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Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread!

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.

##1.## (solved by @tnich ) Where ##\mathbf A \in \mathbb R^{\text{m x m}}## for natural number ##m \geq 2## and ##p \in (0,1)##

$$\mathbf A := \left[\begin{matrix}

- p_{} & 1 - p_{}& 1 - p_{} & \dots &1 - p_{} &1 - p_{} & 1 - p_{}

\\p_{} & -1&0 &\dots & 0 & 0 & 0

\\0 & p_{}&-1 &\dots & 0 & 0 & 0

\\0 & 0& p_{}&\dots & 0 & 0 & 0

\\0 & 0&0 & \ddots & -1&0 &0

\\0 & 0&0 & \dots & p_{}&-1 &0

\\0 & 0&0 & \dots &0 & p_{} & p_{}-1

\end{matrix}\right]$$

prove the minimal polynomial for ##\mathbf A## is

##= \mathbf A^{m} + \binom{m-1}{1}\mathbf A^{m-1} + \binom{m-1}{2}\mathbf A^{m-2} + \binom{m-1}{3}\mathbf A^{m-3} +...+ \mathbf A## ##\space## ##\space##

##2.## (solved by @tnich ) Find the volume of the solid ##S## which is defined by the relations ##z^2 - y \geq 0,## ##\space## ##x^2 - z \geq 0,## ##\space## ##y^2 - x \geq 0,## ##\space## ##z^2 \leq 2y,## ##\space## ##x^2 \leq 2z,## ##\space## ##y^2 \leq 2x## ##\space## ##\space##

##3.## (solved by @julian ) Determine with analytical methods, i.e. with a calculator only, the wavelengths of all local maximal radiation intensities of a black body of temperature ##T## given the following function of radiation up to three digits:

$$

J(\lambda) =\dfrac{c^2h}{\lambda^5\cdot\left(\exp\left(\dfrac{ch}{\lambda \kappa T}\right)-1\right)}

$$

##4.## (solved by @tnich ) 100 gamers walk into an arcade with 8 different video games. Each gamer plays every video game once and only once. For each video game at least 65 of the gamers beat the final level. (These are easy games.) Prove that there must exist at least one collection of 2 gamers who collectively beat every game / every final level. For avoidance of doubt, this means, e.g.

if we record a 1 indicating the player_k beat final level of game i, and a 0 otherwise,

$$\mathbf p_k = \begin{bmatrix}

1\\

0\\

1\\

0\\

0\\

0\\

1\\

1

\end{bmatrix}$$

the task is to prove there must exist (at least one) vector ##\mathbf a## where

##\mathbf a := \mathbf p_k + \mathbf p_j##

and ##\mathbf a \gt 0## (i.e. each component of ##\mathbf a## is strictly positive) ##\space## ##\space##

##5.## (solved by @Math_QED ) Solve the differential equation ##(2x - 4y + 6)dx + (x + y - 3)dy = 0## ##\space## ##\space##

##6.## (resolved in post #56) Consider ##\mathfrak{su}(3)=\operatorname{span}\{\,T_3,Y,T_{\pm},U_{\pm},V_{\pm}\,\}## given by the basis elements

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## $$

\begin{align*}

T_3&=\frac{1}{2}\lambda_3\; , \;Y=\frac{1}{\sqrt{3}}\lambda_8\; ,\\

T_{\pm}&=\frac{1}{2}(\lambda_1\pm i\lambda-2)\; , \;U_{\pm}=\frac{1}{2}(\lambda_6\pm i\lambda_7)\; , \;V_{\pm}=\frac{1}{2}(\lambda_4\pm i\lambda_5)

\end{align*}$$

(cp.

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space##

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\mathfrak{B}:=\langle T_3,Y,T_+,U_+,V_+ \rangle##

Now ##\mathfrak{A(B)}=\{\,\alpha: \mathfrak{g} \to \mathfrak{g}\, : \,[X,\alpha(Y)]=[Y,\alpha(X)]\,\,\forall \,X,Y\in \mathfrak{B}\,\}## is the one-dimensional Lie algebra spanned by ##\operatorname{ad}(V_+)## because ##\mathbb{C}V_+## is a one-dimensional ideal in ##\mathfrak{B}## (Proof?). Then ##\mathfrak{g}:=\mathfrak{B}\ltimes \mathfrak{A(B)}## is again a Lie algebra by the multiplication ##[X,\alpha]=[\operatorname{ad}X,\alpha]## for all ##X\in \mathfrak{B}\; , \;\alpha \in \mathfrak{A(B)}##. (For a proof see problem 9 in

a) Determine the center of ##\mathfrak{g}## , and whether ##\mathfrak{g}## is semisimple, solvable, nilpotent or neither.

b) Show that ##(X,Y) \mapsto \alpha([X,Y])## defines another Lie algebra structure on ##\mathfrak{B}## , which one?

c) Show that ##\mathfrak{A(g)}## is at least two-dimensional. ##\space## ##\space##

##7.## (solved by @julian ) Given m distinct nonzero complex numbers ## x_1, x_2, ..., x_m##, prove that

##\sum_{k=1}^m \frac{1}{x_k} \prod_{j \neq k} \frac{1}{x_k - x_j} = \frac{(-1)^{m+1}}{x_1 x_2 ... x_m}##

##p(x) = -1 + \sum_{k=1}^m \prod_{j\neq k} \frac{x - x_j}{x_k - x_j}## ##\space## ##\space##

##8.## (solved by @julian ) Find the last ##1000## digits of the number ##a = 1 + 50 + 50^2 + 50^3 + \cdots + 50^{999}## ##\space## ##\space##

##9.## (solved by @julian ) Consider the Hilbert space ##\mathcal{H}=L_2([a,b])## of Lebesgue square integrable functions on ##[a,b]## , i.e.

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space##

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\langle \psi,\chi \rangle = \int_{a}^{b}\psi(x)\chi(x)\,dx##

The functions ##\{\,\psi_n:=x^n\, : \,n\in \mathbb{N}_0\,\}## build a system of linear independent functions which can be used to find an orthonormal basis by the Gram-Schmidt procedure. Show that the Legendre polynomials

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##p_n(x):=\dfrac{1}{(b-a)^n\,n!}\,\sqrt{\dfrac{2n+1}{b-a}}\,\dfrac{d^n}{dx^n}[(x-a)(x-b)]^n\; , \;n\in \mathbb{N}_0##

build an orthnormal system. ##\space## ##\space##

##10.## a) (solved by @I like Serena ) Give an example of an integral domain (no field), which has common divisors, but doesn't have greatest common divisors.

b) (solved by @tnich ) Show that there are infinitely many units (invertible elements) in ##\mathbb{Z}[\sqrt{3}]##.

c) (solved by @I like Serena ) Determine the units of ##\{\,\frac{1}{2}a+ \frac{1}{2}b\sqrt{-3}\,\vert \,a+b \text{ even }\}##.

d) (solved by @I like Serena ) The ring ##R## of integers in ##\mathbb{Q}(\sqrt{-19})## is the ring of all elements, which are roots of monic polynomials with integer coefficients. Show that ##R## is built by all elements of the form ##\frac{1}{2}a+\frac{1}{2}b\sqrt{-19}## where ##a,b\in \mathbb{Z}## and both are either even or both are odd. ##\space## ##\space##

**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 @tnich ) Where ##\mathbf A \in \mathbb R^{\text{m x m}}## for natural number ##m \geq 2## and ##p \in (0,1)##

$$\mathbf A := \left[\begin{matrix}

- p_{} & 1 - p_{}& 1 - p_{} & \dots &1 - p_{} &1 - p_{} & 1 - p_{}

\\p_{} & -1&0 &\dots & 0 & 0 & 0

\\0 & p_{}&-1 &\dots & 0 & 0 & 0

\\0 & 0& p_{}&\dots & 0 & 0 & 0

\\0 & 0&0 & \ddots & -1&0 &0

\\0 & 0&0 & \dots & p_{}&-1 &0

\\0 & 0&0 & \dots &0 & p_{} & p_{}-1

\end{matrix}\right]$$

prove the minimal polynomial for ##\mathbf A## is

##= \mathbf A^{m} + \binom{m-1}{1}\mathbf A^{m-1} + \binom{m-1}{2}\mathbf A^{m-2} + \binom{m-1}{3}\mathbf A^{m-3} +...+ \mathbf A## ##\space## ##\space##

**(by**@StoneTemplePython**)**##2.## (solved by @tnich ) Find the volume of the solid ##S## which is defined by the relations ##z^2 - y \geq 0,## ##\space## ##x^2 - z \geq 0,## ##\space## ##y^2 - x \geq 0,## ##\space## ##z^2 \leq 2y,## ##\space## ##x^2 \leq 2z,## ##\space## ##y^2 \leq 2x## ##\space## ##\space##

**(by**@QuantumQuest**)**

##3.## (solved by @julian ) Determine with analytical methods, i.e. with a calculator only, the wavelengths of all local maximal radiation intensities of a black body of temperature ##T## given the following function of radiation up to three digits:

$$

J(\lambda) =\dfrac{c^2h}{\lambda^5\cdot\left(\exp\left(\dfrac{ch}{\lambda \kappa T}\right)-1\right)}

$$

**(by**@fresh_42**)**##4.## (solved by @tnich ) 100 gamers walk into an arcade with 8 different video games. Each gamer plays every video game once and only once. For each video game at least 65 of the gamers beat the final level. (These are easy games.) Prove that there must exist at least one collection of 2 gamers who collectively beat every game / every final level. For avoidance of doubt, this means, e.g.

if we record a 1 indicating the player_k beat final level of game i, and a 0 otherwise,

e.g. if the kth player beat the final level of games 1, 3, 7 and 8 but lost on the others, we'd havee.g. if the kth player beat the final level of games 1, 3, 7 and 8 but lost on the others, we'd have

$$\mathbf p_k = \begin{bmatrix}

1\\

0\\

1\\

0\\

0\\

0\\

1\\

1

\end{bmatrix}$$

the task is to prove there must exist (at least one) vector ##\mathbf a## where

##\mathbf a := \mathbf p_k + \mathbf p_j##

and ##\mathbf a \gt 0## (i.e. each component of ##\mathbf a## is strictly positive) ##\space## ##\space##

**(by**@StoneTemplePython**)**##5.## (solved by @Math_QED ) Solve the differential equation ##(2x - 4y + 6)dx + (x + y - 3)dy = 0## ##\space## ##\space##

**(by**@QuantumQuest**)**##6.## (resolved in post #56) Consider ##\mathfrak{su}(3)=\operatorname{span}\{\,T_3,Y,T_{\pm},U_{\pm},V_{\pm}\,\}## given by the basis elements

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## $$

\begin{align*}

T_3&=\frac{1}{2}\lambda_3\; , \;Y=\frac{1}{\sqrt{3}}\lambda_8\; ,\\

T_{\pm}&=\frac{1}{2}(\lambda_1\pm i\lambda-2)\; , \;U_{\pm}=\frac{1}{2}(\lambda_6\pm i\lambda_7)\; , \;V_{\pm}=\frac{1}{2}(\lambda_4\pm i\lambda_5)

\end{align*}$$

(cp.

__https://www.physicsforums.com/insights/representations-precision-important__) where the ##\lambda_i## are the Gell-Mann matrices and its maximal solvable Borel-subalgebra##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space##

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\mathfrak{B}:=\langle T_3,Y,T_+,U_+,V_+ \rangle##

Now ##\mathfrak{A(B)}=\{\,\alpha: \mathfrak{g} \to \mathfrak{g}\, : \,[X,\alpha(Y)]=[Y,\alpha(X)]\,\,\forall \,X,Y\in \mathfrak{B}\,\}## is the one-dimensional Lie algebra spanned by ##\operatorname{ad}(V_+)## because ##\mathbb{C}V_+## is a one-dimensional ideal in ##\mathfrak{B}## (Proof?). Then ##\mathfrak{g}:=\mathfrak{B}\ltimes \mathfrak{A(B)}## is again a Lie algebra by the multiplication ##[X,\alpha]=[\operatorname{ad}X,\alpha]## for all ##X\in \mathfrak{B}\; , \;\alpha \in \mathfrak{A(B)}##. (For a proof see problem 9 in

__https://www.physicsforums.com/threads/intermediate-math-challenge-may-2018.946386/__)a) Determine the center of ##\mathfrak{g}## , and whether ##\mathfrak{g}## is semisimple, solvable, nilpotent or neither.

b) Show that ##(X,Y) \mapsto \alpha([X,Y])## defines another Lie algebra structure on ##\mathfrak{B}## , which one?

c) Show that ##\mathfrak{A(g)}## is at least two-dimensional. ##\space## ##\space##

**(by**@fresh_42**)**##7.## (solved by @julian ) Given m distinct nonzero complex numbers ## x_1, x_2, ..., x_m##, prove that

##\sum_{k=1}^m \frac{1}{x_k} \prod_{j \neq k} \frac{1}{x_k - x_j} = \frac{(-1)^{m+1}}{x_1 x_2 ... x_m}##

*hint:*first consider the polynomial##p(x) = -1 + \sum_{k=1}^m \prod_{j\neq k} \frac{x - x_j}{x_k - x_j}## ##\space## ##\space##

**(by**@StoneTemplePython**)**##8.## (solved by @julian ) Find the last ##1000## digits of the number ##a = 1 + 50 + 50^2 + 50^3 + \cdots + 50^{999}## ##\space## ##\space##

**(by**@QuantumQuest**)**##9.## (solved by @julian ) Consider the Hilbert space ##\mathcal{H}=L_2([a,b])## of Lebesgue square integrable functions on ##[a,b]## , i.e.

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space##

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\langle \psi,\chi \rangle = \int_{a}^{b}\psi(x)\chi(x)\,dx##

The functions ##\{\,\psi_n:=x^n\, : \,n\in \mathbb{N}_0\,\}## build a system of linear independent functions which can be used to find an orthonormal basis by the Gram-Schmidt procedure. Show that the Legendre polynomials

##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##\space## ##p_n(x):=\dfrac{1}{(b-a)^n\,n!}\,\sqrt{\dfrac{2n+1}{b-a}}\,\dfrac{d^n}{dx^n}[(x-a)(x-b)]^n\; , \;n\in \mathbb{N}_0##

build an orthnormal system. ##\space## ##\space##

**(by**@fresh_42**)**##10.## a) (solved by @I like Serena ) Give an example of an integral domain (no field), which has common divisors, but doesn't have greatest common divisors.

b) (solved by @tnich ) Show that there are infinitely many units (invertible elements) in ##\mathbb{Z}[\sqrt{3}]##.

c) (solved by @I like Serena ) Determine the units of ##\{\,\frac{1}{2}a+ \frac{1}{2}b\sqrt{-3}\,\vert \,a+b \text{ even }\}##.

d) (solved by @I like Serena ) The ring ##R## of integers in ##\mathbb{Q}(\sqrt{-19})## is the ring of all elements, which are roots of monic polynomials with integer coefficients. Show that ##R## is built by all elements of the form ##\frac{1}{2}a+\frac{1}{2}b\sqrt{-19}## where ##a,b\in \mathbb{Z}## and both are either even or both are odd. ##\space## ##\space##

**(by**@fresh_42**)**
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