The Challenge (originally known as Road Rules: All Stars, followed by Real World/Road Rules Challenge and occasionally known as The Real World/Road Rules Challenge during this time), is a reality competition show on MTV that is spun off from two of the network's reality shows, The Real World and Road Rules. Originally featuring alumni from these two shows, casting for The Challenge has slowly expanded to include contestants who debuted on The Challenge itself, alumni from other MTV franchises including Are You the One?, Ex on the Beach (Brazil, UK and US), Geordie Shore and from other non-MTV shows. The contestants compete against one another in various extreme challenges to avoid elimination. The winners of the final challenge win the competition and share a large cash prize. The Challenge is currently hosted by T. J. Lavin.
The series premiered on June 1, 1998. The show was originally titled Road Rules: All Stars (in which notable Real World alumni participated in a Road Rules style road-trip). It was renamed Real World/Road Rules Challenge for the 2nd season, then later abridged to simply The Challenge by the show's 19th season.
Since the fourth season, each season has supplied the show with a unique subtitle, such as Rivals. Each season consists of a format and theme whereby the subtitle is derived. The show's most recent season, Double Agents, premiered on December 9, 2020. A new special limited-series, titled The Challenge: All Stars premiered on April 1, 2021 on the Paramount+ streaming service.
1. (solved by @nuuskur ) Let ##K## be a non-empty compact subset of ##\Bbb{C}##. Construct a bounded operator ##u: H \to H## on some Hilbert space ##H## that has spectrum ##\sigma(u) =K##. (MQ)
2. Let ##f,g:[0,2]\to\mathbb{R}## be continuous functions such that ##f(0)=g(0)=0## and...
The $\triangle ABC$ and $\triangle AEF$ are in the same plane. Between them, the following conditions hold:
1. The midpoint of $AB$ is $E$.
2. The points $A,\,G$ and $F$ are on the same line.
3. There is a point $C$ at which $BG$ and $EF$ intersect.
4. $CE=1$ and $AC=AE=FG$.
Prove that if...
Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
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.
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 $\dfrac{1}{2}<\dfrac{S_Q+S_R}{S_P}<2-\sqrt{2}$
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...
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 $\angle ADE=\angle BDC$.
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...
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)##...
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...
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...
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...
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...
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...