2017 Definition and 169 Threads

2017 (MMXVII) was a common year starting on Sunday of the Gregorian calendar, the 2017th year of the Common Era (CE) and Anno Domini (AD) designations, the 17th year of the 3rd millennium, the 17th year of the 21st century, and the 8th year of the 2010s decade.
2017 was designated as International Year of Sustainable Tourism for Development by the United Nations General Assembly.

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  1. anemone

    MHB What is the value of $\gamma-\alpha$ for a given $\cos$ equation?

    Here is this week's POTW: ----- Assume that $\alpha,\, \beta,\,\gamma$ satisfy $0<\alpha<\beta<\gamma<2\pi$. If $\cos (x+\alpha)+\cos(x+\beta)+\cos(x+\gamma)=0$ for arbitrary $x\in \Bbb{R}$, evaluate $\gamma-\alpha$. ----- Remember to read the...
  2. Euge

    MHB Is Contour Integration the Key to Solving This Integral?

    Here is this week's POTW: ----- Using contour integration, prove $$\int_0^\infty \sin(x^\alpha)\, dx = \sin\left(\frac{\pi}{2\alpha}\right)\,\Gamma\!\left(1 + \frac{1}{\alpha}\right),\quad \alpha > 1.$$----- Remember to read the...
  3. Ackbach

    MHB Evaluate the integral $\displaystyle \int_0^{\infty}\frac{dx}{(1+x^2)^{\alpha/2}}$ for $\alpha>1$

    Here is this week's POTW: ----- Evaluate the integral $\displaystyle \int_0^{\infty}\frac{dx}{(1+x^2)^{\alpha/2}}$ for $\alpha>1$. Express your answer using Gamma functions, where $$\Gamma(x) :=\int_{0}^{\infty}t^{x-1}e^{-t} \, dt.$$ ----- Remember to read the...
  4. anemone

    MHB Prove AB CD + AD BC < AC(AB+AD) for Convex Quadrilateral ABCD

    Here is this week's POTW: ----- The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^\circ$. Prove that $AB\cdot CD+AD \cdot BC <AC(AB+AD)$. ----- Remember to read the...
  5. P

    Are there cert courses/MOOCs about Electromagnetism in 2017?

    Hello everyone, I'm currently doing a CSE bachelor (europe) in which I'll focus on physics. Now, I'd like to also put in some extra work to strengthen my focus on physics. This will put me ahead of others but it might also give me the chance of doing a masters in physics. Anyway, currently...
  6. Greg Bernhardt

    Art Can Equations Be Aesthetic Art?

    The goal is to create the most beautiful or interesting equation aesthetically (pleasing to the eye). This is not about it's mathematical significance. Get your inner designer on! Each member is allowed to post one equation The equation can be completely made up Must use LaTeX Be creative...
  7. Ackbach

    MHB What is the solution to the ODE with coefficients involving x and y?

    Here is this week's POTW (I will identify the problem source when I post the solution): ----- Solve the ODE $(y^3+xy^2+y) \, dx + (x^3+x^2y+x) \, dy=0$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  8. ISamson

    2017 Nobel Prize in Physics - Opinions and Expectations

    Hello, The Nobel Prize in Physics is upcoming soon and I wanted to hear your opinions and expectations about some important discoveries or inventions which might end up receiving this most major prize in science. I look forward to hearing your thoughts! Thanks.
  9. ISamson

    2017 Nobel Prize in Chemistry - Expectations and Opinions

    Hello, I have heard that the 2017 Nobel Prize in Chemistry is going to be announced in the next few months. Do you know about any interesting and/or useful discoveries that chemistry achieved worth the Nobel Prize in your opinion? Thanks.
  10. anemone

    MHB What Are the Possible Values of (ab+cd)/(ad+bc)?

    Here is this week's POTW: ----- Given positive real numbers $a,\,b,\,c,\,d$ satisfy the equalities $a^2-ad+d^2=b^2+bc+c^2$ and $a^2+b^2=c^2+d^2$, find all possible values of the expression $\dfrac{ab+cd}{ad+bc}$. ----- Remember to read the...
  11. Ackbach

    MHB How can we show that a continuous function satisfies a specific inequality?

    Here is this week's POTW: ----- Let $f : [a,\infty)\to \Bbb R$ be a continuous function that satisfies the inequality $\displaystyle f(x) \le A + B\int_a^x f(t)\, dt$, where $A$ and $B$ are constants with $B < 0$. If $\displaystyle \int_a^\infty f(x)\, dx$ exists, show that $\displaystyle...
  12. BillTre

    Why are the 2017 Lasker Prizes causing controversy?

    The Lasker Prizes (regarded as the US’s most prestigious biomedical research awards) were announced (Science article) (The Lasker Foundation's Awards Announcement) and are interesting to me for two reasons: 1) Michael Hall, 64, of the University of Basel’s Biozentrum in Switzerland...
  13. Euge

    MHB Can Even Maps Between n-Spheres Have Odd Homological Degrees?

    Here is this week's POTW: ----- Prove that if $n > 0$, an even map between $n$-spheres has even homological degree.----- 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...
  14. Ackbach

    MHB Is the commutator of two complex matrices nilpotent?

    Here is this week's POTW: ----- If the commutator, $Z$, of two complex $n\times n$ matrices commutes with one of those matrices, must $Z$ be nilpotent? ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  15. anemone

    MHB How Can You Paint 16 Seats in Consecutive Odd Red or Green Colors?

    Here is this week's POTW: ----- In how many ways can we paint 16 seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same color is always odd? ----- Remember to read the...
  16. davenn

    Stargazing Sunspots visible for 2 Sept 2017

    A quick shot of the sun this morning Nth lower left, East limb lower right 400mm telephoto = x2 teleconverter, 500th sec, f11
  17. Ackbach

    MHB Find all pairs of real numbers

    Here is this week's POTW: ----- Find all pairs of real numbers $(x,y)$ satisfying the system of equations \begin{align*} \frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4). \end{align*} ----- Remember to read the...
  18. anemone

    MHB How to Solve This Week's POTW Quadruple Equations?

    Here is this week's POTW: ----- Find all quadruples $(a,\,b,\,c,\,d)$ of real numbers that simultaneously satisfy the following equations: \begin{array}{rcr}ab^2+cd^2\hspace{-10px} & = & \hspace{-10px}-6 \\ a^2b+c^2d\hspace{-12px} & = & 0 \\ a^3+c^3\hspace{-10px} & = & 2 \\...
  19. Greg Bernhardt

    Courses 2017 Students: List your Semester 1 courses

    What courses are you taking first semester?
  20. Ackbach

    MHB Problem of the Week # 277 - Aug 22, 2017

    Here is this week's POTW: ----- Let $n$ be an even positive integer. Write the numbers $1,2,\ldots,n^2$ in the squares of an $n\times n$ grid so that the $k$-th row, from left to right, is \[(k-1)n+1,(k-1)n+2,\ldots, (k-1)n+n.\] Color the squares of the grid so that half of the squares in...
  21. Chronos

    Stargazing Stunning 2017 Eclipse Photos of the Corona Chromosphere

    corona chromosphere
  22. anemone

    MHB Quadrilateral Circumscribed Circle Diagonals and Chords Concurrency Proof

    Here is this week's POTW: ----- If a quadrilateral is circumscribed about a circle, prove that its diagonals and the two chords joining the points of contact of opposite sides are all concurrent. ----- Remember to read the...
  23. Euge

    MHB Can You Solve the Harmonic Function Challenge?

    Here is this week's POTW: ----- Let $\Bbb D$ be the open unit disc in the complex plane, and let $f$ be a continuous complex function on $\partial\Bbb D$. Consider the function $$F(re^{i\phi}) \,\dot{=}\, \frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})\frac{1-r^2}{1-2r\cos(\theta-\phi) + r^2}\...
  24. Ackbach

    MHB Problem of the Week # 276 - Aug 15, 2017

    Here is this week's POTW: ----- Can an arc of a parabola inside a circle of radius 1 have a length greater than 4? ----- 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...
  25. anemone

    MHB Can You Prove This Challenging Inequality for x Between 1.5 and 5?

    Here is this week's POTW: ----- Suppose \frac{3}{2}\le x \le 5. Prove that 2\sqrt{x+1}+\sqrt{2x-3}+\sqrt{15-3x}<2\sqrt{19}. ----- 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...
  26. Ackbach

    MHB Problem of the Week # 275 - Aug 07, 2017

    Here is this week's POTW: ----- Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$. ----- 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...
  27. anemone

    MHB How Do You Calculate the Integral of f(x) from 0 to e?

    Here is this week's POTW: ----- Let $f$ satisfy $x=f(x)e^{f(x)}$. Calculate \int_{0}^{e} f(x)\,dx. ----- 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...
  28. Euge

    MHB Is the Tensor Algebra of $\Bbb Z/n\Bbb Z$ Isomorphic to $\Bbb Z[x]/(nx)$?

    Here is this week's POTW: ----- Show that the tensor algebra of $\Bbb Z/n\Bbb Z$ is isomorphic to $\Bbb Z[x]/(nx)$. ----- 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...
  29. Ackbach

    MHB Problem of the Week # 274 - Aug 01, 2017

    Here is this week's POTW: ----- Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$. ----- Remember to read...
  30. anemone

    MHB How Can You Minimize |x|-|y| Given Logarithmic Constraints?

    Here is this week's POTW: ----- Minimize $|x|-|y|$, given \log_4{(x+2y)}+\log_4{(x-2y)}=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 http://www.mathhelpboards.com/forms.php?do=form&fid=2!
  31. MarkFL

    MHB Solar Eclipse 2017 - Share Your Viewing Plans & Memories

    Hello MHB Community! :D As I'm sure most of you who live in the U.S. are aware, we will be witness to a solar eclipse on the 21st of August and the path of totality will cut across the continental U.S.: I live in NE Florida, and will experience almost 90% obscurity, but I have family in TN...
  32. Ackbach

    MHB Which values of $m$ make $P_m(x)$ factorable?

    Here is this week's POTW: ----- For each integer $m$, consider the polynomial \[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\] For what values of $m$ is $P_m(x)$ the product of two non-constant polynomials with integer coefficients? ----- Remember to read the...
  33. anemone

    MHB Integral Sum Problem #272: Find the Integral Part of a Sum

    Here is this week's POTW: ----- Find the integral part of \large \sum_{n=1}^{10^9} n^{\small-\dfrac{2}{3}}. ----- 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...
  34. Euge

    MHB What Does the Integral Evaluate To?

    Here's this week's POTW: ----- Determine the value of the definite integral $$\int_0^\infty \frac{dt}{(1+t^2)t^{1/2}}$$----- 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...
  35. Ackbach

    MHB What is the probability of an odd number of heads when tossing $n$ biased coins?

    Here is this week's POTW: ----- You have coins $C_1,C_2,\ldots,C_n$. For each $k$, $C_k$ is biased so that, when tossed, it has probability $\displaystyle \frac{1}{2k+1}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the...
  36. anemone

    MHB How Do You Prove Equidistance in This Isosceles Right Triangle Geometry Problem?

    Here is this week's POTW: ----- On the sides $AC$ and $BC$ of an isosceles right-angled triangle $ABC$, points $D$ and $E$ are chosen such that $|CD|=|CE|$. The perpendiculars from $C$ and $D$ on $AE$ intersect the hypotenuse $AB$ at $L$ and $K$ respectively. Prove that $|LK|=|LB|$. -----...
  37. Ackbach

    MHB What is the binary operation in set $S$ and what is the given property?

    Here is this week's POTW: ----- Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$. ----- Remember to read the...
  38. davenn

    Stargazing The Sun today - 9 July 2017 - nice spot group

    AR2665 ... largest spot group for some time Canon 6D, 800mm, f11, 125th, ISO100 ( the 800mm is a 100-400mm L lens with a x2 teleconverter) With my eyesight going downhill, I have really been struggling of late to be able to get sharp manual focus Dave
  39. anemone

    MHB What Is the Smallest k for a Cubic Function to Have Exactly One Real Root?

    Here is this week's POTW: ----- Compute the smallest value $k$ such that for all $n>k$, the cubic function $x^3+x^2+nx+9$ has exactly one real root.----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out...
  40. Ackbach

    MHB Is an Equilateral Triangle Possible with Distinct Points in n-Dimensional Space?

    Here is this week's POTW: ----- Let $B$ be a set of more than $2^{n+1}/n$ distinct points with coordinates of the form $(\pm 1,\pm 1,\ldots,\pm 1)$ in $n$-dimensional space with $n\geq 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle...
  41. Drakkith

    Summer Games Done Quick (SGDQ) 2017

    It's that time again! Summer Games Done Quick is starting tomorrow! What is SGDQ you ask? I'll let wiki do the talking: Games Done Quick is a semiannual video game speedrun charity marathon held in the United States, originally organized by the Speed Demos Archive and Speedruns Live...
  42. anemone

    MHB How Many Digits Are in $19^{9^9}$?

    Here is this week's POTW: ----- How many digits are there in (decimal representation of) the integer $19^{9^9}$? ----- 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...
  43. Ackbach

    MHB Is there a solution to the equation $x^2+2y^2=3z^2$?

    Here is this week's POTW: ----- Let $S_0$ be a finite set of positive integers. We define finite sets $S_1,S_2,\ldots$ of positive integers as follows: the integer $a$ is in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$...
  44. jedishrfu

    June 2017 Petya Ransomware Virus Hits Ukraine

    This new virus used multiple means of attack to infect machines on a network. The initial attack came from a legitimate software updater program: http://www.zdnet.com/article/microsoft-petya-ransomware-attacks-were-spread-by-hacked-software-updater/
  45. Euge

    MHB Is the sequence $(X_n)$ of $L^1$ random variables uniformly integrable?

    Here is this week's POTW: ----- Let $(X_n)$ be a sequence of $L^1$ random variables on a probability space $(\Omega, \Bbb P)$. Let $f$ be a continuous, nondecreasing function from $[0,\infty)$ onto itself such that 1. $\Bbb E[f(|X_n|)]$ is uniformly bounded 2. $\dfrac{f(x)}{x}\to \infty$ as...
  46. anemone

    MHB Can You Solve the Sum of Reciprocals for the Fourth Root Function?

    Here is this week's POTW: ----- Evaluate \sum_{i=1}^{1995}\dfrac{1}{f(i)}, given that f(k) be the integer closest to \sqrt[4]{k}. ----- 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...
  47. Ackbach

    MHB What is the solution to Problem B-4 in the 2000 Putnam Mathematical Competition?

    Here is this week's POTW: ----- Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\leq x\leq 1$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  48. anemone

    MHB What is the Sum of Squares in POTW #267?

    Here is this week's POTW: ----- Determine $a^2+b^2+c^2+d^2$ if...
  49. Euge

    MHB Why are nondegenerate fixed points important for self-maps of a smooth manifold?

    Here is this week's POTW: ----- Let $f : M \to M$ be a self-map of a smooth manifold $M$. Prove that the graph of $f$ is transversal to the diagonal of $M$ if and only if the fixed points of $M$ are nondegenerate, i.e., for all fixed points $p$, $+1$ is not an eigenvalue of $df_p$. -----...
  50. Ackbach

    MHB Can the Number of Zeroes of a Derivative Be Controlled?

    Here is this week's POTW: ----- Let $\displaystyle f(t)=\sum_{j=1}^N a_j \sin(2\pi jt)$, where each $a_j$ is real and $a_N$ is not equal to 0. Let $N_k$ denote the number of zeroes (including multiplicities) of $\dfrac{d^k f}{dt^k}$. Prove that \[N_0\leq N_1\leq N_2\leq \cdots \mbox{ and }...
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