2016 Definition and 204 Threads

Here is this week's POTW: ----- Find all positive integers $n$ such that $n^8+n^4+1$ is a prime. ----- 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...

For those that are interested, the annual charity event "Summer Games Done Quick" starts tomorrow (July 3rd) and runs through July 10th. From the event website: Summer Games Done Quick is an annual charity video game marathon streamed live here and on Twitch.TV, with donations going directly to...

Here is this week's POTW: ----- Suppose that each of 20 students has made a choice of anywhere from 0 to 6 courses from a total of 6 courses offered. Prove or disprove: there are 5 students and 2 courses such that all 5 have chosen both courses or all 5 have chosen neither course. -----...

Here is this week's POTW: ----- Let $A$ be a commutative ring with unity. Prove that a polynomial $p(x) = a_0 + a_1 x + \cdots + a_n x^n$ over $A$ is a unit in $A[x]$ if and only if $a_0$ is an $A$-unit and $a_1,\ldots, a_n$ are nilpotent in $A$. ----- Remember to read the...

Here is this week's POTW: ----- Given that $$\frac{\sin (x-a)}{\sin (x-b)}=m$$ and $$\frac{\cos (x-a)}{\cos (x-b)}=n$$ where $$0\lt x \lt \frac{\pi}{2}$$ and $m$ and $n$ are two positive real numbers. Express $$\cos (a − b)$$ in terms of $m$ and $n$. ----- Remember to read the...

according to blogger Tommaso Dorigo the LHC "What to expect for ICHEP I believe that if the machine keeps delivering at this pace, by mid-July CMS and ATLAS could have in their hands some 10 inverse femtobarns of 13-TeV collisions. Those data, three to four times larger in size that what was...

Here is this week's POTW: ----- Let $C_1$ and $C_2$ be circles whose centers are 10 units apart, and whose radii are 1 and 3. Find, with proof, the locus of all points $M$ for which there exists points $X$ on $C_1$ and $Y$ on $C_2$ such that $M$ is the midpoint of the line segment $XY$. -----...

Here is this week's POTW: ----- Evaluate the infinite series $$\sum_{n = 1}^\infty \frac{(-1)^{n+1} n^2}{n^3 + 1}$$ ----- 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...

I would like to say a big thank you to MarkFL, who stood in for me during the last month to take care of the POTW duty while I was away in another country.Here is this week's POTW: ----- Let $b\gt a \gt 0$. Prove that $$\int_{a}^{b} (x^2+1)e^{-x^2} \,dx\ge e^{-a^2}-e^{-b^2}$$. -----...

From a spreadsheet I developed during the Ceres mission, I've calculated that the Hubble Space Telescope would barely be able to see it, with a resolution of only 0.026 pixels. For comparison: Object _____________Pixels 2016 HO3__________0.026 Pluto_______________2.1 Ceres______________18...

Here is this week's POTW: ----- Find the least number $A$ such that for any two squares of combined area $1$, a rectangle of area $A$ exists such that the two squares can be packed into the rectangle (without interior overlap). You may assume that the sides of the squares are parallel to the...

Here is this week's POTW: ----- Show that the complex function $$F(z) = \frac{1}{\pi}\int_0^1 \int_{-\pi}^\pi \frac{r}{re^{i\theta} + z}\, d\theta\, dr$$ is anti-holomorhpic (i.e., the conjugate $\bar{F}$ is holomorphic) in the open unit disc, $\Bbb D$, and holomoprhic in complement $\Bbb C...

Here is this week's POTW: ----- Suppose you are at the top of a building of height $h$, and you have one rope of mass $m_1$ hanging over the side tied to a second rope of mass $m_2$ on the ground. Both ropes are of length $h$. Show that the work required to haul the second rope up to you, such...

Here is this week's POTW: ----- For a positive real number $\alpha$ define $$S(\alpha)=\{\lfloor n\alpha\rfloor: n=1,2,3,\dots\}.$$ Prove that $\{1,2,3,\dots\}$ cannot be expressed as the disjoint union of three sets $S(\alpha), S(\beta),$ and $S(\gamma)$. (As usual, $\lfloor x\rfloor$ is...

Here is this week's POTW: ----- The positive rational numbers may be arranged in the form of a simple series as follows: $$\frac{1}{1},\,\frac{2}{1},\,\frac{1}{2},\,\frac{3}{1},\,\frac{2}{2},\,\frac{1}{3},\,\frac{4}{1},\,\frac{3}{2},\,\frac{2}{3},\,\frac{1}{4},\,\cdots$$ In this series...

Here is this week's POTW: ----- Evaluate the abelianization of the fundamental group of the $n$-fold connected sum $\underbrace{\Bbb RP^2\, \# \cdots \#\, \Bbb RP^2}_{n}$. ----- Remember to read the...

Here is this week's POTW: ----- Evaluate $\displaystyle\sqrt[8]{2207-\dfrac{1}{2207-\dfrac{1}{2207-\cdots}}}$. Express your answer in the form $\dfrac{a+b\sqrt{c}}{d}$, where $a,b,c,d$ are integers. ----- Remember to read the...

Here is this week's POTW: ----- Define the logarithm of an $n\times n$ matrix $A$ by the power series $$\sum_{k = 1}^\infty \frac{(-1)^{k-1}(A - I)^k}{k}$$ which converges for $\|A - I\| < 1$ (the standard matrix norm is being used here). Prove that for all $n\times n$ matrices $A$ and $B$...

anemone has asked me to fill in for her this week. Here is this week's POTW: ----- How many distinct numbers are in the list $$\frac{1^2-1+4}{1^2+1},\,\frac{2^2-2+4}{2^2+1},\,\frac{3^2-3+4}{3^2+1},\,\cdots,\frac{2011^2-2011+4}{2011^2+1}$$ ----- Remember to read the...

In January 2016, Dr. Attila Krasznahorkay (at the Hungarian Academy of Sciences’s Institute for Nuclear Research in Debrecen, Hungary) and his colleagues published a paper announcing he had found a dark photon by firing protons at lithium-7, which created unstable beryllium-8 nuclei that then...

Here is this week's POTW: ----- Suppose that each of $n$ people writes down the numbers $1,2,3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a,b,c$ of the resulting...

The following problem is a nice exercise that both the undergrad and grad members here can enjoy: ----- Let $R$ be a ring such that for all $r\in R$, $r^3 = r$. Prove $R$ is commutative. -----Remember to read the...

Here is this week's POTW: ----- Prove that $ab+ 2a^2b^2\le a^2 +b^2 +ab^3$ for all reals $0\le a \le 1$ and $0\le b\le 1$. ----- 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 $x_1, x_2,\dots,x_n$ be differentiable (real-valued) functions of a single variable $t$ which satisfy \begin{align*} \d{x_1}{t}&=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n \\ \d{x_2}{t}&=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n \\ \vdots \\...

Here is this week's POTW: ----- Given that $a,\,b$ and $c$ are different real numbers. Prove that the expression $\dfrac{1}{(a-b)^2}+\dfrac{1}{(b-c)^2}+\dfrac{1}{(c-a)^2}$ is a square. ----- Remember to read the...

Here is this week's POTW: ----- Prove the following theorem of Tate: If $\phi : X\to Y$ is a mapping of real Banach spaces such that for some positive number $M$, $\lvert \phi(x + y) - \phi(x) - \phi(y)\rvert \le M$ ($x,y\in X$), then there is a unique additive mapping $\psi : X \to Y$ such...

Here is this week's POTW: ----- By method of contour integration, find the values of the integrals $$\int_{-\infty + i\alpha}^{\infty + i\alpha} e^{-x^2}\, dx$$ for all $\alpha \ge 0$. ----- Remember to read the...

Here is this week's POTW: ----- Suppose we have a necklace of $n$ beads. Each bead is labeled with an integer and the sum of all these labels is $n-1$. Prove that we can cut the necklace to form a string whose consecutive labels $x_1, x_2, \dots, x_n$ satisfy $$\sum_{i=1}^k x_i\le k-1 \qquad...

Here is this week's POTW: ----- Suppose that 4 real numbers $a,\,b,\,c,\,d$ satisfy the conditions as shown below: $a^2+b^2=4$ $c^2+d^2=4$ $ac+bd=2$ Evaluate all possible values for $ab+cd$. ----- Remember to read the...

Here is this week's POTW: ----- The number $d_{1}d_{2}\dots d_{9}$ has nine (not necessarily distinct) decimal digits. The number $e_{1}e_{2}\dots e_{9}$ is such that each of the nine 9-digit numbers formed by replacing just one of the digits $d_{i}$ in $d_{1}d_{2}\dots d_{9}$ by the...

Here is this week's POTW: ----- Let $q$ be a complex number with $\lvert q \rvert < 1$. Show that $$\prod_{n = 1}^\infty (1 - q^n) \sum_{n = -\infty}^\infty q^{n+2n^2} = \prod_{n = 1}^\infty (1 - q^{2n})^2$$ ----- Note: Do not worry about arguments of convergence.Remember to read the...

Here is this week's POTW: ----- Given $x,\,y,\,u,\,v$ are real numbers that satisfy the following system: $$x+y+u+v=\frac{1}{2}$$ $$8x+4y+2u+v=\frac{1}{3}$$ $$27x+9y+3u+v=\frac{1}{4}$$ $$64x+16y+4u+v=\frac{1}{5}$$ Evaluate $$343x+49y+7u+v.$$ ----- Remember to read the...

Here is this week's POTW: ----- Why must a group with cyclic automorphism group be abelian? -----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!

Here is this week's POTW: ----- For what pairs $(a,b)$ of positive real numbers does $$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\,\right) dx$$ converge? ----- Remember to read the...

Here is this week's POTW: ----- Let $0\le x \le 1$ and $0\le y \le 1$ . Prove the inequality $$\frac{x}{\sqrt{2y^2+5}}+\frac{y}{\sqrt{2x^2+5}}\le \frac{2}{\sqrt{7}}.$$ ----- Remember to read the...

Here is this week's POTW: ----- Show that the holomorphic mappings on a compact connected Riemann surface are constant. ----- 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: ----- An ellipse, whose semi-axes have lengths $a$ and $b$, rolls without slipping on the curve $y=c\sin\left(\dfrac{x}{a}\right).$ How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve? ----- Remember to...

Here is this week's POTW: ----- Prove that $\sin(2^{25})^{\circ}=-\cos 2^{\circ}$ ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

I didn't expect that this year will be too hot here in the Philippines... D: Welp... Here's an update :https://web.pagasa.dost.gov.ph/index.php/climate/climate-advisories Is climate change taking effect here? I guess so. Will somebody care for an ice cream? lol

Here is this week's POTW: ----- The real numbers $x,\,y$ and $z$ are such that $x^2+y^2=2z^2$, $x\ne y,\,z\ne -x,\,z\ne -y$. Prove that $$\frac{(x+y+2z)(2x^2-y^2-z^2)}{(x-y)(x+z)(y+z)}$$ is an integer. ----- Remember to read the...

Here's this week's problem! __________________ Let $u$ be an $H^1(\Bbb R^d)$-solution of the semi-linear PDE $$-\Delta u + au = b|u|^{\alpha}u\quad (a > 0,\, \alpha > 0,\, b\in \Bbb R)$$ Derive the Pohozaev identity $$(d - 2)\int_{\Bbb R^d} \lvert \nabla_xu\rvert^2\, dx + da\int_{\Bbb R^d}...

Here is this week's POTW: ----- Two players play the following game. The first player selects any integer from 1 to 11 inclusive. The second player adds any positive integer from 1 to 11 inclusive to the number selected by the first player. They continue in this manner alternately. The player...

Here is this week's POTW: ----- Let $S$ be a set of real numbers which is closed under multiplication (that is, if $a$ and $b$ are in $S$, then so is $ab$). Let $T$ and $U$ be disjoint subsets of $S$ whose union is $S$. Given that the product of any three (not necessarily distinct) elements of...

Problem: Compute the trace and norm of $\sqrt{2} + \sqrt{3}$ in the Galois extension $\Bbb Q(\sqrt{2}, \sqrt{3})/\Bbb Q$. ----- 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...

Last week, activity at Popocatepetl began increasing. http://www.upi.com/Top_News/US/2016/03/31/Volcanic-activity-in-Mexicos-Popocatpetl-continues-to-increase/5931459434946/...

Here is this week's POTW: ----- Let $P(x)=a_1\sin x+a_2\sin 2x+\cdots+a_n\sin nx$ where $a_1,\,a_2,\,\cdots,\,a_n$ are real numbers. Suppose that $|P(x)|\le |\sin x|$ for all real $x$, prove that $$|a_1+2a_2+\cdots+na_n|\le 1$$ ----- Remember to read the...

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/1603.08658 The Atoms Of Space, Gravity and the Cosmological...

Here is this week's POTW: ----- Show that if $5$ points are all in, or on, a square of side length $1$, then some pair of them will be no further than $\dfrac{\sqrt{2}}{2}$ apart. ----- Remember to read the...

Here is this week's POTW: ----- Let $\Bbb D$ denote the open unit disc in the complex plane. Given a holomorphic function $f$ on $\Bbb D$, define $$N_p(f) := \sup_{0 < r < 1} \left[\frac{1}{2\pi}\int_{-\pi}^\pi \lvert f(re^{i\theta})\rvert^p\, d\theta\right]^{1/p},\quad 0 < p < \infty$$ The...

Here is this week's POTW: -----Let $ABC$ be a triangle with $AC$ is longer than $AB$. The point $X$ lies on the side $BA$ extended through $A$, and the point $Y$ lies on the side $CA$ in such a way that $BX=CA$ and $CY=BA$. The line $XY$ meets the perpendicular bisector of the side $BC$ at $P$...