2015 Definition and 219 Threads

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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. -----...

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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...

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!

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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...