Challenge Definition and 911 Threads

Find $a_{2017}$, if $a_1 = 1$, and $$\frac{a_n}{n+1}=\frac{\sum_{i=1}^{n-1}a_i}{n-1}.$$

Calculate the following definite trigonometric integral: \[\int_{0}^{\frac{\pi}{2}} \cos^{2017}x \sin^{2017}x dx\].

Here's your challenge - is $\displaystyle \begin{align*} \cos{ \left( \frac{\pi}{60} \right) } \end{align*}$ transcendental, or does it have an exact surd value? If it has an exact surd value, what is it?

In basic car aerodynamics car manufactures know the basic design in every car creates lift. Let's take the Prius for example, very aerodynamic but does not have any downforce due to the shape of the vehicle (air travels farther on top of the car) being shaped like a wing of an airplane. My...

Prove that for $m=2,3,...$ $$\sin\frac{\pi}{m}\sin\frac{2\pi}{m}\sin\frac{3\pi}{m}\cdots\,\sin\frac{(m-1)\pi}{m}=\frac{m}{2^{m-1}}$$

Submitted by: @QuantumQuest Challenge Level: High School 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 without proof as long as you cite them and as long as it is "common...

New Florida law let's any resident challenge what’s taught in science classes. https://www.washingtonpost.com/news/speaking-of-science/wp/2017/07/01/new-florida-law-lets-any-resident-challenge-whats-taught-in-science-classes/?utm_term=.0104dd426ae7 I'm wondering if soon teaching actual science...

Prove the following identity:\[\sum_{n =1}^{\infty }\frac{1}{\binom{n+r}{r+1}}=\frac{r+1}{r},\: \: \: \: r,n \in \mathbb{N}.\]

I had been meaning to go into Feynman's derivation of Maxwell's Equations for a while now. I finally got around to it: http://signallake.com/innovation/DysonMaxwell041989.pdf He didn't make use of gauge invariance which Schwinger showed is its real basis and I know that derivation, as well as...

Greg asked me to post it myself. 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 without proof as long as you cite them and as long as it is "common knowledge to all...

Submitted and judged by: @QuantumQuest Solution credit awarded to: @ddddd28 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 without proof as long as you cite them and as long as...

Hi everyone: I need the brain of an engineer if anyone out there cares to help. I have a masters in Math from UofT but could use some knowledge from the smartest people- Engineers, I am now an experienced builder and yes everyone, we need smart people doing construction too- my math degree has...

Determine the area of the painted hexagon, knowing that the area of triangle ABC is 120cm^2 IMG Link: https://m.imgur.com/a/WtdsW I tried using Heron´s formula, but just ended up with a bunch of terms and one more variable. Sidenote: I guess part of it is figuring out that the side lenghts...

I have a reversible chemical reaction described by the balanced equation: ##aA+bB=cC+dD##. I devise a reversible process to take a closed system containing these species (and its surroundings) from thermodynamic equilibrium state 1 to thermodynamic equilibrium state 2: State 1: a moles of...

Submitted and judged by: @andrewkirk Solution credit: 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 without proof as long as you cite them and as long as it is "common...

Correct me if I am wrong, but my basic understanding of how the Chronology Protection Conjecture (CPC) would work is that, as virtual particles created from the quantum fields of the vacuum would traverse a wormhole and arrive in the past, they would then travel back into the wormhole alongside...

Submitted and judged by: @QuantumQuest Solution credit: @MAGNIBORO 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 without proof as long as you cite them and as long as it is...

Homework Statement Given n>=0, create an array length n*n with the following pattern, shown here for n=3 : {0, 0, 1, 0, 2, 1, 3, 2, 1} (spaces added to show the 3 groups). Homework EquationsThe Attempt at a Solution public int[] squareUp(int n) { int length = n*n; int[] completeArry...

Submitted and Judged by @QuantumQuest Solution credited to: @TSny 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 without proof as long as you cite them and as long as it is...

Submitted and judged by: @Erland Solution Credit: @SSequence for 2a, 2B, C, D 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 without proof as long as you cite them and as long...

Submitted by @Erland Solution Credit: @mfb 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 without proof as long as you cite them and as long as it is "common knowledge to all...

Submitted by: @QuantumQuest Credit to: @stevendaryl 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 without proof as long as you cite them and as long as it is "common...

I composed a problem and propose it here. I know the solution so it just for fun of the participants. There is a cylindrical bobbin of radius ##r##; the bobbin rotates about its central axis with angular velocity ##\omega=const>0##. An inextensible weightless string is coiled around the...

Submitted by @Charles Link Solved by: @MAGNIBORO and @maline 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 without proof as long as you cite them and as long as it is "common...

Assuming that the photon packet size is generally related to the wavelength of the light, say 500 nanometers and the electron approximate size of 2.82x10<-15 meters, how does the huge wavelength funnel it's packet energy to an electron approximately 1/17,730 th of it's size?

Acute triangle ABC Prove :Sin A +Sin B +Sin C>Cos A + Cos B + Cos C

I have figured out a nice way to prove that if the complex numbers z_1,z_2,\ldots, z_N\in\mathbb{C} are all distinct, then the equation \prod_{n=1}^N \frac{1}{z - z_n} = \sum_{n=1}^N \frac{\alpha_n}{z-z_n} is true for all z\in\mathbb{C}\setminus\{z_1,z_2,\ldots, z_N\}, where the alpha...

Prove the inequality: \[\left | \cos x \right |+ \left | \cos 2x \right |+\left | \cos 2^2x \right |+...+ \left | \cos 2^nx \right |\geq \frac{n}{2\sqrt{2}}\] - for any real x and any natural number, n.

Let's have a snack challenge for a while. ^^ Let $$x$$ and $$y$$ be real numbers (with restrictions $$y \ne 0, \ y \ne -x$$) and $$\frac{x - y}{x + y} = \frac{x + y}{y}$$. Find $$\frac{\mathrm{d}y}{\mathrm{d}x}$$ in whatever form you like most. I mean, for example forms...

Factorize $x^2+y^2+z^2-2xy-2yz-2zx$.

Hey all, I have a unit test tomorrow on Gravitational and Electric Fields. If you have any good practice questions for Grade 12 University level please leave them below! :) In my class we've learned gravitational force, gravitational field strength and orbits (omit geosynchronous) as well as...

For my physics assignment we have to design and test a way to allow an egg to fall from a three storey building and not crack. The only requirments are the egg must be visible in atleast one place and the smallest design wins. I tried making a crumple area and then protecting the egg with...

This is a problem that I thought I 'solved' many years ago. In actual fact there are many things about it that are not clear to me, and I would like to hear your opinion, please. Very briefly, there was this TV programme where a (supposedly psychic) guy had to match 5 (husband-wife) couples...

Nice little puzzle: http://www.bbc.co.uk/programmes/articles/5m5cv4NM5dWx108YgCQXj9J/can-you-crack-gchqs-code-breaker-challenge I didn't find it particularly difficult (less than a minute to spot how to do it, and a few minutes to actually work through it), although Google translate thinks the...

Given that $x,\,y$ and $z$ are non-zero real numbers such that $x + y + z = 3$ and $xy + yz + zx = −1$. Evaluate $$\frac{(x + 1)(y + 1)}{x + y}+ \frac{(y + 1)(z + 1)}{y + z}+ \frac{(z + 1)(x + 1)}{z + x}$$.

I took a test for a scholarship that had mathematical "riddles" just like Micromass' challenges. It was multiple choice and I guessed for some, but others I was able to do or at least use process of elimination. I didn't think I did that well, but I advanced to the second round. Could I have...

In 2012 actor Alan Alda started a competition in which scientists are asked to explain by whatever means a designated phenomenon or concept to 11 year olds. The explanations are judged by students whose schools are participating in the competition worldwide some 26,000 students so far. This...

Prove the inequality: $\sqrt{a(1-b)(1-c)}+\sqrt{b(1-a)(1-c)}+\sqrt{c(1-a)(1-b)} \le 1 + \sqrt{abc}, \;\;\;\;a,b,c \in [0;1].$

Time for the october challenge! This time a lot of people sent me suggestions for challenges. I wish to thank them a lot! If you think of a good challenge that could be included here, don't hesitate to send me! Ranking [and previous challenges] here...

Calculate the definite integral\[ \int_{2}^{7} \frac{x}{1-\sqrt{2+x}}\,dx \]- without the use of an integral calculator

Solve the indefinite integral \[\int \frac{dx}{\cos x+\sin x}\] - in three different ways.

September, schools restart, summer ends, but a new challenge is here: Ranking here: https://www.physicsforums.com/threads/micromass-big-challenge-ranking.879070/ 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...

Prove, that \[3(x^2y^2+x^2z^2+y^2z^2)-2xyz(x+y+z) \leq 3,\: \: \: \forall x,y,z \in \left [ 0;1 \right ]\]

Derive an expression for the definite integral:\[I = \int_{0}^{\frac{\pi}{4}}sec^m(x)dx, \;\;\;\;m = 2,4,6,...\]

August is already well underway, so time for some nice challenges! This thread contains both challenges for high schoolers and college freshmen, and for more advanced people. Also some previously unsolved challenges are omitted. Ranking here...

Given that $$a,\,b\in\Bbb{N}$$ such that $$\sqrt{a^2+2b+1}+\sqrt[3]{b^3+3a^2+3a+1}$$ is a rational number. Find the relationship between $a$ and $b$.

hi gang this came up in one of those crazy facebook challenges and you can imagine the arguments that ensued My maths knowledge isn't brilliant ... so in the above, if there are no brackets, do you still do the multiplication first or do you just work your way through from end to end ? I...

$x,y,z,w>0$ prove: $(1+x)(1+y)(1+z)(1+w)\geq (\sqrt[3]{1+xyz}\,\,\,)(\sqrt[3]{1+yzw}\,\,\,)(\sqrt[3]{1+zwx}\,\,\,)(\sqrt[3]{1+wxy}\,\,\,)$

Let $a,\,b$ and $c$ be non-negative real numbers such that $a+b+c=1$. Prove that $$\sum_{cyclic}\sqrt{4a+1} \ge \sqrt{5}+2$$.

Homework Statement You have been employed but(sic) the Mathematics Football League (MFL) to design a football. Using the volume of revolution technique, your football design must have a capacity of 5L ± 100mL. You must present a statement considering the brief below. Just a quick side note, I...