What is 2017: Definition and 169 Discussions

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 Minimal Possible Values of Triangle Point Distances - POTW #266 Jun 7th, 2017

    Here is this week's POTW: ----- Let $PQR$ be a triangle such that $PQ=3,\,QR=4$ and $PR=5$. Let $X$ be a point in the triangle. Find the minimal possible values of $PX^2+QX^2+RX^2$. ----- Remember to read the...
  2. Ackbach

    MHB Is the expression $\frac{\gcd(m,n)}{n}\binom{n}{m}$ always an integer?

    Here is this week's POTW: ----- Prove that the expression \[ \frac{\gcd(m,n)}{n}\binom{n}{m} \] is an integer for all pairs of integers $n\geq m\geq 1$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  3. Ackbach

    MHB Can you solve this week's POTW with a clever solution?

    Here is this week's POTW: ----- Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$. ----- Remember to...
  4. anemone

    MHB POTW #265: Solve the Cubic Root Equation for Rational Numbers | May 29, 2017

    Here is this week's POTW: ----- Find a triple of rational numbers $(x,\,y,\,z)$ such that \sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  5. Euge

    MHB How to Prove $X$ Has Lebesgue Measure Zero?

    Here is this week's POTW: ----- Consider a strictly increasing sequence of natural numbers $(n_k)_{k = 1}^\infty$, and suppose $X$ is the subset of $[0,2\pi]$ consisting of all $x$ such that the sequence $(\sin(n_k x))_{k = 1}^\infty$ is convergent. Prove $X$ has Lebesgue measure zero.-----...
  6. mfb

    I What are the challenges faced by LHC in the initial data-taking of 2017?

    "Stable beams" has been declared 30 minutes ago. Similar to 2016, the initial collision rate is low (0.2% the design rate). The machine operators have to check that everything works and nothing presents a danger to the machine before more protons can be filled in. It will probably take a few...
  7. Ackbach

    MHB Is there a way to guarantee that the sequence will eventually reach 0?

    Here is this week's POTW: ----- Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$...
  8. anemone

    MHB Find the Sum of Binomial Coefficients in POTW #264 - May 20th, 2017

    Here is this week's POTW: ----- Without using a calculator, evaluate {2000 \choose 2}+{2000 \choose 5}+{2000 \choose 8}+\cdots+{2000 \choose 2000}. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out...
  9. Ackbach

    MHB Can three points on a circle have a distance between them of less than r^(1/3)?

    Here is this week's POTW: ----- Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$. ----- Remember to read the...
  10. anemone

    MHB What is the solution to this week's challenging integral problem?

    Here is this week's POTW: ----- Compute \int_{0}^{\pi} \dfrac{2\sin x+3\cos x-3}{13\cos x-5} \,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...
  11. Euge

    MHB How can we prove the tail bound for a standard Gaussian random variable?

    Here is this week's POTW: ----- Suppose $Z$ is a standard Gaussian random variable. Prove $\Bbb P(\lvert Z\rvert \ge z) = O[\exp(-z^2/2)]$ as $z\to \infty$.----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to...
  12. Ackbach

    MHB Can the improper integral involving the sine function and a polynomial converge?

    Here is this week's POTW: ----- Show that the improper integral \[ \lim_{B\to\infty}\int_{0}^B \sin(x) \sin\left(x^2\right) \, dx\] converges. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...
  13. anemone

    MHB Acute Triangle ABC: Proving Perpendicular Lines AD and EF

    Here is this week's POTW: ----- Acute triangle $ABC$ has $\angle BAC <45^\circ$. Point $D$ lies in the interior of triangle $ABC$ such that $BD=CD$ and $\angle BDC=4\angle BAC$. Point $E$ is the reflection of $C$ across line $AB$, and point $F$ is the reflection of $B$ across line $AC$...
  14. Ackbach

    MHB What is the maximum area of an octagon inscribed in a circle?

    Here is this week's POTW: ----- The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the...
  15. Euge

    MHB How to Prove the Integral of Log Absolute Value is 0?

    Here is this week's POTW: ----- Prove that $$\int_{-\pi}^{\pi}\ln\lvert 1 - e^{it}\rvert\, dt = 0$$ ----- 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...
  16. Ackbach

    MHB Problem of the Week # 260 - Apr 25, 2017

    Here is this week's POTW: ----- Prove that there exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.] ----- Remember to read the...
  17. anemone

    MHB What Values of k Make the Equation Have One Non-Negative Root?

    Hi MHB, sorry for missing one week of high school's POTW, I guess I can make it up by posting two POTWs this week.(Blush) Here is this week's another POTW: ----- Find all values of $k$ such that the equation $\left(\dfrac{1}{x+k}+\dfrac{k}{x-k}-\dfrac{2k}{k^2-x^2}\right)(|x-k|-k)=0$ has...
  18. anemone

    MHB Triangle Ratio Problem: Find AC/AB with Tan Ratios 1:2:3 | POTW #260

    Here is this week's POTW: ----- In a triangle $ABC$, $\tan A:\tan B: \tan C=1:2:3$. Find $\dfrac{AC}{AB}$. ----- 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...
  19. Greg Bernhardt

    Happy Earth Day 2017: Activities to Care for Our Home

    Time to think about what you can personally do to be a good steward of our home Earth. What activities are you doing today? I was hoping I could attend a sister science march in my city, but alas I have a wedding to attend. Instead this morning I will pick up trash I find in my community during...
  20. S

    LaTeX Making Tables in LaTeX - 2017 Update

    Is there a way to do tables in the forum implementation of Latex? ( I asked this question in 2012 and the "tabular" environment wasn't implemented.)
  21. Jameson

    MHB What is the Kaggle Data Science Bowl 2017 competition about?

    There is a neat site called Kaggle that is home to lots of data science info and the place of very featured competitions with large cash prizes. The goal of this year's competition was to create a model to detect lung cancer. It just wrapped up last week and the results are being verified right...
  22. Ackbach

    MHB What are the possible values of the sum of squares, given a specific sum?

    Here is this week's POTW: ----- Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\displaystyle\sum_{j=0}^\infty x_j=A$? ----- Remember to read the...
  23. kaliprasad

    MHB Find remainder when (2∗4∗6∗8⋯∗2016)−(1∗3∗5∗7⋯∗2015) is divided by 2017

    $ ( 2 * 4 * 6 * 8 \cdots * 2016) - ( 1 * 3 * 5 * 7 \cdots * 2015)$ is divided by 2017 what is the remainder
  24. Euge

    MHB Is Continuously Differentiable Imply Fourier Coefficients in l^1?

    Here is this week's POTW: ----- Let $f : \Bbb S^1\subset \Bbb C \to \Bbb C$ be a continuous map. Show that if $f$ is continuously differentiable on $\Bbb S^1$, then its Fourier coefficient sequence $\{\hat{f}_n\}_{n\in \Bbb Z}$ belongs to $\ell^1(\Bbb Z)$.----- Remember to read the...
  25. Ackbach

    MHB What is the determinant of a special matrix involving trigonometric functions?

    Here is this week's POTW, shamefully late. I can only say I will promise to do better in the next few weeks, and even try to catch up with the missed week: ----- For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times...
  26. anemone

    MHB Solution for POTW #259: Finding the Value of a Trigonometric Expression

    Here is this week's POTW: ----- Suppose $\tan A$ and $\tan B$ are the roots of $x^2+\pi x+\sqrt{2}=0$. Evaluate $\sin^2 (A+B) +\pi\sin (A+B)\cos (A+B) +\sqrt{2}\cos^2 (A+B)$ ----- Remember to read the...
  27. anemone

    MHB Triangle Built from Set of Numbers Must be Isosceles - POTW #258 Mar 29th, 2017

    Sorry for being late in coming to do the POTW, I just didn't feel well last week due to the prolonged fever, flu, sore throat and cough. To help make up for being late, I will present to you an intriguing problem which I truly hope you are going to have lots of fun solving the problem! Without...
  28. Euge

    MHB Is $\Bbb R$ homeomorphic to a cartesian product?

    Here is this week's POTW: ----- Prove that if $\Bbb R$ is homeomorphic to a cartesian product $A\times B$, then either $A$ or $B$ is a singleton. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...
  29. Ackbach

    MHB Can we prove that $f'(x)<2f(x)$ for all $x$ using the given conditions?

    Here is this week's POTW: ----- Let $f$ be a real function with a continuous third derivative such that $f(x),f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$. ----- Remember to read the...
  30. K

    I Moriond EW 2017 results -- No supersymmetry

    52nd Rencontres de Moriond EW 2017 présidé par Lydia Iconomidou-Fayard (LAL), Jean Marie Frere (ULB Brussels) has released results based on 36 fb-1 data @ 13 TEV they compare predictions from SUSY i.e squarks gluinos stops etc, with SM all results consistent with SM 95% No supersymmetry...
  31. anemone

    MHB Comparing $X$ and $Y$: POTW #257 (Mar 20, 2017)

    Here is this week's POTW: ----- Which number, $X$ or $Y$, is larger? $X=\dfrac{1}{2016}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2016}\right)$ $Y=\dfrac{1}{2017}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2017}\right)$ ----- Remember to read the...
  32. Ackbach

    MHB What is the limit of a special sum at the point (1,1)?

    Here is this week's POTW: ----- Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1)...
  33. Chestermiller

    Celebrate Pi Day 2017: A Big Day of Fun!

    Today is the bi...g day. Happy pi day everyone.
  34. anemone

    MHB Real Number Triples: Solving Problem #256 Mar 13th, 2017 POTW

    Here is this week's POTW: ----- Find all triples $(a,\,b,\,c)$ of real numbers that satisfy $a^2+b^2+c^2+1=ab+bc+ca+|a-2b+c|$. ----- 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 Can a Polynomial Have More Roots Than Its Degree?

    Here is this week's POTW: ----- Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$, where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots. -----...
  36. ZapperZ

    March For Science, April 22, 2017

    It seems that the March for Science is getting more traction and endorsements. The APS just released a statement endorsing this event. @Greg Bernhardt : Is PhysicsForums participating? :) Zz.
  37. Euge

    MHB How can $L^1(\Bbb R)$ be isomorphic to an ideal in $M(\Bbb R)$?

    Here is this week's POTW: ----- Consider the Lebesgue space $L^1(\Bbb R)$ as an algebra with product given by convolution. Prove that $L^1(\Bbb R)$ is isomorphic as an algebra to an ideal in the algebra $M(\Bbb R)$ of complex Borel measures on $\Bbb R$, and identify the ideal. Note the product...
  38. anemone

    MHB Triangle Centroid and Circumcenter Relationship: POTW #255 Mar 4th, 2017

    Here is this week's POTW: ----- Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Prove that if $BC$ is its largest side, then $G$ lies in the interior of the circle with diameter $OA$. ----- Remember to read the...
  39. Ackbach

    MHB Problem of the Week # 254 - Feb 27, 2017

    Here is this week's POTW: ----- Right triangle $ABC$ has right angle at $C$ and $\angle BAC =\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE = \theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate...
  40. Euge

    MHB How can we construct a chain homotopy between homotopic chain maps?

    Here is this week's POTW: ----- Let $(\mathcal{C}, \partial)$ be a chain complex of abelian groups. Suppose $f, g : \mathcal{C} \to \mathcal{C}$ are homotopic chain maps. Construct an explicit chain homotopy between the $n$-fold compositions $f^n$ and $g^n$.----- Remember to read the...
  41. robphy

    AAPT 2017 Summer Meeting-Cincinnati, OH (Jul 22-26, 2017)

    The AAPT Winter Meeting in Atlanta ( https://www.aapt.org/Conferences/wm2017/ ) is going on now. Unfortunately, this meeting was scheduled for now [in the middle of the semester], rather than its usual time in January [during my break]. So, I wasn't able to make it. However, the website for the...
  42. anemone

    MHB Real Root of x^3+3x-2=0 | POTW #254 Feb 19, 2017 - Math Help Boards

    Here is this week's POTW: ----- Find the exact value for the real root of the equation $x^3+3x-2=0$. ----- 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. N

    Schools Math Reu Season 2016: Results & Questions

    Please post results or questions on this Math Reu season. See last years discussionhttps://www.physicsforums.com/threads/math-reus-2016.858490/
  44. Finesagan

    Schools Physics REUs 2017: Have You Heard Back?

    I feel like it's about time we started this thread once again. Has anyone heard back from the schools they applied to? Furthermore, does anyone have questions about the REUs they have been accepted into?
  45. Ackbach

    MHB Can You Solve the 1999 Putnam Competition Problem A-5?

    Here is this week's POTW: ----- Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $2017,$ then \[|p(0)|\leq C \int_{-1}^1 |p(x)|\,dx.\] ----- Remember to read the...
  46. anemone

    MHB POTW #253: Feb 9th, 2017 - Solve for x in $\sqrt{x^3+1648}-\sqrt{4949-x^3}=75$

    Here is this week's POTW: ----- Given that $\sqrt{x^3+1648}-\sqrt{4949-x^3}=75$ for $x\in\Bbb{N}$. Find $x$. ----- 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. Euge

    MHB Is $\mathcal{M}(X)$ a Banach space?

    Here is this week's POTW: ----- Consider the normed space $\mathcal{M}(X)$ of all complex regular Borel measures on a locally compact Hausdorff space $X$, with total variation norm $\|\mu\| := \lvert \mu\rvert (X)$, for all $\mu\in \mathcal{M}(X)$. Prove that $\mathcal{M}(X)$ is a Banach...
  48. Ackbach

    MHB Is $\theta(N)$ a normal subgroup of $G$?

    Here is this week's POTW: ----- Let $G$ be a group. If $\theta$ is an automorphism of $G$ and $N \vartriangleleft G$, prove that $\theta(N) \vartriangleleft G$. ----- Remember to read the...
  49. Ackbach

    MHB What is the solution to Problem A-4 in the 1999 Putnam Mathematical Competition?

    Here is this week's POTW: ----- Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\] ----- 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...
  50. anemone

    MHB Maximize Square Roots with POTW #252 - Feb 1st, 2017

    Here is this week's POTW: ----- Maximize $\sqrt{x^4-3x^2-6x+13}-\sqrt{x^4-x^2+1}$ for all $x\in \Bbb{R}$. ----- 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...
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