Challenge Definition and 911 Threads

Let $F$ be the set of infinite sequences $(a_1,a_2,a_3...)$, where $a_i \in \Bbb{R}$ that satisfy $a_{i+3}=a_i+a_{i+1}+a_{i+2}$ This describes a finite-dimensional vector space. Determine a basis for $F$.

If both a transmitter and receiver antennae were submerged, and within a defined range and depth, what range might be available to a civilian operator? Are two divers able to communicate by radio? B

Determine the value of $\dfrac{2k^2}{k-1}$ given $\dfrac{k^2}{k-1}=k^2-8$.

Here are some of my thoughts: - thermal management of individual LEDs in a RBG system - photon absorption in phosphor coating - exponential decay in intensity - CCT/ CRT?

For $0<x<\dfrac{\pi}{2}$, prove that $\dfrac{\pi^2-x^2}{\pi^2+x^2}>\left(\dfrac{\sin x}{x}\right)^2$. I personally find this challenge very intriguing and I solved it, and feel good about it, hehehe...

Let $P$ be a function defined on $[0, 1]$ such that $P(0)=P(1)=1$ and $|P(a)-P(b)|<|a-b|$, for all $a\ne b$ in the interval $[0, 1]$. Prove that $|P(a)-P(b)|<\dfrac{1}{2}$.

Let $P,\,Q,\,R$ be the angles of an acute-angled triangle. Prove that $\sin P+\sin Q> \cos P+\cos Q +\cos R$.

Given a cyclic quadrilateral $PQRS$ where $PQ=p,\,QR=q,\,RS=r$, $\angle PQR=120^{\circ}$ and $\angle PQS=30^{\circ}$. Prove that $|\sqrt{r+p}-\sqrt{r+q}|=\sqrt{r-p-q}$

Can a person suspended in space with nothing to push off of, turn themselves around?

Solve the equation $\cos^k x-\sin^k x=1$, where $k$ is a given positive integer.

Homework Statement Combustion of hydrocarbon X in excess oxygen produces 0.66g of carbon dioxide, and 0.27 g of water. At room tempertaure and pressure , X is a gas with density 1.75gdm^-3 . What could the molecular of X be? What is the ans ? i only managed to get the empirical formula is CH2...

How many solutions does the equation $\sin a \sin (2a) \sin (3a) \cdots \sin (11a) \sin (12a) =0$ have in the interval $(0,\,\pi]$?

The image shows a network of rooms. A ball starts in room P. If the ball moves from one room to another adjacent one every second (assume no time is spent between the rooms) and it randomly chooses a room to go to, find the probability that it reaches room Q after n seconds. A room is adjacent...

Prove that for all integers $k>1$: $\left(\dfrac{1+(k+1)^{k+1}}{k+2}\right)^{k-1}>\left(\dfrac{1+k^k}{k+1}\right)^{k}$

Suppose $f$ is a continuous function on $(-\infty,\infty)$. Calculating the following in terms of $f$. $$\lim_{{x}\to{0}}f\left(\int_{0}^{\int_{0}^{x}f(y) \,dy} f(t)\,dt\right)$$

Homework Statement http://postimg.org/image/t2uxlnsdh/ What is the IUPAC name of this compound? 2. The attempt at a solution I have tried: 5-chloro-1,4-dimethyl-cyclohexene, 4-chloro-2,5-dimethyl-cyclohex-1-ene, and 5-chloro-1,4-dimethyl-cyclohex-1-ene

In a triangle $PQR$ right-angled at $R$, the median through $Q$ bisects the angle between $QP$ and the bisector of $\angle Q$. Prove that $2.5<\dfrac{PQ}{QR}<3$.

You have a malfunctioning calculator that cannot perform multiplication. However, it can add, subtract, and compute the reciprocal $\frac{1}{x}$ of any number $x$. Can you nevertheless use this defective calculator to multiply numbers?

Suppose that $PQRS$ is a square with side $p$. Let $A$ and $B$ be points on side $QR$ and $RS$ respectively, such that $\angle APB=45^{\circ}$. Let $T$ and $U$ be the intersections of $AB$ with $PQ$ and $PS$ respectively. Prove that $PT+PU\ge 2\sqrt{2}p$.

Find $y(x)$ to satisfy $$ y(x)=y'(x)+\int e^{2x}y(x) \, dx+\lim_{{x}\to{-\infty}}y(x)$$ given $$\lim_{{x}\to{0}}y(x)=0$$ and $$\lim_{{x}\to{\ln\left({\pi/2}\right)}}y(x)=1.$$

Given that $f(x)=x^4+x^3+x^2+x+1$. Show that there exist polynomials $P(y)$ and $Q(y)$ of positive degrees, with integer coefficients, such that $f(5y^2)=P(y)\cdot Q(y)$ for all $y$.

Let $ax^2+bx+c$ be a quadratic polynomial with complex coefficients such that $a$ and $b$ are non-zero. Prove that the roots of this quadratic polynomial lie in the region $|x|\le\left|\dfrac{b}{a}\right|+\left|\dfrac{c}{b}\right|$.

Prove that $\dfrac{\ln x}{x^3-1}<\dfrac{x+1}{3(x^3+x)}$ for $x>0,\,x\ne 1$.

What is the smallest positive integer n such that there are exactly 3 nonisomorphic Abelian group of order n

Let $a,\,b,\,c$ be integers such that $(a-b)^2+(b-c)^2+(c-a)^2=abc$. Prove that $a^3+b^3+c^3$ is divisible by $a+b+c+6$.

If $a,\,b$ are non-zero numbers with $a^2+ab+b^2=0$. Evaluate $\left(\dfrac{a}{a+b}\right)^{2015}+\left(\dfrac{b}{a+b}\right)^{2015}$.

Let $a,\,b,\,c,\,d$ be real numbers such that $abcd=1$ and $a+b+c+d>\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{d}+\dfrac{d}{a}$. Prove that $a+b+c+d<\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{d}{c}+\dfrac{a}{d}$.

Show that if $\dfrac{\cos x}{\cos y}+\dfrac{\sin x}{\sin y}=-1$, then $\dfrac{\cos^3 y}{\cos x}+\dfrac{\sin^3 y}{\sin x}=1$.

There is a 3-volume set of books $\quad$on a bookshelf in the usual manner. Each volume has two covers, each $\frac{1}{4}$ inch thick, $\quad$and the printed portion is $1$ inch thick. A bookworm starts at the first page of Volume 1 $\quad$and eats his way to the last page of Volume 3. How...

Submitted by @PeroK Consider all 7-digit numbers which are a permutation of the digits 1-7. How many of these are divisible by 7? Can you prove the answer algebraically, rather than simply counting them? Please make use of the spoiler tag

This challenge has been provided by @Joffan A magic square has rows, columns and diagonals summing to the same number. For a 3x3 magic square there are 8 such sums. Given a set of 9 distinct integers which has at least 8 subsets of 3 all with a common sum, is it always possible to make a magic...

Google has recently created a challenge to find new employees by having link appear in people's google searches saying "you know our language". This happens for people with large amounts of programming searches in their history. Read all about it...

Homework Statement Here it is: Let Ω be a convex region in R2 and let L be a line segment of length ι that connects points on the boundary of Ω. As we move one end of L around the boundary, the other end will also move about this boundary, and the midpoint of L will trace out a curve within Ω...

With only only paper & pencil (no calculator or logarithmic tables), figure out which of the following expressions has a greater value: 101/10 or 31/3. Please make use of the spoiler tag and write out your full explanation, not just the answer.

What is the smallest integer such that if you rotate the number to the left you get a number that is exactly one and a half times the original number? (To rotate the number left, take the first digit off the front and append it to the end of the number. 2591 rotated to the left is 5912.)...

for positive a , b, c prove that $a^4+b^4+c^4 \ge abc(a+b+c)$

There are six villages along the coast of the only perfectly round island in the known universe. The villages are evenly distributed along the coastline so that the distance between any two neighboring coastal villages is always the same. There is an absolutely straight path through the jungle...

Homework Statement Parameters: 1. Design and build an egg capsule within a volume of 10cm x 10cm x 10cm. 2. There must be a lid opening to insert the egg. 3.The capsule must be drop ready within 60 seconds or less after obtaining the egg from your instructor. 4. Drop time is non-negotiable, be...

Prove that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots+\sqrt{2006}}}}<2$.

Assuming the axiom of choice, show that a group $G$ has trivial automorphism group if and only if $G$ has less than three elements.

$a_1=2 ,$ and $a_{n+1}=\dfrac{a_n+4}{2a_n+3},\,\, n\in N$ find :$a_n$

For any positive integer $n\;,$ prove that$\sqrt{4n+1}<\sqrt{n}+\sqrt{n+1}<\sqrt{4n+2}$. Hence or otherwise, prove that $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+1}}\right\rfloor$ for any positive integer $n$.

Evaluate $2\cos^3 \dfrac{\pi}{7}-\cos^2 \dfrac{\pi}{7}-\cos \dfrac{\pi}{7}$.

Place the digits 1 through 8 in the boxes so that no two consecutive digits are adjacent (not vertically, horizontally or diagonally). . . \begin{array}{cccccccccc}&& * & - & * & - & * \\ && | && | && | \\ * &-& * &-& * &-& * &-& * \\ | && | && | && | && | \\ * &-& * &-& * &-& * &-& * \\ && | &&...

Prove that $\left\lfloor{\sqrt{n}+\sqrt{n+1}}\right\rfloor=\left\lfloor{\sqrt{4n+2}}\right\rfloor$ for any positive integer $n$.

Hi, I am doing an exercise practice samples for the upcoming quiz, and stumbled across two questions I'm having trouble solving... First question is to integrate integral e-x2 dx ...where the solution is equal to pi1/2 Also... As for the second question (of a different equation) how can one...

Dear Readers, I'm facing a pretty interesting challenge, which is: "How to connect multiple smartphone mic's in the quickest and simplest way?" Here is a use case. I go into a meeting with a group of random (strangers) that all have smartphones, Android, iOS, Windows. Now I want to record this...

Find the minimum of $\sqrt{a^2-12a+40}+\sqrt{b^2-8b+20}+\sqrt{a^2+b^2}$.

One thing I find consistently disturbing about the way printing shows up on web forums is that it's hard to perceive the customary double-space between two sentences.as being much of a physical division. The period is usually a tiny speck, so it doesn't, by itself, do the job of separation...

Hello, Before anyone thinks this is a coursework question, it is not. It is a challenge problem, which I found online, and seems worth discussing. (Question) In a 400-m relay race the anchorman (the person who runs the last 100 m) for team A can run 100 m in 9.8 s. His rival, the anchorman...