Challenge Definition and 911 Threads

https://www.zdnet.com/article/a-student-has-rebuilt-the-machine-that-first-cracked-german-enigma-codes/

Evaluate $\displaystyle\int\limits_0^{\infty} \dfrac{x^2+2}{x^6+1} \, dx$.

Evaluate $\dfrac{\sin^2 \dfrac{\pi}{7}}{\sin^4 \dfrac{2\pi}{7}}+\dfrac{\sin^2 \dfrac{2\pi}{7}}{\sin^4 \dfrac{3\pi}{7}}+\dfrac{\sin^2 \dfrac{3\pi}{7}}{\sin^4 \dfrac{\pi}{7}}$ without the help of a calculator.

It is given that the ratio of angles $A,\,B$ and $C$ is $1:2:4$ in a $\triangle ABC$, prove that $(a^2-b^2)(b^2-c^2)(c^2-a^2)=(abc)^2$.

1. (solved by @nuuskur ) Let ##V## be an infinite dimensional topological vector space. Show that the weak topology on ##V## is not induced by a norm. (MQ) 2. The matrix groups ##U(n)## and ##SL_n(\mathbb{C})## are submanifolds of ##\mathbb{C}^{n^2}=\mathbb{R}^{2n^2}##. Do they intersect...

Let $P_1P_2P_3P_4P_5P_6P_7,\,Q_1Q_2Q_3Q_4Q_5Q_6Q_7,\,R_1R_2R_3R_4R_5R_6R_7$ be regular heptagons with areas $S_P,\,S_Q$ and $S_R$ respectively. Let $P_1P_2=Q_1Q_3=R_1R_4$. Prove that $\dfrac{1}{2}<\dfrac{S_Q+S_R}{S_P}<2-\sqrt{2}$

Let $ABC$ be an isosceles triangle such that $AB=AC$. Find the angles of $\triangle ABC$ if $\dfrac{AB}{BC}=1+2\cos\dfrac{2\pi}{7}$.

Heh heh, unfortunately I can’t do that. However, many of my posts in here do (sometimes) contain exercises. I will try to make it a habit in the future. :smile: Here is one relevant for relativity forum: Use the definition T^{\mu\nu} = \frac{1}{\sqrt{-g}} \frac{\delta...

If the equation $ax^2+(c-b)x+e-d=0$ has real roots greater than 1, show that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real root.

In a recent https://mathhelpboards.com/threads/inequality-challenge.27634/#post-121156, anemone asked for a proof that $1-x + x^4 - x^9 + x^{16} - x^{25} + x^{36} > 0$. When I graphed that function, I noticed that in fact it is never less than $\frac12$. If you add more terms to the series, this...

In convex quadrilateral $ADBE$, there is a point $C$ within $\triangle ABE$ such that $\angle EAD+\angle CAB=\angle EBD+\angle CBA=180^{\circ}$. Prove that $\angle ADE=\angle BDC$.

Prove $x+x^9+x^{25}<1+x^4+x^{16}+x^{36}$ for $x>0$.

Questions 1. (solved by @nuuskur ) Let ##H_1, H_2## be Hilbert spaces and ##T: H_1 \to H_2## a linear map. Suppose that there is a linear map ##S: H_2 \to H_1## such that for all ##x\in H_2## and all ##y \in H_1## we have $$\langle Sx,y \rangle = \langle x, Ty \rangle$$ Show that ##T## is...

Prove $e^{-x}\le \ln(e^x-x-\ln x)$ for $x>0$.

Solve $\{ \sin \lfloor x \rfloor \}+\{ \cos \lfloor x \rfloor \}=\{ \tan \lfloor x \rfloor \}$ for real solution(s).

Find all positive integer solutions $(a,\,b,\,c,\,n)$ of the equation $2^n=a!+b!+c!$.

Hey everyone, This is more of a motivational thread, and of course if anyone wants to join in, please do! Any comments are welcome. It's also fine if no one comments. Maybe don't remove the thread though please. I hope this might be useful later on for others as motivation. So the challenge is...

Questions 1. (solved by @benorin ) Let ##1<p<4## and ##f\in L^p((1,\infty))## with the Lebesgue measure ##\lambda##. We define ##g\, : \,(1,\infty)\longrightarrow \mathbb{R}## by $$ g(x)=\dfrac{1}{x}\int_x^{10x}\dfrac{f(t)}{t^{1/4}}\,d\lambda(t). $$ Show that there exists a constant ##C=C(p)##...

Prove that $\dfrac{378^3+392^3+1053^3}{2579}$ is an integer.

Problem 1 (@wrobel ) (solved by @TSny ) There is a perfectly rough horizontal table. This table is pretty wide (actually it is a plane) and it rotates about some vertical axis. Angular velocity is a given constant: ##\Omega\ne 0##. Somebody throws a homogeneous ball on the table. The ball has a...

Questions 1. (solved by @nuuskur ) Let ##U\subseteq X## be a dense subset of a normed vector space, ##Y## a Banach space and ##A\in L(U,Y)## a linear, bounded operator. Show that there is a unique continuation ##\tilde{A}\in L(X,Y)## with ##\left.\tilde{A}\right|_U = A## and...

Questions 1. (solved by @hilbert2 ) Let ##\sum_{k=1}^\infty a_k## be a given convergent series with ##|a_{k+1}|\leq |a_k|## for all ##k##. Assume we use a computer to sum its value until the partial sum is closer than ##\varepsilon## to the actual value of the series. Does it make sense to use...

Questions 1. (solved by @Antarres, @Not anonymous ) Prove the inequality ##\cos(\theta)^p\leq\cos(p\theta)## for ##0\leq\theta\leq\pi/2## and ##0<p<1##. (IR) 2. (solved by @suremarc ) Let ##F:\mathbb{R}^n\to\mathbb{R}^n## be a continuous function such that ##||F(x)-F(y)||\geq ||x-y||## for all...

Hi Y'all For the purpose of exploring COMSOL, I challenged my self to plot the E/M-fields of a piece of current carrying wire in 3D. It's quite a simple task to plot the fields inside the wire, but I fail when plotting the fields outside the wire. For plotting the outside fields I have...

Questions 1. (solved by @archaic ) Determine ##\lim_{n\to \infty}\cos\left(t/\sqrt{n}\right)^n## for ##t\in \mathbb{R}##. 2. (solved by @Antarres ) Let ##a_0,\ldots,a_n## be distinct real numbers. Show that for any ##b_0,\ldots,b_n\in\mathbb{R}##, there exists a unique polynomial ##p## of...

John has baked 31 cupcakes for 5 different students. He wants to give them all to his students but he wants to give an odd number of cupcakes to each one. How many ways can he do this? Brute-forcing will take about a whole day, I think. If 4 students receive 1 cupcake and the other one receive...

Hello everyone, I hope I'm not intruding with too simple of a request for help. I have this math problem: A rack space with 100 slots for plastic crates. I have two type of crates, one with 20 dividers weighing 5 grams and one with 60 dividers weighing 25 grams. I want to add crates to...

Questions 1. (solved by @PeroK ) Let ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## be a smooth, ##2\pi-##periodic function with square integrable derivative, and ##\displaystyle{\int_0^{2\pi}}f(x)\,dx = 0\,.## Prove $$ \int_0^{2\pi} \left[f(x)\right]^2\,dx \leq \int_0^{2\pi}...

Questions 1. Let ##(X,d)## be a metric space. The open ball with center ##z\in X## of radius ##r > 0## is defined as $$ B_r(z) :=\{\,x\in X\,|\,d(x,z)<r\,\} $$ a.) Give an example for $$ \overline{B_r(z)} \neq K_r(z) :=\{\,x\in X\,|\,d(x,z)\leq r\,\} $$ Does at least one of the inclusions...

Homework Statement: Problem: The planet X is far 48 light-years from Earth. Suppose that we want to travel from Earth to planet X in a time no more than 23 years, as reckoned by clocks aboard our spaceship. At what constant speed would we have to travel? How long would the trip take as reckoned...

Questions 1. (solved by @tnich ) Show that ##\sin\dfrac{\pi}{m} \sin\dfrac{2\pi}{m}\sin\dfrac{3\pi}{m}\cdots \sin\dfrac{(m - 1)\pi}{m} = \dfrac{m}{2^{m - 1}}## for ##m## = ##2, 3, \dots##(@QuantumQuest) 2. (solved by @PeroK ) Show that when a quantity grows or decays exponentially, the rate of...

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I've seen a thread posted on another forum which described a thermodynamic situation that captured my interest, so I though I would introduce a challenge problem on it. The other forum was not able to adequately specify or address how to approach a problem like this. I know how to solve this...

1.Prove that f'(x) is strictly decreasing at (- ##\infty##,a) and strictly increasing at (a,##\infty##). 2.Prove that f'(x) has exactly two roots. I tried to find f''(x)=0, but I'm not able to solve the equation. What should I do?

Questions 1. (solved by @MathematicalPhysicist ) Show that the difference of the square roots of two consecutive natural numbers which are greater than ##k^2##, is less than ##\dfrac{1}{2k}##, ##k \in \mathbb{N} - \{0\}##. (@QuantumQuest ) 2. (solved by @tnich ) Let A, B, C and D be four...

Show that \[\tan^{-1}(k) = \sum_{n=0}^{k-1}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ),\;\;\;\;\; k \geq 1,\] - and deduce that \[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]

Questions 1. Consider the ring ##R= C([0,1], \mathbb{R})## of continuous functions ##[0,1]\to \mathbb{R}## with pointwise addition and multiplication of functions. Set ##M_c:=\{f \in R\mid f(c)=0\}##. (a) (solved by @mathwonk ) Show that the map ##c \mapsto M_c, c \in [0,1]## is a bijection...

I was thinking about giving the bond energy to calculate the enthalpy change of some exothermic and spontaneous reaction. Than using that exothermic enthalpy to heat the own products and reagents. That would change the Gibbs free energy of the equation (as the elements will be in a different...

Show, that the identity \[\int_{0}^{1}\frac{x^{n-1}+x^{n-\frac{1}{2}}-2x^{2n-1}}{1-x}dx = 2\ln2\] - holds for any natural number $n$.

Start with some pennies. Flip each penny until a head comes up on that penny. The winner(s) are the penny(s) which were flipped the most times. Prove that the probability there is only one winner is at least $\frac{2}{3}$.

Questions 1. (solved by @Pi-is-3 )The maximum value of ##f## with ##f(x) = x^a e^{2a - x}## is minimal for which values of positive numbers ##a## ? 2. (solved by @KnotTheorist ) Find the equation of a curve such that ##y''## is always ##2## and the slope of the tangent line is ##10## at the...

Find \[\lim_{n\rightarrow \infty}\int_{0}^{\infty}\frac{e^{-x}\cos x}{\frac{1}{n}+nx^2}dx.\]

Trying to follow and learn from the solution and did not want to clutter up the original thread My naive question is why doesn't Jensen's Inequality prevent this step? Where you are swapping the expectation of a function for applying the function to the expectation which according to the...

Questions 1. (solved by @Flatlanderr , solved by @lriuui0x0 ) Show that ##\frac{\pi}{4} + \frac{3}{25} \lt \arctan \frac{4}{3} \lt \frac{\pi}{4} + \frac{1}{6}## 2. (solved by @nuuskur ) Show that the equation ##x + x^3 + x^5 + x^7 = {c_1}^2 (c_1 - x) + {c_2}^2 (c_2 - x)## where ##c_1, c_2 \in...

[FONT=Times New Roman]Let $S_n$ be the group of all permutations of the set $\{1,\ldots,n\}$. Determine whether the following assertions are true or false. 1. For each $\pi\in S_n$, $$\sum_{i=1}^n\,(\pi(i)-i)\ =\ 0.$$ 2. If $$\sigma_\pi\ =\ \sum_{i=1}^n\,\left|\pi(i)-i\right|$$ for each...

We have a prize this month donated by one of our most valued members, and that's what the points are for. The first who achieves 6 points, will win a Gold Membership. Questions 1. Let ##\mathfrak{g}## be a Lie algebra. Define $$ \mathfrak{A(g)} = \{\,\alpha\, : \,\mathfrak{g}\longrightarrow...

[FONT=Times New Roman]Let $a,b,c$ be positive real numbers such that $a+b+c=2$. Prove that $$3\left(\frac1{\sqrt{a^3+1}}+\frac1{\sqrt{b^3+1}}+\frac1{\sqrt{c^3+1}}\right)\ \geqslant\ 2\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right).$$

[FONT=Times New Roman]Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$

Dear all, I am momentarily designing a robot arm and I will use the Denavit-Hartenberg convention to model my arm (see picture). The problem I have, however, is that I cannot seem to define the robot arm in the parameters of a, d, alpha and theta. It could be that my coordinates are not...

Questions 1. a. Let ##(\mathfrak{su}(2,\mathbb{C}),\varphi,V)## be a finite dimensional representation of the Lie algebra ##\mathfrak{g}=\mathfrak{su}(2,\mathbb{C})##. Calculate ##H\,^0(\mathfrak{g},\varphi)## and ##H\,^1(\mathfrak{g},\varphi)## for the Chevalley-Eilenberg complex in the cases...