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. O

    How can a degenerate gambler escape his eternal punishment in hell?

    Sorry about missing a week guys. A degenerate gambler dies and is sentenced to hell. The devil informs him that his punishment is that he must play a slot machine. He starts with one coin, and each time he puts in a coin he gets countably infinite many coins out of the machine. He must...
  2. anemone

    MHB Polynomial Challenge: Find $f(p)+f(q)+f(r)+f(s)$

    The roots of $x^4-x^3-x^2-1=0$ are $p, q, r, s$. Find $f(p)+f(q)+f(r)+f(s)$, where $f(x)=x^6-x^5-x^3-x^2-x$.
  3. DreamWeaver

    MHB Can You Crack This Advanced Integral Problem?

    OK, OK, so I'll stop soon... lol This'll be the last one for a while. But hey, you all know what it's like; you just can't log on here and find too many interesting threads, so forgive me for getting carried away. I'm sorry... [liar] (Heidy) For 0 < a < \pi, and b \in \mathbb{R} > -1, show...
  4. DreamWeaver

    MHB Integral Challenge #3: Proving Li$_{2m+1}$ w/ Clausen Function

    Prove the following integral representation of Polylogarithm, in terms of the Clausen function:\text{Li}_{2m+1}(e^{-\theta})=\frac{2}{\pi}\int_0^{\pi /2}\text{Cl}_{2m+1}(\theta \tan x)\, dx NB. You're unlikely to find this is any books... I worked it out a while back, and haven't seen it...
  5. DreamWeaver

    MHB Evaluating the Integral Using the Clausen Function

    Find a closed form evaluation for the following trigonometric integral, where the 0 < \theta \le \pi/2:\int_0^{\theta}\frac{x^2}{\sin x} \, dx= \text{?} Hint:
  6. DreamWeaver

    MHB Integral Challenge #1: Define & Prove Special Functions

    Define the special functions:\text{Ti}_1(z)=\tan^{-1}z \text{Ti}_{m+1}(z)=\int_0^z\frac{ \text{Ti}_{m+1}(x)}{x}\,dxand\text{Thi}_1(z)=\tanh^{-1}z \text{Thi}_{m+1}(z)=\int_0^z\frac{ \text{Thi}_{m+1}(x)}{x}\,dx Now, for a, b \in \mathbb{R}^{+}, prove the following:\int_0^{\infty}\frac{x^m}{a...
  7. anemone

    MHB Solve Exponent Challenge: Prove Equality

    Prove that \left( 6+845^{\frac{1}{3}}+325^{\frac{1}{3}} \right)^{\frac{1}{3}}+\left( 6+847^{\frac{1}{3}}+539^{\frac{1}{3}} \right)^{\frac{1}{3}}=\left( 4+245^{\frac{1}{3}}+175^{\frac{1}{3}} \right)^{\frac{1}{3}}+\left( 8+1859^{\frac{1}{3}}+1573^{\frac{1}{3}} \right)^{\frac{1}{3}}
  8. O

    Can You Find the Best Constant for Sum-Free Subsets?

    A set A of non-zero integers is called sum-free if for all choices of a,b\in A, a+b is not contained in A. The Challenge: Find a constant c > 0 such that for every finite set of integers B not containing 0, there is a subset A of B such that A is sum-free and |A| ≥ c|B|, where |A| means the...
  9. O

    Challenge 4: There's no app for that: integration

    a.) Poor Wolfram Alpha got asked to calculate the following integral \int_{0}^{\infty} e^{-ax} \frac{\sin(x)}{x} dx but couldn't handle it! http://www.wolframalpha.com/input/?i=int_%7B0%7D%5E%7Binfty%7D+e%5E%7B-ax%7D+sin%28x%29+%2Fxdx (Results are not guaranteed if you use wolfram alpha...
  10. O

    Weekly Math Challenge Point Standings

    Here are the total point standings as of 10:33am on 12/20/2013: mfb 9 Citan Uzuki 7 Boorglar 3 jbunniii 3 Mandelbroth 2 jk22 2 hilbert2 2 economicsnerd 2 HS-Scientist 1 D H 1 jackmell 1 verty 1 Perok 1
  11. O

    Challenge 3b: What's in a polynomial?

    In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The new challenge: Are there any...
  12. O

    Challenge 3a: What's in a polynomial?

    In order to challenge a broader section of the forum, there is a part a and a part b to this challenge - if you feel that part b is an appropriate challenge, then I request you do not post a solution to part a as part a is a strictly easier question than part b. The challenge: Prove that the...
  13. 0

    Solve Min Velocity for Ball Not to Touch Hemispherical Rock

    Homework Statement A person standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it a horizontal velocity v. What is the minimum initial speed to ensure the ball doesn't touch the rock? Homework Equations x^2 + y^2 = r^2...
  14. O

    Challenge 2: Covering the Triangle

    Suppose A1, A2 and A3 are closed convex sets, and let Δ be a triangle with edges F1, F2 and F3 such that A_1 \cup A_2 \cup A_3 = \Delta and F_i \cap A_i = \emptyset \text{ for } i=1,2,3 Prove there exists some point x\in \Delta such that x\in A_1 \cap A_2 \cap A_3 Figurative bonus...
  15. anemone

    MHB Can $3^{2008}+4^{2009}$ Be Factored into Two Numbers Larger Than $2009^{182}$?

    Show that $3^{2008}+4^{2009}$ can be written as product of two positive integers each of which is larger than $2009^{182}$
  16. DreamWeaver

    MHB Definite integral challenge....

    For m \in \mathbb{Z}^+, and a, \, z \in \mathbb{R} > 0, evaluate the definite integral:\int_0^z\frac{x^m}{(a+\log x)}\,dx[I'll be adding a few generalized forms like this in the logarithmic integrals thread, in Maths Notes, shortly... (Heidy) ]
  17. O

    Challenge 1: Multiple Zeta Values

    A multiple zeta value is defined as \zeta(s_1,...,s_k) = \sum_{n_1 > n_2 ... > n_k > 0} \frac{1}{n_1^{s_1} n_2^{s_2}...n_k^{s_k}} . For example, \zeta(4) = \sum_{n = 1}^{\infty} \frac{1}{n^4} and \zeta(2,2) = \sum_{m =1}^{\infty} \sum_{n = 1}^{m-1} \frac{1}{ m^2 n^2} . Prove the...
  18. O

    Is there a forum for weekly engineering math challenges?

    What is this place? This forum is for people to come together and stretch their brains on math puzzles. Each week there will be a new challenge for the forum to try. What do I get for answering challenges? We're going to try a point system. The first person to post a solution will be awarded...
  19. anemone

    MHB Find the Maximal Value Challenge

    It's given that $p+m+n=12$ and that $p, m, n$ are non-negative integers. What is the maximal value of $pmn+pm+pn+mn$?
  20. anemone

    MHB What are the four roots of this challenging equation?

    Find the four roots of the equation $(x-3)^4+(x-5)^4+8=0$.
  21. anemone

    MHB Inequality Challenge: Prove $x^x \ge (x+1/2)^{x+1}$ for $x>0$

    Prove x^x \ge \left( \frac{x+1}{2} \right)^{x+1} for $x>0$.
  22. anemone

    MHB How Challenging Is This Definite Integral with a Tangent and Pi Power?

    Evaluate \int_0^{\frac{\pi}{2}} \frac{dx}{1+(\tan x)^{\pi e}}.
  23. anemone

    MHB Can you prove \tan 20^{\circ}+4 \sin 20^{\circ}=\sqrt{3}?

    Prove that \tan 20^{\circ}+4 \sin 20^{\circ}=\sqrt{3}.
  24. A

    Proving Uncountability of (0,1): A Puzzling Challenge

    Homework Statement The problem is attached as a picture. Homework Equations ... The Attempt at a Solution I have been trying a lot to prove this without any really fruitful approach. At first I thought that the statement was false, or that you could at least construct a sequence...
  25. anemone

    MHB Find Real Solution(s) Challenge

    Find the real solution(s) to the equation \frac{36}{\sqrt{x}}+\frac{9}{\sqrt{y}}=42-9\sqrt{x}-\sqrt{y}.
  26. anemone

    MHB Find the Smallest Integer Challenge

    Determine the smallest integer that is square and starts with the first four figure 3005. Calculator may be used but solution by computers will not be accepted.(Tongueout)
  27. N

    Overcoming Loss: My Most Significant Challenge

    Hi all. I'm starting my college apps and started with MIT. Here's the prompt: "Tell us about the most significant challenge you've faced or something important that didn't go according to plan. How did you manage the situation?(*) (200-250 words)." Below is my response. What I want to know...
  28. anemone

    MHB Can the Inequality Challenge be Proven: 2^{\frac{1}{3}}+2^{\frac{2}{3}}<3?

    Prove 2^{\frac{1}{3}}+2^{\frac{2}{3}}<3.
  29. anemone

    MHB Can you prove the combinatorics challenge and find the value of |S_n-3T_n|?

    For $n=1,2,...,$ set S_n=\sum_{k=0}^{3n} {3n\choose k} and T_n=\sum_{k=0}^{n} {3n\choose 3k}. Prove that $|S_n-3T_n|=2$.
  30. anemone

    MHB Evaluate Definite Integral Challenge

    Evaluate \int_0^{\pi} \frac{\cos 4x-\cos 4 \alpha}{\cos x-\cos \alpha} dx
  31. Borg

    News FTC Robocall Challenge Winner Announced: NoMoRobo Goes Live Soon

    I didn't even know this was going on. The FTC declared a winner in April for its FTC Robocall Challenge deal with the problem of illegal Robo calls. The winner will be going live soon and it will be free. They even have a easy to remember name - NoMoRobo. Although I am waiting to see how...
  32. anemone

    MHB Solve Polynomial Challenge: Prove $(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=720$

    Prove that there are only two real numbers such that $(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)=720$.
  33. anemone

    MHB Absolute Value Function Challenge

    Solve $||||||| x^2 – x –1 |–3|–5|–7|–9| – 11|–13| = x^2 – 2x – 48$.
  34. micromass

    Challenge XI: Harmonic Numbers

    This challenge was suggested by jgens. The ##n##th harmonic number is defined by H_n = \sum_{k=1}^n \frac{1}{k} Show that ##H_n## is never an integer if ##n\geq 2##.
  35. micromass

    What is the Limit of e^-n times n^k over k! for n Approaching Infinity?

    Find the following limit: \lim_{n\rightarrow +\infty} e^{-n} \sum_{k=0}^n \frac{n^k}{k!}
  36. alyafey22

    MHB Find Residues for f(z) at $z=-n$

    Find Residue at $z =0 $ of f(z) = \Gamma(z) \Gamma(z-1) x^{-z} Try to find Residues for $ z=-n $
  37. micromass

    Physics Challenge II: Bouncing out of the atmosphere

    Part 1: Consider ##n## balls ##B_1##, ##B_2##, ..., ##B_n## having masses ##m_1##, ..., ##m_n##, such that ##m_1\ll m_2\ll ...\ll m_n##. The ##n## balls are stacked above each other. The bottom of ##B_1## is a height ##h## above the ground, and the bottom ##B_n## is a height ##\ell## above the...
  38. micromass

    Challenge IX: Dealing with Mod 1 solved by mfb

    This challenge was proposed by Boorglar. Many thanks to him! Let n be a natural number larger than 1, and a be a positive real number. Prove that if the sequence \{a\}, \{an\}, \{an^2\},... does not eventually become 0, then it will exceed 1/n infinitely many times. Here {x} means x -...
  39. micromass

    Challenge VIII: Discontinuities of a function solved by Theorem.

    An open set in ##\mathbb{R}## is any set which can be written as the union of open intervals ##(a,b)## with ##a<b##. A subset of ##\mathbb{R}## is called a ##G_\delta## set if it is the countable intersection of open sets. Prove that if a set ##A\subseteq \mathbb{R}## is a ##G_\delta## set...
  40. micromass

    Physics Challenge I: The Raindrop solved by mfb and voko

    Assume that a cloud consists of tiny water droplets suspended (uniformly distributed, and at rest) in air, and consider a raindrop falling through them. What is the acceleration of the raindrop? Assume that the raindrop is initially of negligible size and that when it hits a water droplet, the...
  41. micromass

    Challenge VII: A bit of number theory solved by Boorglar

    This new challenge was suggested by jostpuur. It is rather number theoretic. Assume that q\in \mathbb{Q} is an arbitrary positive rational number. Does there exist a natural number L\in \mathbb{N} such that Lq=99…9900…00 with some amounts of nines and zeros? Prove or find a counterexample.
  42. Greg Bernhardt

    Challenge II: Almost Disjoint Sets, solved by HS-Scientist

    Written by micromass: The newest challenge was the following: This was solved by HS-Scientist. Here's his solution: This is a very beautiful construction. Here's yet another way of showing it. Definition: Let ##X## be a countable set. Let ##A,B\subseteq X##, we say that ##A## and...
  43. Greg Bernhardt

    Challenge I: Concavity, solved by Millenial

    Written by micromass: I have recently posted a challenge in my signature. The challenge read as follows: The first answer I got was from Millenial. He gave the following correct solution: This solution is very elegant. But there are other solutions. For example, we can prove the...
  44. micromass

    Challenge VI: 15 Puzzle solved by Boorglar and mfb

    NEW CHALLENGE: This challenge was a suggestion by jgens. I am very thankful that he provided me with this neat problem. A 15-puzzle has the following form: The puzzle above is solved. The object of the game is to take an unsolved puzzle, such as and to make a combination of...
  45. micromass

    Challenge V: Sylvester-Gallai Theorem, solved by Mandelbroth

    Let's put up a new challenge: This is called the Fano plane: This is a geometric figure consisting of 7 points and 7 lines. However, it is a so-called projective plane. This means that it satisfies the following axioms: 1) Through any two points, there is exactly one line 2) Any two...
  46. micromass

    Challenge IV: Complex Square Roots, solved by jgens

    This is a well-known result in complex analysis. But let's see what people come up with anyway: Challenge: Prove that there is no continuous function ##f:\mathbb{C}\rightarrow \mathbb{C}## such that ##(f(x))^2 = x## for each ##x\in \mathbb{C}##.
  47. micromass

    Challenge III: Rational Tangles, solved by pwsnafu

    The newest challenge is the following: As an example, we can easily go from ##0## to ##-1/3##. Indeed, we can apply ##T## to ##0## to go to ##1##, we apply ##T## to go to ##2##, we apply ##T## to go to ##3##, and then we apply ##R## to go to ##-1/3##.
  48. anemone

    MHB How Can You Effectively Evaluate a Challenging Definite Integral?

    Evaluate \int_2^4 \frac{\sqrt{\ln (9-x)}\,dx}{\sqrt{\ln (9-x)}+\sqrt{\ln (x+3)}}
  49. anemone

    MHB Sequence of Positive Integers Challenge

    Consider the sequence of positive integers which satisfies a_n=a_{n-1}^2+a_{n-2}^2+a_{n-3}^2 for all $n \ge 3$. Prove that if $a_k=1997$, then $k \le 3$.
  50. MarkFL

    MHB Optimization Challenge - Poles and Wires

    Suppose you have two poles separated by the distance $w$, the first of height $h_1$ and the second of $h_2$, where $0<h_1<h_2$. You wish to attach two wires to the ground in between the poles, one to the top of each pole, such that the angle subtended by the two wires is a maximum. What portion...
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