What is Math challenge: Definition and 82 Discussions

MathWorks Math Modeling (M3) Challenge is a mathematical modeling competition open to high school students across the United States (including US territories and DoDEA schools). It is sponsored by MathWorks (a developer of mathematical computing software) based in Boston and organized by the Society for Industrial and Applied Mathematics (SIAM) based in Philadelphia.The M3 Challenge awards $100,000 in scholarship prizes each year to the top teams. An additional incentive is the recognition that the winning teams receive. The winning paper from 2008 was published in the College Mathematics Journal. A representative from High Tech's team appeared on FOX Business Channel, 2010 winners were interviewed by Pimm Fox of Bloomberg radio, presented its findings at Lockheed Martin's Data Capture Center, met with U.S. Census Bureau Director Dr. Robert Groves, and had their research published in SIAM's undergraduate publication, SIAM Undergraduate Research Online (SIURO). The 2011 and 2012 winners were interviewed by Pimm Fox of Bloomberg radio, and the 2014 winners were interviewed by both Pimm Fox and Carol Massar on Bloomberg radio.MathWorks took over sponsorship of the competition, formerly known as the Moody's Mega Math (M³) Challenge, from Moody's Foundation in 2017.

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  1. TomVassos

    I Calculating Statistical Likelihood Knowing Standard Deviations

    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...
  2. TomVassos

    I Calculating the End of the Universe Using Standard Deviation Statistics

    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...
  3. L

    I How is it possible to win one-in-infinity odds?

    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...
  4. Infrared

    Challenge Math Challenge - July 2023

    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...
  5. M

    A What are your insights on The Hardest Logic Puzzle Ever?

    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...
  6. Infrared

    Challenge Math Challenge - June 2023

    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...
  7. PeaceMartian

    How to find integrals of parent functions without any horizontal/vertical shift?

    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?
  8. M

    Engineering Calculating a) and Baffled by b): A Math Challenge

    I was able to calculate a), and got 0.7mm But I have no idea where to even start with b)
  9. fresh_42

    Challenge Math Challenge - December 2021

    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...
  10. fresh_42

    Challenge Math Challenge - November 2021

    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...
  11. fresh_42

    Challenge Math Challenge - October 2021

    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...
  12. fresh_42

    Challenge Math Challenge - September 2021

    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.##...
  13. fresh_42

    Challenge Math Challenge - August 2021

    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...
  14. fresh_42

    Challenge Math Challenge - July 2021

    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...
  15. fresh_42

    Challenge Math Challenge - June 2021

    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...
  16. fresh_42

    Challenge Math Challenge - May 2021

    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...
  17. fresh_42

    Challenge Math Challenge - April 2021

    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...
  18. fresh_42

    Challenge Math Challenge - March 2021

    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...
  19. fresh_42

    Challenge Math Challenge - February 2021

    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)##...
  20. fresh_42

    Challenge Math Challenge - January 2021

    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...
  21. fresh_42

    Challenge Math Challenge - December 2020

    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\, ...
  22. fresh_42

    Challenge Math Challenge - November 2020

    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...
  23. fresh_42

    Challenge Math Challenge - October 2020

    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...
  24. fresh_42

    Challenge Math Challenge - September 2020

    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...
  25. fresh_42

    Challenge Math Challenge - August 2020

    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...
  26. fresh_42

    Challenge Math Challenge - July 2020

    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...
  27. fresh_42

    Challenge Math Challenge - June 2020

    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...
  28. fresh_42

    Challenge Math Challenge - May 2020

    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)##...
  29. fresh_42

    Challenge Math Challenge - April 2020

    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...
  30. fresh_42

    Challenge Math Challenge - March 2020 (Part II)

    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...
  31. fresh_42

    Challenge Math Challenge - March 2020

    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...
  32. fresh_42

    Challenge Math Challenge - February 2020

    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...
  33. fresh_42

    Challenge Math Challenge - January 2020

    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}...
  34. fresh_42

    Challenge Math Challenge - December 2019

    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...
  35. fresh_42

    Challenge Math Challenge - November 2019

    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...
  36. fresh_42

    Challenge Math Challenge - October 2019

    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...
  37. fresh_42

    Challenge Math Challenge - September 2019

    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...
  38. fresh_42

    Challenge Math Challenge - August 2019

    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...
  39. fresh_42

    Challenge Math Challenge - July 2019

    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...
  40. fresh_42

    Challenge Math Challenge - June 2019

    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...
  41. fresh_42

    Challenge Math Challenge - May 2019

    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...
  42. fresh_42

    Challenge Math Challenge - April 2019

    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...
  43. fresh_42

    Challenge Math Challenge - March 2019

    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...
  44. QuantumQuest

    Challenge Math Challenge - February 2019

    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...
  45. fresh_42

    Challenge Math Challenge - January 2019

    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...
  46. mfb

    B Daily math challenge calendar (external)

    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.
  47. fresh_42

    Challenge Math Challenge - December 2018

    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...
  48. fresh_42

    Challenge Math Challenge - November 2018

    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...
  49. fresh_42

    Challenge Math Challenge - October 2018

    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...
  50. fresh_42

    Challenge Intermediate Math Challenge - September 2018

    Summer is coming and brings a new basic math challenge! Enjoy! For more advanced problems you can check our other basic level math challenge thread! 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...
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