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.
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 $\int_{a}^{b} f(x)\,dx=\int_{a}^{b} g(x)\,dx$.
Find all values of $x$ which satisfy $\tan \left( x+\dfrac{\pi}{18}\right)\tan \left( x+\dfrac{\pi}{9}\right)\tan \left( x+\dfrac{\pi}{6}\right)=\tan x$.
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.
Prove that
$\dfrac{2007}{2}-\dfrac{2006}{3}+\dfrac{2005}{4}-\cdots-\dfrac{2}{2007}+\dfrac{1}{2008}=\dfrac{1}{1005}+\dfrac{3}{1006}+\dfrac{5}{1007}+\cdots+\dfrac{2007}{2008}$
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...
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 $x$.
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...
Happily Married
QUESTION:
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 is allowed.
Points will be given as follows:
1)...
Hi MHB,
Problem:
Find all real $x$ which satisfy $\dfrac{x^3+a^3}{(x+a)^3}+ \dfrac{x^3+b^3}{(x+b)^3}+\dfrac{x^3+c^3}{(x+c)^3} + \dfrac{3(x-a)(x-b)(x-c)}{2(x+a)(x+b)(x+c)}=\dfrac{3}{2}$.
I tried my very best to solve this intriguing problem, but failed. Now I'm even clueless than I was...
There is a sequence which has the first 3 terms listed as $1,\,94095,\,5265679\cdots$.
The 50th term has all but one digit. If the missing digit is $a$, find the $a$th term from this sequence.
The cubic polynomial $x^3+mx^2+nx+k=0$ has three distinct real roots but the other polynomial $(x^2+x+2014)^3+m(x^2+x+2014)^2+n(x^2+x+2014)+k=0$ has no real roots. Show that $k+2014n+2014^2m+2014^3>\dfrac{1}{64}$.
Consider the letter T (written as such: thus we have two line segments).
1) Prove that it is impossible to to place uncountably many copies of the letter T disjointly in the plane ##\mathbb{R}^2##.
2) Prove that it is impossible to place uncountably many homeomorphic copies of the letter T...
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...
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...
"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...
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?
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...
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 $k$.
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?