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. ranga519

    MHB Challenge. Difficulty level - medium

    In a regular quadrangular pyramid the area of a side surface is twice as much as the area of the base. Find the ratio of the height to the length of the side of the base of the pyramid.
  2. Olinguito

    MHB Challenge Problem #9 [Olinguito]

    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 $\pi\in S_n$, then...
  3. fresh_42

    Challenge Math Challenge - June 2019

    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...
  4. Olinguito

    MHB Challenge Problem #8: 3Σ(1/(√(a^3+1))≥2Σ(√(a+b))

    Let $a,b,c$ be positive real numbers such that $a+b+c=2$. Prove that $$3\left(\frac1{\sqrt{a^3+1}}+\frac1{\sqrt{b^3+1}}+\frac1{\sqrt{c^3+1}}\right)\ \geqslant\ 2\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right).$$
  5. Olinguito

    MHB Challenge Problem #7: Σ(x/(y^3+2))≥1

    Let $x,y,z$ be positive real numbers such that $xyz=1$. Prove that $$\frac x{y^3+2}+\frac y{z^3+2}+\frac z{x^3+2}\ \geqslant\ 1.$$
  6. C

    Challenge defining these DH parameters

    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...
  7. fresh_42

    Challenge Math Challenge - May 2019

    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...
  8. fresh_42

    Challenge Math Challenge - April 2019

    As (almost) always: have a look on previous challenge threads, too. E.g. in https://www.physicsforums.com/threads/math-challenge-march-2019.967174/ are still problems to solve, and some of them easy, which I find, and in any case useful to know or at least useful to have seen. As a general...
  9. lfdahl

    MHB Probability Challenge: Find Interval of Integers Drawn from Urn

    An urn contains $n$ balls numbered $1, 2, . . . , n$. They are drawn one at a time at random until the urn is empty. Find the probability that throughout this process the numbers on the balls which have been drawn is an interval of integers. (That is, for $1 \leq k \leq n$, after the $k$th draw...
  10. fresh_42

    Challenge Math Challenge - March 2019

    Questions 1.) (disclosed by @Demystifier ) Using the notion of double integrals prove that $$B(m,n) = \frac{\Gamma (m) \Gamma (n)}{\Gamma (m + n)}\; \;(m \gt 0\,,\, n\gt 0)$$ where ##B## and ##\Gamma## are the Beta and Gamma functions respectively. 2.) (solved by @Math_QED ) Show that the...
  11. Olinguito

    MHB Challenge Problem #6: Prove tan 18°=√(1-2/√5)

    Prove that $$\tan18^\circ\ =\ \sqrt{1-\dfrac2{\sqrt5}}.$$ No calculator, computer program, Excel, Google, or any other kind of cheating tool allowed. (Smirk) Have fun!
  12. anemone

    MHB Challenge question on equilateral triangle: Prove ∠DBA=42°

    In an equilateral triangle $ABC$, let $D$ be a point inside the triangle such that $\angle BAD=54^\circ$ and $\angle BCD=48^\circ$. Prove that $\angle DBA=42^\circ$.
  13. robphy

    B General Relativity as a Challenge for Physics Education

    This week I am at "General Relativity as a Challenge for Physics Education" 690. WE-Heraeus-Seminar https://www.we-heraeus-stiftung.de/veranstaltungen/seminare/2019/general-relativity-as-a-challenge-for-physics-education/ (...
  14. QuantumQuest

    Challenge Math Challenge - February 2019

    Time for our new winter challenge! This time our challenge has also two Computer Science related questions and a separate section with five High School math problems. Enjoy! Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be...
  15. fresh_42

    Challenge Math Challenge - January 2019

    Merry Christmas to all who celebrate it today! Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month. b) It is fine to use nontrivial results without proof as long...
  16. mfb

    B Daily math challenge calendar (external)

    I found this calendar with daily math puzzles. Based on the first three puzzles it seems to be much easier than the math challenges here, and require no advanced mathematics. The answer is always a three-digit number, and the answers to all 24 puzzles together create a larger puzzle.
  17. fresh_42

    Challenge Math Challenge - December 2018

    It's December and we like to do a Special this month. The challenges will be posted like an Advent Calendar. We will add a new problem each day, from 12/1 to 12/25. They vary between relatively easy logical and numerical problems, calculations, to little proofs which hopefully add some...
  18. fresh_42

    Challenge Math Challenge - November 2018

    Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month. b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common...
  19. fresh_42

    Challenge Math Challenge - October 2018

    Summer is coming and brings ... Oops, time for a change! Fall (Spring) is here and what's better than to solve some tricky problems on a long dark evening (with the power of returning vitality all around). RULES: a) In order for a solution to count, a full derivation or proof must be given...
  20. T

    NASA NASA Challenge - CO2 Conversion

    NASA is looking for a process to use CO2 as a Carbon source on Mars; ultimate goal is to use the Carbon in the synthesis of other products. $50,000 prize. Open to U.S. citizens, permanent residents, and U.S. business entities, work must be done in the U.S...
  21. fresh_42

    Challenge Intermediate Math Challenge - September 2018

    Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other basic level math challenge thread! RULES: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be...
  22. fresh_42

    Challenge Basic Math Challenge - September 2018

    Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions...
  23. anemone

    MHB System of Equation Challenge (a+b)(b+c)=-1

    Given that a,\,b and c are real numbers that satisfy the system of equations below: (a+b)(b+c)=-1\\(a-b)^2+(a^2-b^2)^2=85\\(b-c)^2+(b^2-c^2)^2=75 Find (a-c)^2+(a^2-c^2)^2.
  24. fresh_42

    Challenge Intermediate Math Challenge - August 2018

    Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no...
  25. fresh_42

    Challenge Are You Ready for the Basic Math Challenge This August?

    Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions...
  26. W

    MHB Figuring Out the Value of N: Jack & John's CWS Challenge

    Jack, John and CWS's ============== Canadian Wild Strawberries (CWS) are tiny but tasty. A and B each have a jar containing 400 CWS; they decide to have a CWS eating race; A wins, swallowing his last CWS when B still has 23 left. Took A 13.2 seconds; burp! Next, B takes on C, each with a jar...
  27. W

    MHB Bingo! A Number-Caller's Challenge

    BINGO! ====== The number-caller announced: under the G...n! Gertrude, Josephine and Waltzing Mathilda all yelled "BINGO!". Happens that all 3 filled the top line of their bingo cards. The 15 numbers are all odd, plus numbers under each letter are like this: B: > 10 and none...
  28. fresh_42

    Challenge Intermediate Math Challenge - July 2018

    Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no...
  29. fresh_42

    Challenge Ready for a Summer Math Challenge?

    Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will...
  30. Olinguito

    MHB Challenge problem #5 [Olinguito]

    If $A$ and $B$ are nonempty sets of complex numbers, define $$A\circ B\ =\ \{z_1z_2:z_1\in A,\,z_2\in B\}.$$ Further define $A^{[1]}=A$ and recursively $A^{[n]}=A^{[n-1]}\circ A$ for $n>1$. Let $\zeta_n=\{z\in\mathbb C:z^n=1\}$. Given a fixed integer $n\geqslant2$ and any positive integer $r$...
  31. QuantumQuest

    Challenge Intermediate Math Challenge - June 2018

    Summer is coming and brings a new intermediate math challenge! Enjoy! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no...
  32. QuantumQuest

    Challenge Can You Solve These Summer Math Challenges?

    Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine...
  33. Olinguito

    MHB Challenge problem #4 Is it possible for all n switches to be on at the end

    $n$ lights are arranged in a circle, each operated by exactly one of $n$ switches (with each switch operating exactly one light). Flicking a switch turns the light it is operating on if it is off, and off if it is on. Initially all the lights are off. The first person comes and flicks one of the...
  34. Olinguito

    MHB Challenge problem Find k if x=k is tangent to the curve y=x+√(2).e^[(x+y)/√(2)]

    If the line $x=k$ is tangent to the curve $$\large y\:=\:x+\sqrt2\,e^{\frac{x+y}{\sqrt2}}$$ what is the value of $k$?
  35. L

    MHB Why is Angle C Assumed to be Acute and What is the Value of Sin C in ∆ABC?

    Given that √5 tanA=-2 and CosB=8/17 in ∆ABC State why we may assume that angle C is acute and determine the value of Sin CAttempt made: tanA=-2/√5 CosB=8/17 A is obtuse angle of 138° or reflex angle 318.19° B is an acute angle of61.9° or reflex angle298.1 °.Since it is a right...
  36. QuantumQuest

    Challenge Intermediate Math Challenge - May 2018

    It's time for an intermediate math challenge! If you find the problems difficult to solve don't be disappointed! Just check our other basic level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored...
  37. QuantumQuest

    Challenge Can You Solve These Math Challenges?

    It's time for a basic math challenge! For more advanced problems you can check our other intermediate level math challenge thread! RULES: 1) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. 2) It is fine to use nontrivial results...
  38. fresh_42

    Reviving Math Challenges: Can You Solve These Tricky Problems?

    We, a small group of currently four members, want to try a new version of the math-challenges-threads once a month. It turned out to be not as easy as we thought, to find good problems. So what we've gathered are ten questions on "B" level and ten on "I" level for May, and plan to do the same...
  39. Olinguito

    MHB Challenge problem #2 Show that 5φ^2n+4(−1)^n is a perfect square

    Define a Fibonacci sequence by $$\varphi_0=0,\,\varphi_1=1;\ \varphi_{n+2}=\varphi_{n+1}+\varphi_n\ \forall \,n\in\mathbb Z^+\cup\{0\}.$$ Show that $$5\varphi_n^2+4(-1)^n$$ is a perfect square for all non-negative integers $n$.
  40. Olinguito

    MHB Challenge problem #1 Solve 2x^2y^2−2xy+x^2+y^2−2x−2y+3=0

    Hi all. I would like to post some challenge problems from time to time. I’ll start with a simple one. :) Find all real numbers $x,y$ satisfying the following equation: $$2x^2y^2-2xy+x^2+y^2-2x-2y+3=0.$$
  41. lfdahl

    MHB Fish Pond Challenge: Show Equilibrium Variation with $R_f$

    Suppose a pond contains $x(t)$ fish at time $t$, and $x(t)$ changes according to the DE: \[\frac{\mathrm{d} x}{\mathrm{d} t} = x\left ( 1-\frac{x}{x_0} \right )-R_f\] where $x_0$ is the equilibrium amount with no fishing and $R_f > 0$ is the constant rate of removal due to fishing. Assume $x(0)...
  42. lfdahl

    MHB Trigonometric product challenge

    Prove, that $$\prod_{j = 1}^{n}\left(1+2\cos \left(\frac{3^j}{3^n+1}2\pi\right)\right) = 1.$$
  43. lfdahl

    MHB Integral challenge ∫ln2(1+x^(−1))dx

    Evaluate the definite integral $$\int_{0}^{1} \ln^2(1+x^{-1}) \,dx$$
  44. castor28

    MHB Polynomial challenge: Show that not all the coefficients of f(x) are integers.

    $f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. Show that not all the coefficients of $f(x)$ are integers.
  45. lfdahl

    MHB Definite integral challenge ∫ln(2−2cosx)dx=0

    Prove, that the definite integral $$\int_{0}^{\pi}\ln (2-2\cos x)dx = 0.$$
  46. lfdahl

    MHB Finding the Product of Distinct Roots: A Complex Challenge

    Let $r_1,r_2, …,r_7$ be the distinct roots (one real and six complex) of the equation $x^7-7= 0$. Let \[p = (r_1+r_2)(r_1+r_3)…(r_1+r_7)(r_2+r_3)(r_2+r_4)…(r_2+r_7)…(r_6+r_7) = \prod_{1\leq i<j\leq 7}(r_i+r_j).\] Evaluate $p^2$.
  47. lfdahl

    MHB Calculus inequality challenge prove ∫10f(x)/f(x+1/2)dx≥1

    Let $f$ be a positive and continuous function on the real line which satisfies $f(x + 1) = f(x)$ for all numbers $x$. Prove \[\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})}dx \geq 1.\]
  48. lfdahl

    MHB Sequence Challenge: Proving Periodicity of $\left\{x_n\right\}$ (Mod 11)

    Let the sequence $\left\{x_n\right\}$ of integers (modulo $11$) be defined by the recurrence relation: $x_{n+3} \equiv \frac{1}{3}(x_{n+2}+x_{n+1}+x_n)$ (mod $11$), for $n=1,2,..$ Show, that every such sequence $\left\{x_n\right\}$ is either constant or periodic with period $10$.
  49. lfdahl

    MHB Integral challenge ∫(sin^2θ)/(1−2acosθ+a^2)dθ, 0<a<1

    Solve the definite integral \[I(a) = \int_{0}^{2\pi}\frac{\sin^2 \theta }{1-2a\cos \theta + a^2}\: \: d\theta,\;\;\; 0<a<1.\]
  50. lfdahl

    MHB Series challenge: Evaluate 1/4+4/8+8/12+12/16+....

    Determine the sum: \[\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}+\frac{12!}{16!}+...\]
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