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. anemone

    MHB Can Inequalities Be Proven? A Solution to a Complex Equation

    Prove that $\sqrt{x^4+7x^3+x^2+7x}+3\sqrt{3x}\ge10x-x^2$.
  2. anemone

    MHB What Are the Real Solutions to This Complex Quadratic Equation?

    Solve for the real solutions for $(3x^2-4x-1)(3x^2-4x-2)-3(3x^2-4x+5)-1=18x^2+36x-13$.
  3. anemone

    MHB Prove Integral Equality of Polynomials Degree 2 & 3

    Let $f(x)$ be a polynomial of degree 2 and $g(x)$ a polynomial of degree 3 such that $f(x)=g(x)$ at some three distinct equally spaced points $a,\,\dfrac{a+b}{2}$ and $b$. Prove that $\int_{a}^{b} f(x)\,dx=\int_{a}^{b} g(x)\,dx$.
  4. anemone

    MHB Solve Trig Challenge: Find All Values of x

    Find all values of $x$ which satisfy $\tan \left( x+\dfrac{\pi}{18}\right)\tan \left( x+\dfrac{\pi}{9}\right)\tan \left( x+\dfrac{\pi}{6}\right)=\tan x$.
  5. anemone

    MHB Can the Polynomial $x^7-2x^5+10x^2-1$ Have a Root Greater Than 1?

    Prove that $x^7-2x^5+10x^2-1$ has no root greater than 1. This is one of my all time favorite challenge problems! :o
  6. anemone

    MHB Probability of Male Picking Wife as Dance Partner

    There are 6 married couples (12 people) in a party. If every male has to pick a female as his dancing partner, find the probability that at least one male pick his own wife as his dancing partner.
  7. anemone

    MHB Algebra Challenge: Prove $\dfrac{2007}{2}-\ldots=\dfrac{2007}{2008}$

    Prove that $\dfrac{2007}{2}-\dfrac{2006}{3}+\dfrac{2005}{4}-\cdots-\dfrac{2}{2007}+\dfrac{1}{2008}=\dfrac{1}{1005}+\dfrac{3}{1006}+\dfrac{5}{1007}+\cdots+\dfrac{2007}{2008}$
  8. anemone

    MHB Sequence Challenge: Find $a_{61}+a_{63}$

    A sequence is defined recursively by $a_1=2007$, $a_2=2008$, $a_3=-2009$, and for $n>3$, $a_n=a_{n-1}-a_{n-2}+a_{n-3}+n$. Find $a_{61}+a_{63}$.
  9. anemone

    MHB Integral Compute: $\int \sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}\ dx$

    Compute the integral $I=\int \sqrt{2+\sqrt{2+\cdots+\sqrt{2+x}}}\,dx$ where the expression contains $n\ge 1$ square roots.
  10. anemone

    MHB Cubic Equation Challenge: What is the value of $mn^2+nk^2+km^2$?

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  11. anemone

    MHB Geometry and Trigonometry Challenge

    A rectangle with sides $x$ and $y$ is circumscribed by another rectangle of area $A^2$. Find all possible values of $A$ in terms of $x$ and $y$.
  12. Saitama

    MHB Summation #2 Prove: $\sum_{k=1}^n (2^k\sin^2\frac{x}{2^k})^2$

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  13. Saitama

    MHB Summation Challenge #1: Evaluate $\sum$

    Evaluate the following: $$\Large \sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k}$$
  14. micromass

    Challenge 19: Infinite products

    Infinite Products This weeks challenge is a short one: Please list any sources that you have used to solve this question. Using google or other search engines is forbidden. Wikipedia is allowed but not its search engine. Wolfram alpha and other mathematical software are not allowed...
  15. anemone

    MHB Can You Prove $6^{33}>3^{33}+4^{33}+5^{33}?

    Prove that $6^{33}>3^{33}+4^{33}+5^{33}$.
  16. anemone

    MHB Proving $2P(x)>P'(x)$ with Continuous Third Derivative of $P(x)$

    Let $P$ be a real function with a continuous third derivative such that $P(x),\,P'(x),\,P''(x),\,P'''(x)$ are greater than zero for all $x$. Suppose that $P(x)>P'''(x)$ for all $x$, prove that $2P(x)>P'(x)$ for all $x$.
  17. anemone

    MHB Is There a Solution to the Challenge of Inequality?

    Given that $0<k,\,l,\,m,\,n<1$ and $klmn=(1-k)(1-l)(1-m)(1-n)$, show that $(k+l+m+n)-(k+m)(l+n)\ge1$.
  18. anemone

    MHB What is the Simplified Form of the Trigonometric Expression?

    Evaluate $\dfrac{1}{\sin^2 \dfrac{\pi}{10}}+\dfrac{1}{\sin^2 \dfrac{3\pi}{10}}$.
  19. anemone

    MHB Polynomial Challenge: Find $k$ Integral Values

    Find all integral values of $k$ such that $q(a)=a^3+2a+k$ divides $p(a)=a^{12}-a^{11}+3a^{10}+11a^3-a^2+23a+30$.
  20. Saitama

    MHB What is the Maximum Value of Lambda in this Vector Algebra Challenge?

    Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors such that $\left|\vec{a}+\vec{b}+\vec{c}\right|=\sqrt{3}$ and $\left(\vec{a}\times\vec{b}\right)\cdot \left(\vec{b}\times\vec{c}\right)+\left(\vec{b}\times\vec{c}\right)\cdot...
  21. micromass

    Challenge 18: Happily Married

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  22. anemone

    MHB Triangle Inequality: $a^4-1, a^4+a^3+2a^2+a+1, 2a^3+a^2+2a+1$

    Show that for all $a>1$, there is a triangle with sides $a^4-1$, $a^4+a^3+2a^2+a+1$, and $2a^3+a^2+2a+1$.
  23. anemone

    MHB Solving a Challenge: Finding Real x to Satisfy an Equation

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  24. DreamWeaver

    MHB Convergence of Improper Integral with Hyperbolic Functions?

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  25. anemone

    MHB Solve the Sequence Challenge: Find the Missing Digit & a Term

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  26. anemone

    MHB Can You Prove Inequality Challenge II?

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  27. anemone

    MHB Inequality Challenge: Prove $b^3+a^3 \le 2$

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  28. micromass

    Challenge 17: T and other letters

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  29. anemone

    MHB Trigonometric Identity Correction: Solving a Complex Equation

    If $\dfrac{\sin 4x}{a}=\dfrac{\sin 3x}{b}=\dfrac{\sin 2x}{c}=\dfrac{\sin x}{d}$, show that $2d^3(2c^3-a^2)=c^4(3d-b)$.
  30. micromass

    Challenge 16: About the equation x^y = y^x

    The idea of this challenge is to investigate the equation x^y = y^x Prove the following parts: If ##0<x<1## or if ##x=e##, then there is a unique real number ##y## such that ##y^x = x^y##. However, if ##x>1## and ##x\neq e##, then there is precisely one number ##g(x)\neq x## such...
  31. anemone

    MHB Can You Prove the Average of Trigonometric Numbers Equals Cot 1^o?

    Prove that the average of the numbers $n\sin n^{\circ}$ (where $n=2,\,4,\,6,\, \cdots,\,180$) is $\cot 1^{\circ}$.
  32. T

    Finding Matrices E & F: A Matrix Challenge

    Homework Statement Find two matrices E and F such that: EA= \begin{bmatrix} 2 & 1 & 2\\ 0 & 2 & 1\\ 0 & 3 & 0\\ \end{bmatrix} FA= \begin{bmatrix} 0 & 2 & 1\\ 0 & 3 & 0\\ 2 & 7 & 2\\ \end{bmatrix} Homework Equations The Attempt at a Solution So I know how to get...
  33. anemone

    MHB How is the Equation Derived for an Isosceles Triangle with a specific Angle?

    Given a triangle $PQR$ where $QR=m$, $PQ=PR=n$ and $\angle P=\dfrac{\pi}{7}$. Show that $m^4-3m^2n^2-mn^3+n^4=0$.
  34. anemone

    MHB Triangle $PQR$: Find $\tan P,\,\tan Q,\,\tan R$ Values

    In triangle $PQR$, $\tan P,\,\tan Q,\,\tan R$ are integers, find their values.
  35. anemone

    MHB Construct $\sqrt[4]{x^4+y^4}$ Segment with Straightedge & Compass

    Given two segments of lengths $x$ and $y$, construct with a straightedge and a compass a segment of length $\sqrt[4]{x^4+y^4}$.
  36. lfdahl

    MHB Divisibility Challenge: Find Smallest Integer for $f(x)$

    Let $f(x) = 5x^{13}+13x^5+9\cdot a \cdot x$ Find the smallest possible integer, $a$, such that $65$ divides $f(x)$ for every integer $x$.
  37. E

    Optic and waves design challenge about lasers

    "A diode laser has a divergence of 5mrad in the p-direction and 1 mrad in the s-direction. Design an optical system in front of the laser which will make the output circular, and calculate the resulting divergence." Attempt; I am taking the course optic and waves, and the instructor did some...
  38. anemone

    MHB Can You Prove the Inequality Challenge VI for Arctan Sequences?

    If $\alpha_n=\arctan n$, prove that $\alpha_{n+1}-\alpha_n<\dfrac{1}{n^2+n}$ for $n=1,\,2,\,\cdots$.
  39. anemone

    MHB Inequality Challenge V: Prove $(a+b)^{a+b} \le (2a)^a(2b)^b$

    Prove that for any real numbers $a$ and $b$ in $(0,\,1)$, that $(a+b)^{a+b}\le (2a)^a(2b)^b$.
  40. anemone

    MHB Polynomial Challenge: Find # of Int Roots of Degree 3 w/ Coeffs

    If $P(0)=3$ and $P(1)=11$ where $P$ is a polynomial of degree 3 with integer coefficients and $P$ has only 2 integer roots, find how many such polynomials $P$ exist?
  41. anemone

    MHB What is the Solution to This Definite Integral Challenge?

    Evaluate $\displaystyle\int^{\dfrac{\pi}{4}}_0 \dfrac{x}{(\sin x+\cos x)\cos x}\ dx$.
  42. lfdahl

    MHB What is the value of $a_{2013}$ in the sequence challenge II?

    Let $a_1 = 1$, $a_2 = a_3 = 2$, $a_4 = a_5 = a_6 = 3$, $a_7 = a_8 = a_9 = a_{10} = 4$, and so on. That is, $a_n ∶ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, . . . . $ What is $a_{2013}$?
  43. I

    Reaching the Moon in 0.9 Seconds: A Physics Challenge

    Homework Statement Nothing travels faster than light, which manages to get to the moon from the Earth in 1 second. However, we can still get there in a shorter amount of time. How fast would we have to travel to reach the moon in 0.9 seconds? Homework Equations I know the question is weird but...
  44. Saitama

    MHB Uncovering the Hidden Identity in Solving Quadratic Equation Challenge

    If the quadratic equation $x^2+(2 – \tan \theta)x – (1 + \tan \theta) = 0$ has two integral roots, then sum of all possible values of $\theta$ in the interval $(0, 2\pi)$ is $k\pi$. Find $k$.
  45. anemone

    MHB How Can the Sum of Sines Be Expressed Using a Trigonometric Identity?

    Show that $\displaystyle \sum_{k=0}^n \sin k=\dfrac{\sin \dfrac{n}{2} \sin\dfrac{n+1}{2}}{\sin \dfrac{1}{2}}$.
  46. anemone

    MHB Can We Prove This Inequality Challenge IV?

    Prove that $\dfrac{1}{\sqrt{4x}}\le\left( \dfrac{1}{2} \right)\left( \dfrac{3}{4} \right)\cdots\left( \dfrac{2x-1}{2x} \right)<\dfrac{1}{\sqrt{2x}}$.
  47. anemone

    MHB Can Jensen's Inequality Solve the Inequality Challenge III?

    Show that $e^\dfrac{1}{e}_{\phantom{i}}+e^{\dfrac{1}{\pi}}_{\phantom{i}} \ge2e^{\dfrac{1}{3}}_{\phantom{i}}$.
  48. anemone

    MHB Can you prove the cosine rule for three angles in a triangle?

    For all $x,\,y,\,z \in R$ with $x+y+z=2\pi$, prove that $\cos^2 x+\cos^2 y+\cos^2 z+2\cos x\cos y \cos z=1$
  49. anemone

    MHB Does the equation $a^2=b^4+b^2+1$ have integer solutions?

    Show that the equation $a^2=b^4+b^2+1$ does not have integer solutions.
  50. anemone

    MHB Probability Challenge: Jason's 2010 Coin Flips

    Jason has a coin which will come up the same as the last flip $\dfrac{2}{3}$ of the time and the other side $\dfrac{1}{3}$ of the time. He flips it and it comes up heads. He then flips it 2010 more times. What is the probability that the last flip is heads?
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