What is Challenge: Definition and 942 Discussions

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.

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  1. fresh_42

    Challenge Math Challenge - August 2020

    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...
  2. anemone

    MHB Geom Ch: Prove $AB=x^3$ Given $\triangle ABC$ & $\triangle AEF$

    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...
  3. anemone

    MHB Max & Min Values of $S$ for $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$

    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 $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.
  4. anemone

    MHB Can You Prove 2+2√(28n²+1) is a Square Number?

    Let $n$ be a positive integer. Show that if $2+2\sqrt{28n^2+1}$ is an integer, then it is a square.
  5. jedishrfu

    Computer Anyone up to the Challenge of Building Enigma

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

    MHB Integral Challenge: Evaluating $\int_0^\infty \frac{x^2+2}{x^6+1} \, dx$

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

    MHB What is the solution to this trigonometric challenge?

    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.
  8. anemone

    MHB Can You Solve the Triangle Sides Challenge?

    It is given that the ratio of angles $A,\,B$ and $C$ is $1:2:4$ in a $\triangle ABC$, prove that $(a^2-b^2)(b^2-c^2)(c^2-a^2)=(abc)^2$.
  9. fresh_42

    Challenge Math Challenge - July 2020

    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...
  10. anemone

    MHB Heptagon Challenge: Proving $\dfrac{1}{2}<\dfrac{S_Q+S_R}{S_P}<2-\sqrt{2}$

    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 $\dfrac{1}{2}<\dfrac{S_Q+S_R}{S_P}<2-\sqrt{2}$
  11. anemone

    MHB What are the angles of an isosceles triangle with a specific ratio?

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

    I Samalkhaiat's challenge #001

    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...
  13. anemone

    MHB Polynomial Challenge: Show Real Roots >1 Exist

    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.
  14. Opalg

    MHB Can You Prove This Inequality Challenge?

    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...
  15. anemone

    MHB Geometry Challenge: Prove $\angle ADE=\angle BDC$ in Convex Quadrilateral $ADBE$

    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 $\angle ADE=\angle BDC$.
  16. anemone

    MHB Inequality Challenge: Prove $x$ for $x>0$

    Prove $x+x^9+x^{25}<1+x^4+x^{16}+x^{36}$ for $x>0$.
  17. fresh_42

    Challenge Math Challenge - June 2020

    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...
  18. anemone

    MHB Unsolved Challenge: Natural logarithm and Exponent

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

    MHB Can You Solve This Tricky Trigonometric Floor Function Equation?

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

    MHB Factorials and Exponent Challenge

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

    MHB Unsolved Challenge: Σ(cosAcosB)>√(cosAcosBcosC)

    Let $ABC$ be a non-obtuse triangle. Prove that $\cos A \cos B+\cos B \cos C+\cos C \cos A>2\sqrt{\cos A \cos B \cos C}$.
  22. Replusz

    30-day Topological Quantum challenge

    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...
  23. anemone

    MHB Unsolved Challenge: Trigonometric Identity

    Prove $\tan 3x=\tan \left(\dfrac{\pi}{3}-x\right) \tan x \tan \left(\dfrac{\pi}{3}+x\right)$ geometrically.
  24. fresh_42

    Challenge Math Challenge - May 2020

    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)##...
  25. anemone

    MHB Integers and Divisibility Challenge

    Prove that $\dfrac{378^3+392^3+1053^3}{2579}$ is an integer.
  26. fresh_42

    Challenge How do you calculate the motion of a ball rolling on a rotating table?

    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...
  27. fresh_42

    Challenge Math Challenge - April 2020

    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...
  28. fresh_42

    Challenge Math Challenge - March 2020 (Part II)

    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...
  29. fresh_42

    Challenge Math Challenge - March 2020

    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...
  30. vis viva

    I Personal COMSOL challenge (EM)

    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...
  31. fresh_42

    Challenge Math Challenge - February 2020

    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...
  32. Monoxdifly

    MHB John's Cupcake Challenge: Finding the Perfect Distribution

    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...
  33. B

    Solving Math Problem with Crates: A Maximum Dividers Challenge

    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...
  34. fresh_42

    Challenge Math Challenge - January 2020

    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}...
  35. fresh_42

    Challenge Math Challenge - December 2019

    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...
  36. Athenian

    Traveling to Planet X in 23 Years: An SR Challenge

    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...
  37. fresh_42

    Challenge Math Challenge - November 2019

    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...
  38. RUTA

    Insights Answering Mermin’s Challenge with the Relativity Principle

    Continue reading...
  39. Chestermiller

    Thermodynamic Challenge Problem

    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...
  40. K

    Proving f'(x) Properties & Finding Its Roots: A Challenge

    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?
  41. fresh_42

    Challenge Math Challenge - October 2019

    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...
  42. lfdahl

    MHB Trigonometric Sum Challenge Σtan^(-1)(1/(n^2+n+1)=π/2

    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 \[ \sum_{n=0}^{\infty}\tan^{-1} \left ( \frac{1}{n^2+n+1} \right ) = \frac{\pi}{2}.\]
  43. fresh_42

    Challenge Math Challenge - September 2019

    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...
  44. jaumzaum

    Chemistry Create a Gibbs Free Energy challenge question

    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...
  45. lfdahl

    MHB How Can the Given Definite Integral Identity Be Proven for Any Natural Number n?

    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 $n$.
  46. lfdahl

    MHB Solve Probability Challenge: Prove 2/3 Chance of Only 1 Winner

    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 $\frac{2}{3}$.
  47. fresh_42

    Challenge Math Challenge - August 2019

    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...
  48. lfdahl

    MHB Limit of integral challenge of (e^(-x)cosx)/(1/n+nx^2)

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

    I Question on problem 7 on July Challenge

    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...
  50. fresh_42

    Challenge Math Challenge - July 2019

    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...
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