What is 2015: Definition and 219 Discussions

2015 (MMXV) was a common year starting on Thursday of the Gregorian calendar, the 2015th year of the Common Era (CE) and Anno Domini (AD) designations, the 15th year of the 3rd millennium, the 15th year of the 21st century, and the 6th year of the 2010s decade.
2015 was designated by the United Nations as:

International Year of Light
International Year of Soil

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  1. anemone

    MHB What Are All the Real Solutions to the Equation \(a^4+b^4+c^4+1=4abc\)?

    Here is this week's POTW: ----- Find all real solutions of the equation $a^4+b^4+c^4+1=4abc$. ----- 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...
  2. Ackbach

    MHB What is the solution to Problem of the Week #172?

    Here is this week's POTW: ----- A number of different objects has been distributed into $n$ boxes $B_{1}, B_{2}, \dots ,B_{n}.$ All the objects from these boxes are removed and redistributed into $n+1$ new boxes $B_{1}^{*}, B_{2}^{*}, \dots , B_{n+1}^{*},$ with no new box empty (so the total...
  3. Euge

    MHB Can you determine the stability of a fixed point in a system of ODEs?

    Here is this week's POTW: ----- Consider the following system of ODE on $\Bbb R^2$. \begin{align} \dot{x} &= y\\ \dot{y} &= \lambda y(1 - x^2) - x \end{align} Determine a condition(s) on $\lambda$ such that the fixed point $(0,0)$ is asymptotically stable. ----- Remember to read the...
  4. anemone

    MHB Can $\sqrt{k-1} + \sqrt{k+1}$ Be a Rational Number for Any Integer k?

    Here is this week's POTW: ----- Is there an integer $k$ such that $\sqrt{k-1}+\sqrt{k+1}$ is a rational number? ----- 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...
  5. Euge

    MHB Can a normal subgroup of a transitive group also be transitive?

    Here is this week's POTW: ----- Let $p$ be a prime, $G$ a transitive subgroup of the symmetric group $S_p$, and $A$ a nontrivial normal subgroup of $G$. Prove that $A$ is a transitive subgroup of $S_p$. ----- Remember to read the...
  6. anemone

    MHB How Do You Maximize This Complex Expression With Given Constraints?

    Here is this week's POTW: ----- Find the maximum of $a_1+a_2+a_3+a_4-a_1a_2-a_1a_3-a_1a_4-a_2a_3-a_2a_4-a_3a_4+a_1a_2a_3+a_1a_2a_4+a_1a_3a_4+a_2a_3a_4-a_1a_2a_3a_4$ where $|a_i|\le1,\,i=1,\,2,\,3,\,4$. ----- Remember to read the...
  7. Ackbach

    MHB What is the integral of x^2 over 1+x^10 from 0 to infinity?

    Here is this week's POTW: ----- Evaluate $\displaystyle \int_0^{\infty}\frac{x^2}{1+x^{10}} \, dx.$ ----- 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...
  8. Ackbach

    MHB What is the condition for $E$ in the time-independent Schrödinger equation?

    Here is this week's POTW: ----- The time-independent Schrödinger equation in one spatial dimension is $$E \, \psi(x)=\left[-\frac{\hbar^2}{2m} \, \frac{d^2}{dx^2}+V(x)\right]\psi(x).$$ Show that $E$ must exceed the minimum value of $V(x)$ for every normalizable solution. ----- Remember to...
  9. Astronuc

    Women's World Cup 2015: US Topples #1 Germany, Reaches Final

    U.S. Topples Top-Ranked Germany 2-0 To Reach World Cup Final http://www.npr.org/sections/thetwo-way/2015/06/30/418725724/womens-soccer-game-tonight-features-the-worlds-top-2-teams The U.S., which hasn't won a World Cup since 1999, will play for the title on Sunday at 7 p.m. ET, facing either...
  10. marcus

    Our picks for second quarter 2015 MIP (most important QG paper)

    Please indicate the papers you think will prove most significant for future Loop-and-allied QG research. The poll is multiple choice, so it's possible to vote for several. Abstracts follow in the next post. http://arxiv.org/abs/1504.01065 Wilson loops in CDT quantum gravity J. Ambjorn, A...
  11. anemone

    MHB How Do You Find the Coefficient \( a_2 \) in a Polynomial Transformation?

    Here is this week's POTW: ----- The polynomial $1-y+y^2-y^3+\cdots+y^{16}-y^{17}$ may be written in the form $a_0+a_1x+a_2x^2+\cdots+a_{16}x^{16}+a_{17}x^{17}$, where $x=y+1$ and $a_i$ are constants. Find $a_2$. ----- Remember to read the...
  12. Euge

    MHB What is the Limit of the Bessel Function as x Goes to Infinity?

    Here is this week's POTW: ----- Let $n$ be an integer. Show that as $x \to \infty$ on the positive real axis, $$J_n(x) \sim \sqrt{\frac{2}{\pi x}}\left[\cos\left(x - \frac{n\pi}{2} - \frac{\pi}{4}\right)\right],$$ where $J_n(x)$ is the $n$th order Bessel function of the first kind. -----...
  13. anemone

    MHB Real Numbers Inequality Proof: x+y+z > (|x|+|y|+|z|)/3

    Here is this week's POTW: ----- Suppose $x,\,y,\,z$ are real numbers such that $x+y>0$, $y+z>0$ and $z+x>0$. Prove that $x+y+z>\dfrac{|x|+|y|+|z|}{3}$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  14. Euge

    MHB How can you prove the Laplace transform of a periodic function with an integral?

    Here is this week's POTW: ----- Suppose $f$ is a $p$-periodic complex-valued function on $[0,\infty)$. Let $F(s)$ denote the Laplace transform of $f(t)$. Prove $$F(s) = \frac{1}{1 - e^{-ps}}\int_0^p e^{-st}f(t)\, dt, \qquad \operatorname{Re}(s) > 0.$$ ----- Remember to read the...
  15. Ackbach

    MHB How many walks of length $n$ can you find on this graph?

    Here is this week's POTW: ----- For the following graph, find the number of walks of length $n$ from any vertex to any other vertex. Use of technology is allowed (but explain what you did and how you did it). https://www.physicsforums.com/attachments/4470._xfImport ----- Remember to read...
  16. Euge

    MHB Is the Endomorphism Ring of a Finite Dimensional Vector Space Dedekind Finite?

    Here is this week's POTW: ----- Prove that if $V$ is a finite dimensional vector space over field $\Bbb k$, then the endomorphism ring $\text{End}_{\Bbb k}(V)$ is Dedekind finite. ----- Remember to read the...
  17. Ackbach

    MHB Is $EFGH$ a parallelogram in quadrilateral $ABCD$?

    Here is this week's POTW: ----- Given a quadrilateral $ABCD$ with respective midpoints $EFGH$, show that the quadrilateral $EFGH$ is a parallelogram. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  18. Astronuc

    Mount Sinabung eruption - June 2015

    http://news.yahoo.com/thousands-indonesians-refuse-leave-volcano-danger-zone-091348000.html http://www.bbc.com/news/world-asia-33139538 http://www.abc.net.au/news/2015-06-16/residents-evacuated-in-indonesia-as-sinabung-volcano-erupts/6548288...
  19. anemone

    MHB How Do You Solve Complex Root Expressions in Polynomial Equations?

    Here is this week's POTW: ----- Let $u,\,v,\,w$ be the roots of the equation $x^3-6x^2+18x-36=0$. Evaluate $\left(\dfrac{u}{v}+\dfrac{v}{u}+\dfrac{v}{w}+\dfrac{w}{v}+\dfrac{u}{w}+\dfrac{w}{u}+3\right)(3^{u^2+v^2+w^2})^{u^3+v^3+w^3}$. ----- Remember to read the...
  20. T

    Job Skills DoE SULI Fall 2015: Waiting to Hear Back

    I applied for the SULI program for this fall and I'm really antsy to find out if I got in or not. I applied at PNNL and LANL. Has anyone heard back about their acceptance yet?
  21. Euge

    MHB Is it Possible for a Random Variable to Have a Variance of Zero?

    Here is this week's POTW: ----- Let $X$ be random variable whose variance is zero. Prove that with probability one, $X = \Bbb E[X]$. ----- 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...
  22. Ackbach

    MHB What is the Maximum Number of Equivalence Classes for the Eight Queens Problem?

    Here is this week's POTW: ----- The statement of the Eight Queens Problem is to place eight queens on a regular chessboard so that no two queens are attacking each other. For anyone ignorant of the rules of chess, queens attack in any direction vertically or horizontally or in either...
  23. anemone

    MHB Prove One Root in Interval (0,1) for Equation with Real Numbers

    Here is this week's POTW: ----- Let $m,\,n,\,k$ be real numbers such that $m>0$ and $\dfrac{m}{5}+\dfrac{n}{4}+\dfrac{k}{3}=0$. Prove that the equation $mx^2+nx+k=0$ has one root in the interval $(0,\,1)$. ----- Remember to read the...
  24. Ackbach

    MHB Can a Matrix with Only 1s and -1s Have a Determinant Divisible by 2^(n-1)?

    N.B. There has been a correction in this problem, thanks to Opalg. It should read correctly now. Here is this week's POTW: ----- Let $A$ be an $n\times n$ matrix whose entries are only $1$ or $-1$. Show that $2^{n-1}$ divides $\det(A)$. ----- Remember to read the...
  25. Euge

    MHB Is the power series convergent in the field of p-adic numbers?

    Here is this week's POTW: ----- Find the domain of convergence of the power series $$\sum\limits_{n = 1}^\infty \frac{(-1)^{n-1}x^n}{n}$$ in the field $\Bbb Q_p$ of $p$-adic numbers. ----- Remember to read the...
  26. anemone

    MHB Smallest N for Ensuring 3 Colors in 100 Marble Draw

    There are 111 marbles in a box, each being green, yellow, purple and blue. It's known that if 100 marbles are drawn, we can ensure getting marbles of all four colors. Find the smallest integer $N$ such that if $N$ marbles are drawn, we can ensure getting marbles of at least three different...
  27. Euge

    MHB How Do You Solve the Integral from POTW #156?

    Here is this week's POTW: ----- Evaluate the improper integral $$\int_{-\infty}^\infty \frac{\sin t}{t}\cos xt\, dt\quad (x \in \Bbb R).$$ ----- 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...
  28. Ackbach

    MHB Is $e^{\pi}$ greater than $\pi^{e}$?

    Here is this week's POTW: ----- Prove that $e^{\pi}>\pi^{e}$. ----- 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!
  29. anemone

    MHB Real Number Problem: Evaluate xyz with Given Equations

    If $x,\,y,\,z$ are real numbers such that $x+2y+3z=6$ and $x^2+4y^2+9z^2=12$, evaluate $xyz$. _______________________________________________________________________________________________________ Remember to read the...
  30. Ackbach

    MHB How to Check if a Number is Prime in Python?

    Here is this week's POTW: ----- Write a computer program in Python to check a positive integer for primality. You may not use any built-in primitives that look like "IsPrime(n)". Check all the numbers up to $\sqrt{n}$, and skip the evens after 2. Inputs: integer n. Check that the input is a...
  31. Euge

    MHB How to Prove the Decomposition Theorem for Projections on a Vector Space?

    Here is this week's POTW: ----- Let $P_1,\ldots, P_n$ be a sequence of projections on a vector space $V$ such that $P_iP_j = 0$ whenever $i \neq j$ and $P_1 + \cdots + P_n = I$. Prove that $$V = \operatorname{Im}(P_1) \oplus \cdots \oplus \operatorname{Im}(P_n).$$ ----- Note: By a projection...
  32. anemone

    MHB Can You Solve the Equation with Logs: $\log_2 (\cos x)=2\log_3 (\cot x)$?

    Solve the equation $\log_2 (\cos x)=2\log_3 (\cot x)$. _______________________________________________________________________________________________________ Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...
  33. Euge

    MHB Is the Convergence of a Real Number Series Determined by Positive Integers?

    Here is this week's POTW: ----- Let $\sum\limits_{n = 1}^\infty a_n$ be a series of real numbers. Show that $\sum\limits_{n = 1}^\infty a_n$ converges absolutely if and only if to each $\epsilon > 0$, there corresponds a positive integer $N = N(\epsilon)$ such that if $n_1,\ldots, n_m$ are...
  34. Ackbach

    MHB Problem of the Week # 163 - May 12, 2015

    Here is this week's POTW: ----- Construct a formal proof of the following propositional statement: $(P\to (Q\to R)) \leftrightarrow ((P\land Q)\to R)$. Here $\to$ means "implies", $\leftrightarrow$ means "if and only if", and $\land$ means "and". Make sure to mention what deductive system you...
  35. anemone

    MHB What is the Range of x for $|x|^{x^2-3x-4}<1$?

    If the range of values of $x$ that satisfies $|x|^{x^2-3x-4}<1$ is given by $(a,\,b)$, evaluate $b-a$. _______________________________________________________________________________________________________ Remember to read the...
  36. D

    What would you present about light if you have 90-sec time?

    As you know, this year, 2015, is the year of light: http://en.wikipedia.org/wiki/International_Year_of_Light I want to create a video clip limited to 90 seconds to show in our university's TED-like show to make students excited about light. Do you have any idea? any experiment?
  37. Ackbach

    MHB What is the direction of the velocity vector on the rising side of a cycloid?

    Here is this week's POTW: ----- It is known that if you take a circle and roll it without slipping on a flat surface, a single point on the circle traces out the path of a cycloid. Show that the direction of the velocity vector for any point on the rising side of a cycloid is directed toward...
  38. Euge

    MHB Is Any Real Zero of This Polynomial Greater Than M+1?

    Here is this week's POTW: ----- Let $f(x)\in \Bbb R[x]$ be a monic polynomial. Prove that if $M$ is the greatest of the absolute values of its coefficients, then no real zero of $f$ can exceed $M + 1$. ----- Remember to read the...
  39. anemone

    MHB Find the $2015$th Term in Sequence $1,2,-2,3,-3,3,4,-4,4,-4,5,-5,5,-5,5\cdots$

    Find the $2015$th term in the sequence $1,\,2,\,-2,\,3,\,-3,\,3,\,4,\,-4,\,4,\,-4,\,5,\,-5,\,5,\,-5,\,5\cdots$. _______________________________________________________________________________________________________ Remember to read the...
  40. Euge

    MHB What is the integral of cos(x^3)?

    Here is this week's POTW: ----- Compute the integral $$\int_{-\infty}^\infty \cos(x^3)\, dx.$$ ----- 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...
  41. Ackbach

    MHB What is the height of the tennis ball after bouncing off a basketball?

    Here is this week's POTW: ----- (N.B. I will give credit for this problem to where it's due, but only after next week.) A tennis ball with (small) mass $m_2$ sits atop a basketball with (large) mass $m_1$. The bottom of the basketball is a height $h$ above the ground, and the bottom of the...
  42. anemone

    MHB How to Evaluate Trigonometric Cosine Sums Manually?

    Evaluate $\cos 5^{\circ}+\cos 77^{\circ}+\cos 149^{\circ}+\cos 221^{\circ}+\cos 293^{\circ}$ without the help of calculator. _______________________________________________________________________________________________________ Remember to read the...
  43. davenn

    BIG quake coming in NOW - Nepal -April 25, 2015

    Preliminary report is a M 7.5, Nepal, central Himalayas has now been upgraded to a M 7.9 http://www.sydneystormcity.com/seismograms.htm cheers Dave
  44. anemone

    MHB Which values of $a$ satisfy a trigonometric equation in a given interval?

    Find all $a$ in the interval $\left(0,\,\dfrac{\pi}{2}\right)$ such that $\dfrac{\sqrt{3}-1}{\sin a}+\dfrac{\sqrt{3}+1}{\cos a}=4\sqrt{2}$. 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...
  45. Euge

    MHB How can you prove that a matrix is equal to zero if its trace is always zero?

    Here is this week's problem! ----- Let $\Bbb k$ be a field. Suppose $A \in M_n(\Bbb k)$ such that $\operatorname{trace}(AM) = 0$ for all $M \in M_n(\Bbb k)$. Prove $A = 0$. Furthermore, show that the linear transformation $L_B : M_n(\Bbb k) \to M_n(\Bbb k)$ given by $L_B(X) = BX$ is an isometry...
  46. Ackbach

    MHB Problem of the Week # 160 - April 21, 2015

    Here is this week's POTW: ----- Find a one-parameter family of solutions of $$(2x-y+1)\, dx+(x+y) \, dy=0.$$ Extra Credit: Generalize to $$(ax+by+c) \, dx+(fx+gy+h) \, dy=0,$$ where $a,b,c,f,g,h$ are constants, and $\tfrac{b}{a}\not=\tfrac{g}{f}$. ----- Remember to read the...
  47. Ackbach

    MHB Problem of the Week # 159 - April 14, 2015

    Here is this week's POTW: ----- What is the $\text{IQR}$ of the standard normal distribution? Using the $1.5\times\text{IQR}$ rule, what $z$ scores would be considered outliers? ----- Remember to read the...
  48. Euge

    MHB How can you prove the linearity of a functional using a signed Borel measure?

    Here is this week's POTW: ----- Let $T : C^1([0,1]) \to \Bbb R$ be a linear functional such that $|T(f)| \le A\|f\| + B\|f'\|$ for all $f \in C^1[0,1]$, where $A$ and $B$ are constants and $\|\cdot\|$ is the supremum norm. Prove that there is a signed Borel measure $\mu$ on $[0,1]$ and a...
  49. anemone

    MHB Can You Prove This Inequality Involving Fractions and Square Roots?

    Prove that $1-\dfrac{1}{2014}\left(\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2015}\right)>\dfrac{1}{\sqrt[2014]{2015}}$ 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. Ackbach

    MHB Is this Roulette Wheel Fair? Statistical Analysis and Casino Manager's Claim

    My apologies for not posting at all on time! I completely spaced it. I can claim a lot of things going on at home (just sold house). Anyway, here you go. This is an easier one. You have until Tuesday to do it. ----- An American roulette wheel has 18 red slots among its 38 slots. To test if a...
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