Challenge Definition and 942 Threads

Find all pairs $(p, q)$ of integers such that $1+1996p+1998q=pq$.

Evaluate \tan\frac{\pi}{13}\tan\frac{2\pi}{13}\tan\frac{3 \pi}{13}\tan\frac{4\pi}{13}\tan\frac{5\pi}{13} \tan \frac{6\pi}{13}.

Solve 2\sin^4 (x)(\sin((2x)-3)-2\sin^2 (x)(\sin((2x)-3)-1=0.

Find distinct positive integers a,\;b, and c such that a+b+c,\;ab+bc+ac,\;abc forms an arithmetic progression.

Prove the following _2F_1 \left(a,1-a;c; \frac{1}{2} \right) = \frac{\Gamma \left(\frac{c}{2} \right)\Gamma \left(\frac{1+c}{2} \right) } {\Gamma \left(\frac{c+a}{2}\right)\Gamma \left(\frac{1+c-a}{2}\right)}.

Consider a pyramid whose base is an $n$-gon with side length $s$, and whose height is $h$. What is the radius of the largest sphere that will fit entirely within the pyramid?

Prove the following {}_2 F_1 \left( a,b; c ; x \right) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int^1_0 t^{b-1}(1-t)^{c-b-1} (1-xt)^{-a} \, dt Hypergeometric function .

d is deposited in a savings account for 3 years, then 2d for 3years, then 3d for 3 years, and so on similarly; here's an example for 9 years, 1st deposit = $100, rate = 10% annual: YEAR DEPOSIT INTEREST BALANCE 0 .00 1 100.00 .00...

Solve the following system in real numbers: a^2+b^2=2c 1+a^2=2ac c^2=ab

Show that \frac{1}{44}>\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right)\cdots\left( \frac{1997}{1998}\right)>\frac{1}{1999}

Solve \sqrt{x^2-1}+\sqrt{x-1}=x\sqrt{x}.

How many five digit numbers are divisible by 3 and contain the digit 6?

Find the constants a,\;b, \;c,\; d such that 4x^3-3x+\frac{\sqrt{3}}{2}=a(x-b)(x-c)(x-d).

Find all x in the interval [0, 2\pi] which satisfies 2\cos(x) \le \left|\sqrt{1+\sin (2x)}-\sqrt{1-\sin (2x)} \right|\le \sqrt{2}

In a competition there are a contestants and b judges, where b \ge 3 is an odd integer. Each judge rates each contestant as either "pass" or "fail". Suppose k is a number such that for any two judges their ratings coincide for at most k contestants. Prove \frac{k}{a}\ge\frac{b-1}{2b}.

Let: f(x)=x\sin(x) Derive a formula for: f^{(n)}(x) Using this, infer a formula for: \frac{d^n}{dx^n}\left(x\cos(x) \right) edit: I wanted to make sure it is clearly understood that: f^{(n)}(x)\equiv\frac{d^n}{dx^n}\left(f(x) \right)

Here is another challenge that I thought was fun :cool: A man wanted to get into his work building, but he had forgotten his code. However, he did remember five clues. These are what those clues were: The fifth number plus the third number equals fourteen. The fourth number is one more than...

Hello MHB, I wanted to post a challange question that is hopefully not really difficult, if the question is not understandable make sure to write it so I can try explain!:)Calculate the Differential equation for y''+2y'=0 that satisfy \lim_{x->\infty}y(x)=1 and y(0)=0 Regards, |\pi\rangle

A mechanical design question, just wanted to pick your brains about a project I'm working on. I have a fixture that houses a magnet, when an operator loads a part with a screw the magnet jumps and a thru beam on the magnet detects that a screw is present. Without a screw (below) the magnet...

Let n =xy be a positive integer where x,y>2 are co-prime. Show that if a is co-prime to n, then $a^{\frac{\phi(n)}{2}}=1$ mod(n)

This isn't so much a challenge problem as much as it has a startling (at least I think so) answer. Say we tie a rope around the Earth. Now we are going to cut it and add another 6 meters to it. If we pull the new rope tight (in a circle) how high is the new rope above the Earth's surface...

Show that if $x$ is a primitive root of p, and $x^{p-1}$ is not congruent to 1 mod$p^2$, then x is a primitive root of $p^2$

Just an easy one to start off with, find a non-recursive formula for the $n$th term of the following linear homogeneous recurrence: $$a_0 = 2, ~ ~ a_1 = -2, ~ ~ a_n = -2 a_{n - 1} + 2 a_{n - 2} ~ ~ \text{for} ~ n \geq 2$$ Hint:

Dr. Chinese’s Challenge: 0,120,240 Data Set Data Features (Quantum Theory): P(B|A) = 1; P(B|Aʹ) = .25; P(B) = .50 Eq. (1) “A” means “same setting”, “Aʹ” means “different setting”, and “B” means “different outcome”, “Bʹ” means “same outcome”. Here is a quote from David Mermin’s paper (Is the...

Hi, I have to make the classic egg drop contraption but we're only allowed to use plastic drinking straws, index cards, and hot glue. the width also has to be smaller than 4x4 inches but it can be as tall as we want it to be (but preferably small) and it has to have a door of some sort...

Let $n = pq$ such that $p$ and $q$ are distinct primes. Let $a$ be coprime to $n$. Show that the following holds: $$a^{p^k + q^k} \equiv a^{n^k + 1} \pmod{n} ~ ~ ~ ~ ~ \text{for all} ~ ~ k \in \mathbb{Z}$$

A force F is acting on an object whose mass is 1kg. The force in Newtons is 4+4t², where t is the time. The angle in radians that the force does with the displacement is 2πt. If the object was initially at rest, estimate the power due to that force at t=3s. I've tried to solve it but I've...

The speed of a body mass m = 1.6 kg moving in a straight line varies smoothly. It is measured five times, at two second intervals. The values are 7.00, 7.77, 8.89, 9.80, and 10.50 (m/s). What is the maximum value of the force acting on the body? I tried taking the average acceleration of the...

I explored the challenge forums today and found it very interesting. I thought it would be a good idea to share a problem with this excellent community. Show that $$ \int_0^\infty \frac{\ln(\tan^2 (x))}{1+x^2}dx = \pi \ln(\tanh(1))$$

HI : good to see you everybody ,I am a new comer on this board Being a math teacher ,I always trained my students with various difficult problems from now on I am going to post a sequential challenging questions for people to share most of them I know the answer and the solution of it , maybe...

Hello. I have been trying to find this limit: Lim as x --> 1 of (sin((1-x)/2)*tan(Pi*x/2)) Of course I don't want to solve it using L'Hospital. I have tried several ways but ended up in one of these. lim as y -->0 of (y*tan(pi/2-y*pi)) The answer when using L'Hospital is 1/pi...

Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations I have no idea. The Attempt at a Solution I seriously have no idea. This is for my analysis course, and I'm not...

I need help finding the answer for this physics problem that I have to do. A 2,345kg car is traveling down a highway entrance ramp, at an angle of 5.74 degree at 65 miles/hours and slams on its brakes to keep from hitting another car. If the coefficient of friction between the tires and the...

hey everybody :) i am new here and i need ur help. I have to solve this problem for home work tomorrow.Topic is forces in the body. I attached it. I actually need help for the whole first question. thx u in advance. Please try to explain it not too difficulty. thanks u :) Homework...

A golfer lines up for her first putt at a hole that is 13 m exactly northwest of her ball’s location. She hits the ball 13 m and straight, but at the wrong angle, 39° from due north. a) In order for the golfer to have a “two putt green,” determine the angle of the second putt...

Hello everyone, first post on this forum and I'm really excited to hear you feed back. For my first year engineering design course I'm asked to design a vehicle that will undergo a hairpin turn (180 degrees) around a 2 by 4 piece of wood. The vehicle must only harness the power of elastics or...

Homework Statement A car goes over a speedbump, which has the cross-section of a cylinder of radius R embedded in the roadway. If you want a car driving with a speed Vo to be impeded, how large must R be? I'm very confused about this problem because we have not discussed the topic in class...

I'm having trouble with the following question: Construct a polynomial $q(x) \neq 0$ with integer coefficients which has no rational roots but is such that for any prime $p$ we can solve the congruence $q(x) \equiv 0 \mod p$ in the integers. Any hints on how to even start the problem will be...

Homework Statement An even integer can be written as the expression 2n. Find three consecutive even integers with a sum of 54. Homework Equations None, this was a challenge question. The Attempt at a Solution I got that the sum of the digits in 104 (2*52) equals 54 but cannot see...

Well, I found this challenge in another forum (not about math) on the internet, and, originally, there were no 'w', 'z' or 'y' drawn on the pic, it was just "find the x", but I put them on because I know you guys probably would create other variables to solve the problem. Another problem I had...

Hello all, Semester has just started and i am dong a 4th year dynamics unit, so i am reviewing the previous years unit which was a pre-requisite for the unit i am currently doing. I have come across a rather difficult dynamics problem that was given as a challenge question at the beginning of...

Homework Statement Imagine a function: f(x) = x + sin(x)/K where K is pi cubed. Suppose f(x)^n means f(f(f(f(...f(x))))) n times. Find the value of lim n->infinity f(x)^n Homework Equations The Attempt at a Solution I have no idea on where to start. Can anyone give a hint...

I need help solving this problem. To be clear, I AM NOT LOOKING FOR THE ANSWER...just need some hints as to HOW to go about solving. I've tried several paths/steps and I keep going in circles. Can someone please help me get started on a path? Person A left Town X at 10:18 am. He walked at...

I thought that this was an interesting idea. Basically, Alan Alda started a competition of some sorts for people to explain what fire is to eleven year olds. The basis for starting this is that, when he was eleven, he was interested in fire, asked his science teacher what it was, and all that...

Hi all, need some help here. If I wish to contradict a well established theory in physics (namely the Carnot theorem), then what would be a suitable platform? I've tried a few journals but most of those publish experimental papers & won't accept my paper because it’s theoretical. Pls suggest a...

Hey Everyone, Firstly, I did post this in the "Should I be an Engineer" thread, but I feel like this post is more asking "do you think this is the right/smart career/academic choice" and not "do you think my skills would make me a good engineer" tl;dr : I'm a lazy kid who just finished up...

I was wondering if anyone could help with this- I've never seen a question like this before and don't know exactly how to tackle it...(Sweating) any help would be greatly appreciated!

Homework Statement Hello Physics World, This is a question that was presented in my High School Science Challenge by a teacher. It is sort of a brain teaser and we will be told the answer to it by next week. However, I was wanting to know your input. So, looking at the image above...

Lets take that magical number 1 with 100 zeroes at the end known as google and see what we can put it into. My challenge/ curiosity for the ones who love calculations is this: Not counting for gravity, and using any temperature you choose, how large would a cube of pure lead have to be to...

The best known algorithm ('Babylonian') to take the square root of a number requires 3 operations, less than Newton's. I suppose it's also the fastest. Is it so? Do you know a simpler or faster one? Can you find a method that requires only 1 operation?