Math challenge Definition and 83 Threads

TL;DR Summary: Isolating coefficient contribution to outcome change This is a problem I've been trying to solve for almost 3 days now, without any solutions. Say we have x + y + z = 5, and x is 1, y is 1, and z is 3. We have a second equation = 2x + 3y + 4z = 17. Using the outcome difference...

Let’s say that we know, with 95% confidence, that something is likely to occur when the Universe is between 1058 and 10549 years old. What is the statistical likelihood that it has already occurred in the first 13.8 billion years of the Universe’s existence? (1.38 X 1010 years) I know the...

One possible end to the Universe is called vacuum decay, where a Higgs boson could transition from a false vacuum to a true vacuum state. This would create a vacuum decay bubble (known as bubble nucleation) that would expand at light speed, destroying everything in its path. According to Anders...

Imagine a roulette wheel with an infinite amount of numbers. Every number on the wheel has a one-in-infinity chance of being selected. Every time the wheel is spun, one number wins those one-in-infinity odds. How is this possible? Isn't one-in-infinity basically zero? It's infinitely far from...

Welcome to this month's math challenge thread! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by @AndreasC) I start watching a...

Hello! I’m an assistant of a mathematical scientific researcher, and my research programme evolves around finding and developing all the (possible) solutions regarding all unsolved mathematical, logic, exact, and IQ puzzles ever created. If you search on the internet for: “The hardest unsolved...

Welcome to the reinstatement of the monthly math challenge threads! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by...

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?

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

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...

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.##...

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...

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...

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...

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...

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...

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...

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\, ...

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...

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...

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...

1. (solved by @nuuskur ) Let ##V## be an infinite dimensional topological vector space. Show that the weak topology on ##V## is not induced by a norm. (MQ) 2. The matrix groups ##U(n)## and ##SL_n(\mathbb{C})## are submanifolds of ##\mathbb{C}^{n^2}=\mathbb{R}^{2n^2}##. Do they intersect...

Questions 1. (solved by @nuuskur ) Let ##H_1, H_2## be Hilbert spaces and ##T: H_1 \to H_2## a linear map. Suppose that there is a linear map ##S: H_2 \to H_1## such that for all ##x\in H_2## and all ##y \in H_1## we have $$\langle Sx,y \rangle = \langle x, Ty \rangle$$ Show that ##T## is...

Questions 1. (solved by @benorin ) Let ##1<p<4## and ##f\in L^p((1,\infty))## with the Lebesgue measure ##\lambda##. We define ##g\, : \,(1,\infty)\longrightarrow \mathbb{R}## by $$ g(x)=\dfrac{1}{x}\int_x^{10x}\dfrac{f(t)}{t^{1/4}}\,d\lambda(t). $$ Show that there exists a constant ##C=C(p)##...

Questions 1. (solved by @nuuskur ) Let ##U\subseteq X## be a dense subset of a normed vector space, ##Y## a Banach space and ##A\in L(U,Y)## a linear, bounded operator. Show that there is a unique continuation ##\tilde{A}\in L(X,Y)## with ##\left.\tilde{A}\right|_U = A## and...

Questions 1. (solved by @hilbert2 ) Let ##\sum_{k=1}^\infty a_k## be a given convergent series with ##|a_{k+1}|\leq |a_k|## for all ##k##. Assume we use a computer to sum its value until the partial sum is closer than ##\varepsilon## to the actual value of the series. Does it make sense to use...

Questions 1. (solved by @Antarres, @Not anonymous ) Prove the inequality ##\cos(\theta)^p\leq\cos(p\theta)## for ##0\leq\theta\leq\pi/2## and ##0<p<1##. (IR) 2. (solved by @suremarc ) Let ##F:\mathbb{R}^n\to\mathbb{R}^n## be a continuous function such that ##||F(x)-F(y)||\geq ||x-y||## for all...

Questions 1. (solved by @archaic ) Determine ##\lim_{n\to \infty}\cos\left(t/\sqrt{n}\right)^n## for ##t\in \mathbb{R}##. 2. (solved by @Antarres ) Let ##a_0,\ldots,a_n## be distinct real numbers. Show that for any ##b_0,\ldots,b_n\in\mathbb{R}##, there exists a unique polynomial ##p## of...

Questions 1. (solved by @PeroK ) Let ##f\, : \,\mathbb{R}\longrightarrow \mathbb{R}## be a smooth, ##2\pi-##periodic function with square integrable derivative, and ##\displaystyle{\int_0^{2\pi}}f(x)\,dx = 0\,.## Prove $$ \int_0^{2\pi} \left[f(x)\right]^2\,dx \leq \int_0^{2\pi}...

Questions 1. Let ##(X,d)## be a metric space. The open ball with center ##z\in X## of radius ##r > 0## is defined as $$ B_r(z) :=\{\,x\in X\,|\,d(x,z)<r\,\} $$ a.) Give an example for $$ \overline{B_r(z)} \neq K_r(z) :=\{\,x\in X\,|\,d(x,z)\leq r\,\} $$ Does at least one of the inclusions...

Questions 1. (solved by @tnich ) Show that ##\sin\dfrac{\pi}{m} \sin\dfrac{2\pi}{m}\sin\dfrac{3\pi}{m}\cdots \sin\dfrac{(m - 1)\pi}{m} = \dfrac{m}{2^{m - 1}}## for ##m## = ##2, 3, \dots##(@QuantumQuest) 2. (solved by @PeroK ) Show that when a quantity grows or decays exponentially, the rate of...

Questions 1. (solved by @MathematicalPhysicist ) Show that the difference of the square roots of two consecutive natural numbers which are greater than ##k^2##, is less than ##\dfrac{1}{2k}##, ##k \in \mathbb{N} - \{0\}##. (@QuantumQuest ) 2. (solved by @tnich ) Let A, B, C and D be four...

Questions 1. Consider the ring ##R= C([0,1], \mathbb{R})## of continuous functions ##[0,1]\to \mathbb{R}## with pointwise addition and multiplication of functions. Set ##M_c:=\{f \in R\mid f(c)=0\}##. (a) (solved by @mathwonk ) Show that the map ##c \mapsto M_c, c \in [0,1]## is a bijection...

Questions 1. (solved by @Pi-is-3 )The maximum value of ##f## with ##f(x) = x^a e^{2a - x}## is minimal for which values of positive numbers ##a## ? 2. (solved by @KnotTheorist ) Find the equation of a curve such that ##y''## is always ##2## and the slope of the tangent line is ##10## at the...

Questions 1. (solved by @Flatlanderr , solved by @lriuui0x0 ) Show that ##\frac{\pi}{4} + \frac{3}{25} \lt \arctan \frac{4}{3} \lt \frac{\pi}{4} + \frac{1}{6}## 2. (solved by @nuuskur ) Show that the equation ##x + x^3 + x^5 + x^7 = {c_1}^2 (c_1 - x) + {c_2}^2 (c_2 - x)## where ##c_1, c_2 \in...

We have a prize this month donated by one of our most valued members, and that's what the points are for. The first who achieves 6 points, will win a Gold Membership. Questions 1. Let ##\mathfrak{g}## be a Lie algebra. Define $$ \mathfrak{A(g)} = \{\,\alpha\, : \,\mathfrak{g}\longrightarrow...

Questions 1. a. Let ##(\mathfrak{su}(2,\mathbb{C}),\varphi,V)## be a finite dimensional representation of the Lie algebra ##\mathfrak{g}=\mathfrak{su}(2,\mathbb{C})##. Calculate ##H\,^0(\mathfrak{g},\varphi)## and ##H\,^1(\mathfrak{g},\varphi)## for the Chevalley-Eilenberg complex in the cases...

As (almost) always: have a look on previous challenge threads, too. E.g. in https://www.physicsforums.com/threads/math-challenge-march-2019.967174/ are still problems to solve, and some of them easy, which I find, and in any case useful to know or at least useful to have seen. As a general...

Questions 1.) (disclosed by @Demystifier ) Using the notion of double integrals prove that $$B(m,n) = \frac{\Gamma (m) \Gamma (n)}{\Gamma (m + n)}\; \;(m \gt 0\,,\, n\gt 0)$$ where ##B## and ##\Gamma## are the Beta and Gamma functions respectively. 2.) (solved by @Math_QED ) Show that the...

Time for our new winter challenge! This time our challenge has also two Computer Science related questions and a separate section with five High School math problems. Enjoy! Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be...

Merry Christmas to all who celebrate it today! Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month. b) It is fine to use nontrivial results without proof as long...

I found this calendar with daily math puzzles. Based on the first three puzzles it seems to be much easier than the math challenges here, and require no advanced mathematics. The answer is always a three-digit number, and the answers to all 24 puzzles together create a larger puzzle.

It's December and we like to do a Special this month. The challenges will be posted like an Advent Calendar. We will add a new problem each day, from 12/1 to 12/25. They vary between relatively easy logical and numerical problems, calculations, to little proofs which hopefully add some...

Rules: a) In order for a solution to count, a full derivation or proof must be given. Answers with no proof will be ignored. Solutions will be posted around 15th of the following month. b) It is fine to use nontrivial results without proof as long as you cite them and as long as it is "common...

Summer is coming and brings ... Oops, time for a change! Fall (Spring) is here and what's better than to solve some tricky problems on a long dark evening (with the power of returning vitality all around). RULES: a) In order for a solution to count, a full derivation or proof must be given...