2017 Definition and 169 Threads

Please scroll-sown to Question 52: https://www.undergraduate.study.cam.ac.uk/files/publications/engineering_s1_qp_2017.pdf The correct answer is 'B'. This is the working I did: F = (change in momentum) / (change in time) change in momentum = mv - mu, where v = final velocity and u = initial...

Please scroll-down to the end (Question 54): https://www.undergraduate.study.cam.ac.uk/files/publications/engineering_s1_qp_2017.pdf I have also been referring to unofficial worked solutions (http://www.engineeringadmissionsassessment.com/2017-solutions.html), but I didn't understand how it...

Please scroll-down to Q50: https://www.undergraduate.study.cam.ac.uk/files/publications/engineering_s1_qp_2017.pdf The correct answer is 'B', or 'mgsin(Θ)'. I put 'E', or 'μmgcos(Θ)'. There are unofficial worked solutions which I have been referring to when I have attempted the question and...

I watched Ghost in the Shell 1995 the other night. It was cartoon. At first I was hesitant being just cartoon. But found it so intriguing, then I rewatched again the 2017 live version staring Johansson Scarlett. I noticed the 1995 cartoon version was more well-received (Tomato score 96%) than...

Hi All, Trying to do some querying in my (local) SQL Server. Thing is I am having trouble accessing my default instance, though no trouble accessing the one named instance I have. I have been to Stack Overflow , read up, posted a few things, then told to read again after it all failed. I get a...

I found this interesting article on theory work done to create qubit flipflops that can be adjusted via electric fields making them easier to integrate into existing computing systems...

Some new papers appeared about processes in the collision of the two neutron stars, measured in August 2017. Is now more clear, why 2 second delay between gamma rays and gravitational waves happens?

If I worked Dec. 2017 and didn't get paid until January 2018, is that taxable income for a 2017 tax filing? Anyone know? Thanks.

A reference article to other articles on science facts that were discovered this year 2017. Some were theorized and finally proven, others were discovered totally by chance. Which ones most impressed you? http://www.sciencealert.com/23-science-facts-we-didn-t-know-at-the-start-of-2017

Here is this week's POTW: ----- For any continuous real-valued function $f$ defined on the interval $[0,1]$, let \begin{gather*} \mu(f) = \int_0^1 f(x)\,dx, \, \mathrm{Var}(f) = \int_0^1 (f(x) - \mu(f))^2\,dx, \\ M(f) = \max_{0 \leq x \leq 1} \left| f(x) \right|. \end{gather*} Show that if $f$...

I go to the movies quite a bit. I've always loved the experience and find it's fun to go into a new world for 1-3 hours. Usually I have at least one movie that gets me super excited each year, but this year was a big flop unfortunately. These are the ones I saw accord to my purchase history. I...

Here is this week's POTW: ----- Prove that $(x^2+ y^2 + z^2)(x + y + z) + x^3+ y^3+ z^3> 4(xy + yz + zx)$ for all $x,\,y,\,z > 1$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Happy Holidays, everyone! Here is this week's POTW: ----- Consider the open unit disk $\Bbb D\subset \Bbb C$ with Riemannian metric $ds^2 = \dfrac{\lvert dz\rvert^2}{(1 - \lvert z\rvert^2)^2}$. Find a formula for the (Riemannian) distance between two points in $\Bbb D$, and use it to find the...

Here is this week's POTW: ----- Suppose $n \ge 0$ and all the roots of $x^3+ax+4-(2\times 2016^n)=0$ are integers. Find all possible values of $a$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out...

Here is this week's POTW: ----- What is the greatest common divisor of the set of numbers $\left\{16^n+10n-1 \;|\; n=1, 2, 3, \dots\right\}?$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- Minimize $$\frac{2x^3+1}{4y(x-y)}$$ given $$x\ge -\frac{1}{2}$$ and $$\frac{x}{y}>1.$$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to...

Here is this week's POTW: ----- Let $a_0 = \dfrac52$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute $\displaystyle\prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right)$ in closed form. ----- Remember to read the...

Here is this week's POTW: ----- If $\phi : A \to B$ is a local homeomorphism from a compact space $A$ to a connected Hausdorff space $B$, show that $\phi$ is surjective and the fibers of $\phi$ over the points of $B$ are finite sets. ----- Remember to read the...

Here is this week's POTW: ----- Find the minimum value of $\dfrac{1}{a-b}+\dfrac{1}{b-c}+\dfrac{1}{a-c}$ for all reals $a>b>c$, given $(a-b)(b-c)(a-c)=17$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to...

Here is this week's POTW: ----- A novel has 6 chapters. As usual, starting from the first page of the first chapter, the pages of the novel are numbered 1, 2, 3, 4, . . . . Also, each chapter begins on a new page. The last chapter is the longest and the page numbers of its pages add up to...

Some experts are saying the big eruption could happen tonight! :nb):frown: http://www.cnn.com/2017/11/27/asia/bali-volcano-agung-eruption/index.html Here is a live video feed

Here is this week's POTW: ----- Prove that there exists $x\in \Bbb{N}$, where $1\le x \le 89$ such that $\sqrt{3}\tan x^\circ-1=\sec 20^\circ$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...

Happy Thanksgiving, for those of you in the USA! Here is this week's POTW (not a Putnam this week!): ----- If $a_0\ge a_1 \ge a_2\ge \cdots\ge a_n\ge 0,$ prove that any root $r$ of the polynomial $$P(z)\equiv a_0 z^n+a_1 z^{n-1}+\cdots+a_n$$ satisfies $|r|\le 1$; i.e., all the roots lie...

Here is this week's POTW: ----- Suppose $\mu$ is a finite Borel measure on $\Bbb R^n$. Define the maximal function of $\mu$ by $$\mathcal{M}\mu(x) = \sup_{0 < r < \infty} \frac{\mu(B(x;r))}{m(B(x;r))}\quad (x\in \Bbb R^n)$$ Here, $m$ denotes the Lebesgue measure on $\Bbb R^n$. Show that if...

"How do you observe the invisible currents of the atmosphere? By studying the swirling, billowing loads of sand, sea salt and smoke that winds carry. A new simulation created by scientists at NASA’s Goddard Space Flight Center in Greenbelt, Md., reveals just how far around the globe such aerosol...

Hi MHB, The High School POTW should be number 289 this week, but due to the fact that I was a bit late carrying out my duty over several weeks, I fell behind a week, and so I will make it up by posting two POTWs this week. I sincerely apologize for this, and I hope our members can take up...

Here is this week's POTW: ----- Fill in the blank: If $a,\,b$ and $c$ are three distinct real numbers, then the quadratic expression $$\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}+\frac{(x-a)(x-b)}{(c-a)(c-b)}$$ is identically equal to _____. ----- Remember to read the...

Here is this week's POTW: ----- Prove that every nonzero coefficient of the Taylor series of \[ \left(1 - x + x^2\right) e^x \] about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number. ----- Remember to read the...

The classic book on General Relativity by Wheeler, Thorne and Misner is now out: https://www.amazon.com/gp/product/0691177791/?tag=pfamazon01-20 I got to use preprints of the book while doing an independent study of General Relativity in 1973. It brings back fond memories of being tortured by...

Here is this week's POTW: ----- Positive integers $a,\,b, \,c, \,d$, and $e$ satisfy the equations $(a + 1)(3bc + 1) = d + 3e + 1\\ (b + 1)(3ca + 1) = 3d + e + 13\\ (c + 1)(3ab + 1) = 4(26 − d − e) − 1$ Evaluate $d^2 + e^2$. ----- Remember to read the...

Here is this week's POTW: ----- Let $f$ be a polynomial with positive integer coefficients. Prove that if $n$ is a positive integer, then $f(n)$ divides $f(f(n)+1)$ if and only if $n=1$. Assume $f$ is non-constant. ----- Remember to read the...

Here is this week's POTW: ----- Without using a calculator, evaluate $\cos^3 \left(\dfrac{2\pi}{7}\right)+\cos^3 \left(\dfrac{4\pi}{7}\right)+\cos^3 \left(\dfrac{8\pi}{7}\right)$. ----- Remember to read the...

Here is this week's POTW: ----- Given commutative rings with unity $R$ and $S$, let $\phi : R \to S$ be a morphism of rings. It induces a morphism $\phi^* : \operatorname{Spec}(S) \to \operatorname{Spec}(R)$ of prime spectra such that $\phi^*(\mathfrak{q}) = \phi^{-1}(\mathfrak{q})$ for all...

Here is this week's POTW: ----- For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k}\,)$. Evaluate \[ \sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}. \] ----- Remember to read the...

Here is this week's POTW: ----- One of the roots of the quadratic equation $x^2-kx+2n=0$ is equals to $\dfrac{1}{\sqrt{1}}+\dfrac{1}{\sqrt{2}}+\cdots+\dfrac{1}{\sqrt{n}}$, where $n$ is a positive integer. Prove that $2\sqrt{2n} \le k \le 3\sqrt{n}$. ----- Remember to read the...

Here is this week's POTW: ----- Show that for each positive integer $n$, all the roots of the polynomial \[ \sum_{k=0}^n 2^{k(n-k)} x^k \] are real numbers. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to...

Hello, MHB Community! (Wave) anemone has asked me to fill in for her this week. :) Here is this week's POTW: Find all real numbers $k$ that give the three roots of the cubic equation $5x^3-5(k+1)x^2+(71k-1)x-(66k-1)=0$ are positive integers. Remember to read the...

Here is this week's POTW: ----- For any positive integer $n$, let $\langle n\rangle$ denote the closest integer to $\sqrt{n}$. Evaluate \[\sum_{n=1}^\infty \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}.\] ----- Remember to read the...

Here is this week's POTW: ----- Find the norm of the linear operator $T: \mathscr{L}^p(0, \infty) \to \mathscr{L}^p(0, \infty)$ defined by the equation $$(Tf)(x) = \frac{1}{x}\int_0^x f(t)\, dt$$ Here it is assumed that $1 < p < \infty$. Note: The space $\mathscr{L}^p(0,\infty)$ consists of...

Here is this week's POTW: ----- Consider the sequence $\{a_k\}_{k\ge 1}$ is defined by $a_1=1$, $a_2=\dfrac{1}{2}$ and $a_{k+2}=a_k+\dfrac{a_{k+1}}{2}+\dfrac{1}{4a_ka_{k+1}}$ for $k\ge 1.$ Prove that $\dfrac{1}{a_1a_3}+\dfrac{1}{a_2a_4}+\dfrac{1}{a_3a_5}+\cdots+\dfrac{1}{a_{98}a_{100}}<4$...

Did the 2017 eclipse prove Einstein was right or the jury is still out? I can't find any references to the new measurements and what they proved (or did not).

Here is this week's POTW: ----- Let $X$ be a topological group; let $A$ be a subgroup of $X$ such that $A$ and $X/A$ are connected. Show that $X$ is connected. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines...

another couple of M6 events from my seismograph M 6.3 - 36km SSW of Putre, Chile M 6.7 - Bouvet Island region on my seismogram below, the P arrival for the M6.3, Chile event is approx 0652UT the P arrival for the M6.7, Bouvet Is event is approx 1907UT cheers Dave

3 respectable events within 10 hours M 6.1 - 153km NW of Pangai, Tonga M 6.3 - Balleny Islands region M 6.6 - 60km E of Buldir Island, Alaska The first 2 events are on this map. The third event, the M 6.6 in the western Aleutian Islands, is the other end of the pacific from me. The largest...

Here is this week's POTW: ----- Let that $a,\, b,\,c$ be three angles with $0<a,\,b,\,c<90^\circ$ that satisfy $\sin a+\sin b+\sin c=1$. Prove that $\tan^2 a+\tan^2 b+\tan^2 c \ge \dfrac{3}{8}$ ----- Remember to read the...

Ackbach has asked me to step in for him for a while. Here is this week's POTW: ----- Evaluate the sum of the alternating series $$\sum\limits_{n = 1}^\infty \frac{(-1)^{n-1}}{n^4}$$ ----- Remember to read the...

Big surprise? awarded to Rainer Weiss, Barry Barish and Kip Thorne for their detection of gravitational waves. https://phys.org/news/2017-10-nobel-physics-prize-awarded-scientists.html

The 2017 Nobel Prize in Chemistry is being announced tomorrow! I really can't wait to see what it will be awarded for, because it is always interesting and sometimes unexpected. I think it might be awarded for works on nanotechnology, because it is currently a hot field. It might be displayed...

This morning, the 2017 Nobel Prize in Physiology or Medicine was awarded to Jeffrey C. Hall, Michael Rosbash and Michael W. Young for their discoveries of molecular mechanisms controlling the circadian rhythm. https://www.nobelprize.org/nobel_prizes/medicine/laureates/2017/press.html

Can't get to some papers. No response at all from the site. Most other sites are working for me. Anyone else have insight?