = w_3 = w_4 = 0\end{align*}

So ##\vec{u} = \begin{bmatrix} 18 \\ 13 \\ 0 \\ 12 \end{bmatrix}##, ##\vec{v} = \begin{bmatrix} -2 \\ 0 \\ 0 \\ 3 \end{bmatrix}##, ##\vec{w} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}##

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Mentor
Problem 5, I can do the first two parts today, too tired to attempt the harder final parts,\begin{align*} \text{det}\begin{bmatrix}5 - \lambda &0&1&6\\3&3-\lambda&5&2\\0&0&3-\lambda&0\\6&0&3&-\lambda\end{bmatrix} &= \text{det}\begin{bmatrix}0&0&3-\lambda&0\\6&0&3&-\lambda\\5 - \lambda &0&1&6\\3&3-\lambda&5&2\end{bmatrix} \\ \\ &= (3-\lambda) \text{det} \begin{bmatrix} 6 & 0 & -\lambda \\ 3-\lambda & 0 & 6 \\ 3 & 3-\lambda & 2 \end{bmatrix} \\ \\ &= (3-\lambda)\left( 36(\lambda -3) -\lambda(\lambda-3)^2 \right) \\ &= (\lambda - 3)^2 (\lambda + 4)(\lambda -9) = 0 \end{align*}so ##\lambda_1 = 3##, ##\lambda_2 = -4##, ##\lambda_3 = 9## are the eigenvalues of ##A##. Now for the eigenvectors, ##\vec{u}, \vec{v}## and ##\vec{w}##,\begin{align*}\begin{bmatrix} -4 & 0 & 1 & 6 \\ 3 & -6 & 5 &2 \\ 0 & 0 & -6 & 0 \\ 6 & 0 & 3 & 9\end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff v_1 = -\frac{2}{3} v_4, v_2 = \frac{13}{12}u_4, v_3 = 0 \\ \\\begin{bmatrix} 9 & 0 & 1 & 6 \\ 3 & 7 & 5 &2 \\ 0 & 0 & 7 & 0 \\ 6 & 0 & 3 & 4\end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \\ v_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff u_1 = \frac{3}{2}u_4, u_2 =0, u_3 = 0 \\ \\ \begin{bmatrix} 2 & 0 & 1 & 6 \\ 0 & 7 & 5 &2 \\ 0 & 0 & 0 & 0 \\ 6 & 0 & 3 & -3\end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \\ w_4 \end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix} \iff w_1 = w_3 = w_4 = 0\end{align*}So ##\vec{u} = \begin{bmatrix} 18 \\ 13 \\ 0 \\ 12 \end{bmatrix}##, ##\vec{v} = \begin{bmatrix} -2 \\ 0 \\ 0 \\ 3 \end{bmatrix}##, ##\vec{w} = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}##
Sleep over it. It is problem 4. And what is the difference between 3 and -4? And what are 9,12,13,18?

etotheipi
etotheipi
Sleep over it. It is problem 4. And what is the difference between 3 and -4? And what are 9,12,13,18?

Think I got the numbers mixed up, they should be ok now!

Mentor
Think I got the numbers mixed up, they should be ok now!
Let's agree on a single set of representatives, not on many at the same time.

mathwonk
Homework Helper
2020 Award
historical hints: 3-weierstrass, inverse mapping theorem, 7-sylow, 9-nakayama.

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Let ##f(x)=x(x+1)(x+2)(x+3)=x^4 +6x^3 11x^2 + 6x##

1. Finding the stationary points of ##f(x) \rightarrow f'(x)=0 ##
##4x^3 + 18x^2 + 22x + 6 = 0##
##(2x+3)(4x^2+12x+4)=0##
##x=-\frac{3}{2} , x=\frac{-3 \pm \sqrt{5}}{2}##

Checking the sign diagram of first derivative:
a. For ##x<\frac{-3 - \sqrt{5}}{2} \rightarrow f'(x) = (-)##

b. For ##\frac{-3 - \sqrt{5}}{2}<x<-\frac{3}{2} \rightarrow f'(x) = (+)##

c. For ##-\frac{3}{2} <x<\frac{-3 + \sqrt{5}}{2} \rightarrow f'(x) = (-)##

d. For ##x>\frac{-3 + \sqrt{5}}{2} \rightarrow f'(x) = (+)##

##x=\frac{-3 - \sqrt{5}}{2} \rightarrow y=-1## so ##(\frac{-3 - \sqrt{5}}{2} , -1)## is local minimum

##x=-\frac{3}{2} \rightarrow y=\frac{9}{16}## so ##(-\frac{3}{2} , \frac{9}{16})## is local maximum

##x=\frac{-3 + \sqrt{5}}{2} \rightarrow y=-1## so ##(\frac{-3 + \sqrt{5}}{2} , -1)## is local minimum

2. Checking the end behaviour of ##f(x)##
##\lim_{x \to \pm \infty} {f(x)=\infty}##

So the local minimum found above is also global minimum

3.
##f(x)=a## will have no solution if ##a## is below global minimum ##\rightarrow a < -1##

No ##a## will result in ##f(x)## has one solution

##f(x)=a## will have two solutions if ##a## is global minimum or ##a## is above global maximum ##\rightarrow a=-1## or ##a>\frac{9}{16}##

##f(x)=a## will have three solutions if ##a## is global maximum ##\rightarrow a=\frac{9}{16}##

##f(x)=a## will have four solutions if ##a## is between local minimum and local maximum ##\rightarrow -1<a<\frac{9}{16}##

I am not sure how to write good conclusion for final answer

Let ##a=65536## and ##b=32768## so ##z=\frac{(a-3)^3 +(a-2)^3 + (a-1)^3 + a^3 + (a+1)^3 (a+2)^3 + (a+3)^3}{(b-3)(b-2)+(b-1).b+b.(b+1)+(b+2)b+3)}##

After expanding everything: ##z=\frac{7a^3+84a}{4b^2 +12}##

Since ##a=2b## :
##z=\frac{56b^3 +168b}{4b^2 +12}##

##=\frac{56b(b^2+3)}{4(b^2+3)}##

##=14b##

Keith_McClary and PeroK
Note ##351 = 3^3\cdot 13##. Let ##n_3## denote the number of Sylow ##3## subgroups of ##G## and ##n_{13}## denote the number of Sylow ##13## subgroups of ##G##. For a given prime ##p## that divides ##\vert G \vert##, the Sylow ##p## subgroups are conjugates. So, it's enough to show ##n_3 = 1## or ##n_{13} = 1##. By Sylow's theorem, ##n_{13} \vert 27## and ##n_{13} = 1 \text{mod 13}##. So, ##n_{13} = 1## or ##27##. If ##n_{13} = 1##, we are done. Suppose ##n_{13} = 27##. This gives ##27\cdot 12 = 324## distinct elements of order ##13##. This leaves ##351 - 324 = 27## elements unaccounted for. Each Sylow ##3## subgroup has order ##27##, and we know ##n_3 \vert 13##. (And no element of a Sylow ##3## subgroup contains an element of order ##13##). Combining the last two lines, we have ##n_3 = 1##. []
[\spoiler]

fresh_42