Challenge Definition and 911 Threads

The answer is 441N instead of 883N, but why? can anyone help?

As a Bedouin schiek you own twenty-five horses. You want to find the three fastest. You have no clock or other device that measures time. Your racing field is wide enough that five horses can race unimpeded, so you can race five at a time and see how they place. You don't want to abuse your...

TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift? Say you were given the equation : How would you find : with a calculator that can only add, subtract, multiply, divide Is there a general formula?

Hello All, Not sure if this belongs in general math but lets start here and see where it takes us. In mathematics, a combination is a way of selecting items from a collection where the order of selection does not matter. As an example , say we have digits 1 to 10. And we want to select 3...

I was able to calculate a), and got 0.7mm But I have no idea where to even start with b)

Chatgpt is actually pretty good at generating math problems. It's awful at solving them. I guarantee every question posted here cannot be solved by chatgpt, but maybe can be solved by a human? My plan is to spend a couple minutes getting a question I think it's cool and then posting it here -...

Deadline April 18, 2022 Competition Eligibility at: https://www.drivendata.org/competitions/93/nasa-mars-spectrometry/rules/ HeroX Challenge page: https://www.herox.com/MarsSpectrometry Brief article: https://www.space.com/nasa-mars-habitable-herox-competition Good Luck, and have Fun! Tom

OK, I'm stuck on a problem implementing this map. (It'll be built in HTML with JavaScript and CSS and is interactive but that's just context - this is really about map-colouring.) The function of the map is to help users see at-a-glance what regions/counties of the province are serviced by...

Fun question on mathoverflow: How would you have answered Richard Feynman's challenge?

This month's challenges will be my last thread of this kind for a while. Call it a creative break. Therefore, we will have a different format this month. I will post one problem a day, like an advent calendar, only for the entire month. I will try to post the questions as close as possible to...

Summary: Analysis. Projective Geometry. ##C^*##-algebras. Group Theory. Markov Processes. Manifolds. Topology. Galois Theory. Linear Algebra. Commutative Algebra.1.a. (solved by @nuuskur ) Let ##C\subseteq \mathbb{R}^n## be compact and ##f\, : \,C\longrightarrow \mathbb{R}^n## continuous and...

Summary: Functional Analysis. Project Management. Set Theory. Group Theory. Lie Theory. Countability. Banach Algebra. Stochastic. Function Theory. Calculus.1. Prove that ##F\, : \,L^2([0,1])\longrightarrow (C([0,1]),\|.\|_\infty )## defined as $$F(x)(t):=\int_0^1 (t^2+s^2)(x(s))^2\,ds$$ is...

Hi! I can't solve this. Please someone give me a hint and help? I'm unsure what equation to use. Thank you!🙏🙏🙏 An artillery crew demonstrates its skill by firing a shell at an angle of 49 deg and then lowering the gun barrel and firing a second shell at a smaller angle of 20 deg in such a way...

Summary: Gamma function. Combinatorics. Stochastic. Semisimple Modules. Topological Groups. Metric spaces. Logarithmic inequality. Stochastic. Primes. Approximation theory.1. (solved by @julian and @benorin ) Let ##f## be a function defined on ##(0,\infty)## such that ##f(x)>0## for all ##x>0.##...

To get you started I will derive the Lorentz force law from the QED Lagrangian \begin{equation}\mathcal{L} = \frac{i}{2} \bar{\psi}\gamma^{\mu}D_{\mu}\psi + h.c - \frac{1}{16\pi}F_{\mu\nu}F^{\mu\nu} ,\end{equation}D_{\mu} = \partial_{\mu} + ieA_{\mu}, and then, I let you do the same to a SM-like...

Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus1. Let ##(X,\rho)## be a metric space, and suppose that there exists a sequence ##(f_i)_i## of real-valued continuous functions on ##X## with the property that a...

The phenomenon of diffusion is a transport phenomenon based on the thermal motion of molecules, a process in which molecules are transported from a region of high concentration to a region of low concentration by Brownian motion. Let's assume that there is a car, the road under the wheels is...

Hello MHB, I saw one question that really tickles my intellectual fancy and because of the limited spare time that I have, I could not say I have solved it already! But, I will most definitely give the question more thought and will post back if I find a good solution to it. Here goes the...

Summary:: Group Theory, Lie Algebras, Number Theory, Manifolds, Calculus, Topology, Differential Equations. 1. (solved by @Infrared ) Suppose that ##G## is a finite group such that for each subgroup ##H## of ##G## there exists a homomorphism ##\varphi \,:\, G \longrightarrow H## such that...

Hi all I've long been a fan of the nrich site for maths and in recent years it has started to add a section on physics here. I also like IsaacPhysics although I haven't used it much in the past year so am still trying to get used to the new layout. I'm looking for other resources along this...

Trying this out for fun, and seeing if people find this stimulating or not. Feedback appreciated! There's only 3 problems, but I hope you'll get a kick out of them. Have fun!1. Springey Thingies: Two damped, unforced springs are weakly coupled and obey the following equations of motion...

Summary: Lie algebras, Hölder continuity, gases, permutation groups, coding theory, fractals, harmonic numbers, stochastic, number theory. 1. Let ##\mathcal{D}_N:=\left\{x^n \dfrac{d}{dx},|\,\mathbb{Z}\ni n\geq N\right\}## be a set of linear operators on smooth real functions. For which values...

Just for fun! :smile: Feel free to have a go at any of the problems. Problem 1 A spherical ball of radius ##a## and centre ##C## rolls on the rough outer surface of a fixed sphere of radius ##b## and centre ##O##. Show that the radial spin ##\boldsymbol{\omega} \cdot \mathbf{c}## is conserved...

The sequence $\{a_n\}$ and $\{b_n\}$ are such that, for every positive integer $n$, $a_n>0,\,b_n>0,\,a_{n+1}=a_n+\dfrac{1}{b_n}$ and $b_{n+1}=b_n+\dfrac{1}{a_n}$. Prove that $a_{50}+b_{50}>20$.

Summary: Group Theory, Integrals, Representation Theory, Iterations, Geometry, Abstract Algebra, Linear Algebra.1. Integrate $$ \int_{0}^\infty \int_{0}^\infty e^{-\left(x+y+\frac{\lambda^3 }{xy}\right)} x^{-\frac{2}{3}}y^{-\frac{1}{3}}\,dx\,dy $$ 2. (solved by @Infrared , basic solution still...

Prove $\sin^2(x+a)+\sin^2(x+b)-2\cos (a-b)\sin (x+a)\sin (x+b)$ is a constant function of $x$.

Summary: Differential Equations, Linear Algebra, Topology, Algebraic Geometry, Number Theory, Functional Analysis, Integrals, Hilbert Spaces, Algebraic Topology, Calculus.1. (solved by @etotheipi ) Let ##T## be a planet's orbital period, ##a## the length of the semi-major axis of its orbit. Then...

Let $x$ be a real number such that $\dfrac{\sin^4 x}{20}+\dfrac{\cos^4 x}{21}=\dfrac{1}{41}$. If the value of $\dfrac{\sin^6 x}{20^3}+\dfrac{\cos^6 x}{21^3}$ can be expressed as $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find the remainder when $m+n$ is divided by 1000.

Summary: Lie Algebras, Commutative Algebra, Ordering, Differential Geometry, Algebraic Geometry, Gamma Function, Calculus, Analytic Geometry, Functional Analysis, Units. 1. Prove that all derivations ##D:=\operatorname{Der}(L)## of a semisimple Lie algebra ##L## are inner derivations...

Summary: Calculus, Measure Theory, Convergence, Infinite Series, Topology, Functional Analysis, Real Numbers, Algebras, Complex Analysis, Group Theory1. (solved by @Office_Shredder ) Let ##f## be a real, differentiable function such that there is no ##x\in \mathbb{R}## with ##f(x)=0=f'(x)##...

Summary: Linear Programming, Trigonometry, Calculus, PDE, Differential Matrix Equation, Function Theory, Linear Algebra, Irrationality, Group Theory, Ring Theory.1. (solved by @suremarc , 1 other solutions possible) Let ##A\in \mathbb{M}_{m,n}(\mathbb{R})## and ##b\in \mathbb{R}^m##. Then...

Do you consider writing a program that takes 3 integer inputs and orders them in ascending order (accounting for same numbers) difficult? You can only use if statements (if, else if, else). I know there is some thing called "sort" that does the tedious work for you but do you find this simple...

Let $x_1,\,x_2,\,\cdots,\,x_{2014}$ be the roots of the equation $x^{2014}+x^{2013}+\cdots+x+1=0$. Evaluate $\displaystyle \sum_{k=1}^{2014} \dfrac{1}{1-x_k}$.

Hi, my question is that what is the main challenge of high energy physics? what is the best theory that maybe explain it and why it would not be accepted?

Summary: Circulation, Number Theory, Differential Geometry, Functional Equation, Group Theory, Infinite Series, Algorithmic Precision, Function Theory, Coin Flips, Combinatorics.1. (solved by @etotheipi ) Given a vector field $$ F\, : \,\mathbb{R}^3 \longrightarrow \mathbb{R}^3\, ...

In a triangle $ABC$ with $\sqrt{2}\sin A-2\sin B+\sin C=0$, prove that $\dfrac{3}{\sin A}+\dfrac{\sqrt{2}}{\sin C}\ge 2(\sqrt{3}+1)$.

Let $x,\,y,\,z$ be integers such that $(x-y)^2+(y-z)^2+(z-x)^2=xyz$, prove that $x^3+y^3+z^3$ is divisible by $x+y+z+6$.

Summary: Diffusion Equation, Sequence Space, Banach Space, Linear Algebra, Quadratic Forms, Population Distribution, Sylow Subgroups, Lotka-Volterra, Ring Theory, Field Extension. 1. Let ##u(x,t)## satisfy the one dimensional diffusion equation ##u_t=Du_{xx}## in a space-time rectangle...

So I am working on a project where I have a tank, which has a volume of electrolyte liquid inside it. This is coupled to a battery which charges it, and gives it energy. I will have a copperband arround it, so i can measure a potential voltage from the electrical field. So what I need to...

Let $a,\,b$ and $c$ be real numbers such that $\sin a+\sin b+\sin c\ge \dfrac{3}{2}$. Prove that $\sin \left(a-\dfrac{\pi}{6}\right)+\sin \left(b-\dfrac{\pi}{6}\right)+\sin \left(c-\dfrac{\pi}{6}\right)\ge 0$.

Prove that $\sqrt[n]{1+\dfrac{\sqrt[n]{n}}{n}}+\sqrt[n]{1-\dfrac{\sqrt[n]{n}}{n}}<2$ for any positive integer $n>1$.

Summary:: Functional Analysis, Algebras, Measure Theory, Differential Geometry, Calculus, Optimization, Algorithm, Integration. Lie Algebras. 1. (solved by @julian ) Let ##(a_n)\subseteq\mathbb{R}## be a sequence of real numbers such that ##a_n \leq n^{-3}## for all ##n\in \mathbb{N}.## Given...

Summary: group theory, number theory, commutative algebra, topology, calculus, linear algebra Remark: new solution manual (01/20-06/20) attached https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/ 1. Given a group ##G## then the intersection of all maximal...

Let $x$ be an irrational number. Show that there are integers $m$ and $n$ such that $\dfrac{1}{2555}<mx+n<\dfrac{1}{2012}$.

Prove that the polynomial equation $x^8-x^7+x^2-x+15=0$ has no real solution.

Evaluate $\displaystyle \sum_{n=0}^\infty \dfrac{16n^2+20n+7}{(4n+2)!}$.

Find all continuous functions $f:[1,\,8] \rightarrow \mathbb{R} $ such that $\displaystyle \int_1^2 f^2(t^3)dt + 2\int_1^2 f(t^3)dt=\dfrac{2}{3}\int_1^8 f(t)dt-\int_1^2 (t^2-1)^2 dt$

1. (solved by @nuuskur ) Let ##K## be a non-empty compact subset of ##\Bbb{C}##. Construct a bounded operator ##u: H \to H## on some Hilbert space ##H## that has spectrum ##\sigma(u) =K##. (MQ) 2. Let ##f,g:[0,2]\to\mathbb{R}## be continuous functions such that ##f(0)=g(0)=0## and...

The $\triangle ABC$ and $\triangle AEF$ are in the same plane. Between them, the following conditions hold: 1. The midpoint of $AB$ is $E$. 2. The points $A,\,G$ and $F$ are on the same line. 3. There is a point $C$ at which $BG$ and $EF$ intersect. 4. $CE=1$ and $AC=AE=FG$. Prove that if...

Let $n$ be a positive integer. Show that if $2+2\sqrt{28n^2+1}$ is an integer, then it is a square.