2016 Definition and 207 Threads

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 $P$...

http://finance.yahoo.com/news/volcano-erupts-southwest-alaska-sends-ash-20-000-032041050.html https://www.avo.alaska.edu/activity/Pavlof.php https://www.avo.alaska.edu/activity/report.php?id=82221&mode=hans&type=3 Some recent history - http://volcano.si.edu/volcano.cfm?vn=312030...

Hi everyone. On March 11th 2016, there were reports of an alleged coolant leak at the kakrapar atomic power station. The reactor is an Indian PHWR. Articles at the time suggested the first trip on the reactor was high containment pressure, which signaled to me some sort of loss of coolant...

Here is this week's POTW: ----- In honor of Opalg's http://mathhelpboards.com/geometry-11/ask-height-icosahedron-18151.html, here is another problem involving icosahedrons: On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39$...

Euge is feeling under the weather so I'm filling in for this week. Huge thanks to him for his hard work keeping on top of these every week! Problem: Prove that in a second countable topological space, every collection of disjoint open sets is countable. ----- Remember to read the...

Here is this week's POTW: ----- Suppose $a$ is an integer that is a sum of squares of three positive integers. Prove that $a^2$ is also a sum of squares of three positive integers. ----- Remember to read the...

Has anybody received acceptance emails from DOE yet? I applied to LLNL and ORNL, and my application still says pending review.

Here is this week's POTW: ----- A repunit is a positive integer whose digits in base 10 are all ones. Find all polynomials $f$ with real coefficients such that if $n$ is a repunit, then so is $f(n)$. ----- Remember to read the...

Here is this week's POTW: ----- Let $\Lambda :\Bbb R \to \Bbb R$ be a mapping such that for all bounded measurable mappings $f : [0,1]\to \Bbb R$, $$\Lambda\left(\int_0^1 f(x)\, dx\right) \le \int_0^1 \Lambda(f(x))\, dx.$$ Show that $\Lambda$ is a convex mapping. ----- Remember to read the...

Here is this week's POTW: ----- Let $a$ be a positive number and we show decimal part of the $a$ with $\{a\}$. For a positive number $x$ with $\sqrt{2}<x<\sqrt{3}$ such that \{\frac{1}{x}\}=\{x^2\}, evaluate $x^{16}-987x.$ ----- Remember to read the...

Here is this week's POTW: ----- Let $f(x)$ be a continuous real-valued function defined on the interval $[0,1]$. Show that $$\int_0^1 \int_0^1 |f(x)+f(y)| \, dx \, dy \ge \int_0^1 |f(x)| \, dx.$$ ----- Remember to read the...

Here is this week's POTW: ----- Let $D$ be a division ring. Show that if $D$ is not simultaneously of characteristic two and commutative, then $D$ is generated by products of perfect squares. ----- Remember to read the...

Here is this week's POTW: ----- Evaluate \lim_{{x}\to{0}}\left(\frac{1+\tan x}{1+\sin x}\right)^{\frac{1}{x^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 POTW: ----- Evaluate the integral $$\int_0^1 \frac{t^{-1/2}(1-t)^{-1/4}}{(16 - 7t)^{5/4}}\, dt.$$----- 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: ----- Show that if the vertex set of a connected graph $G=(V,E)$ is partitioned into two nonempty sets $X$ and $Y$, the disconnecting set $F=(X,Y)$ consisting of all edges of $G$ joining vertices in $X$ and vertices in $Y$ is a cut set if the subgraph $G'=(V,E-F)$ has...

Here is this week's POTW: ----- Solve for the real solution(s) for the equation below: \sqrt[3]{2+3x^2-15x^3}-x=1+71\sqrt{16x^3+3x-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...

How's everyone doing? Also, anyone hear back from the schools they applied to?

Hi, has anyone applied to Perimeter Institute 2016 Undergraduate summer program? I just wonder how competitive this program is.

I have applied to the SULI program for this summer (2016) and I haven't gotten a response back. I applied to LBNL and NREL. From previous years, it seems like now is the time that the first round of acceptances is sent out. Has anyone else heard anything? Thanks.

Here is this week's POTW: ----- Show that the geodesic on the surface of a right circular cylinder is a segment of a helix. ----- 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 abelian group $G$ is called divisible if for every $n\in \Bbb N$ and $g\in G$, there exists $x\in G$ such that $nx = g$. Show that for abelian groups $G$, $G$ is injective if and only if $G$ is divisible. ----- Remember to read the...

Here is this week's POTW: ----- Cyclic pentagon $PQRST$ has side lengths $PQ=QR=5,\,RS=ST=12$ and $PT=14$. Determine the radius of its circumcircle. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...

Going off last years thread, 2016 math reu acceptances (or general discussion) Has anyone heard back yet?

Here is this week's POTW: ----- Twin primes are primes that differ by 2. Find the fraction of the primes up to ten million that are twin primes. For example, 3 and 5 are twin primes, as well as 5 and 7, and all the primes less than 10 are 2, 3, 5, and 7. Hence the fraction of twin primes in...

Here is this week's POTW: ----- Let $(X,\mathcal{M},\mu)$ be a positive measure space. The essential range of a measurable function $f : X \to \Bbb C$ consists of all complex numbers $c$ such that for every $\epsilon > 0$, $\mu(\{x\in X : \lvert f(x) - c\rvert < \epsilon\}) > 0$. Prove that $f$...

Here is this week's POTW: ----- Evaluate $\sin^2 x^\circ+\sin^2 (x+1)^\circ+\sin^2 (x+2)^\circ+\cdots+\sin^2 (x+179)^\circ$. ----- 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...

Hello everybody! I want to know where my fellow PF'ers stand on the upcoming US election. I found this neat website which calculates the candidate that would theoretically be your best choice: http://www.isidewith.com So please fill out the quiz and indicate your outcome on the poll, not the...

Does anyone know where we can watch the live stream of the conference held at Washington DC on 11 feb 2016 10:00 am

Here is this week's POTW: ----- Find all real numbers for $a$ and $b$ that satisfy the system of equations below: \frac{1}{a}+\frac{1}{2b}=(a^2+3b^2)(3a^2+b^2) \frac{1}{a}-\frac{1}{2b}=2(b^4-a^4) ----- Remember to read the...

Here is this week's POTW: ----- Why can't an infinite group that has a subgroup of finite index $> 1$ be simple? ----- 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 palindromic number is one that is the same when written backwards (base 10) like 197791. Find all the palindromic primes that have an even number of digits. ----- Remember to read the...

Here is this week's POTW: ----- Show that if a metric space $X$ has the property that every real-valued continuous function on $X$ is bounded, then $X$ is compact. ----- Remember to read the...

Here is this week's POTW: ----- Find a real number $c$ and a positive number $L$ for which $$\displaystyle\lim_{r\to\infty} \frac{\displaystyle r^c \int_0^{\pi/2} x^r \sin(x) \,dx}{\displaystyle \int_0^{\pi/2} x^r \cos(x) \,dx} = L.$$ ----- Remember to read the...

Here is this week's POTW: ----- If $x^2+x+1=0$, compute the numerical value of \left(x+\frac{1}{x}\right)^2+\left(x^2+\frac{1}{x^2}\right)^2+\left(x^3+\frac{1}{x^3}\right)^2+\cdots+\left(x^{29}+\frac{1}{x^{29}}\right)^2. ----- Remember to read the...

Here is this week's POTW: ----- Construct a formal proof of validity for the following argument: \begin{align*} A &\to B \\ \therefore A &\to (B \lor C). \end{align*} ----- Remember to read the...

Here is this week's POTW: ----- Explain why a uniformly continuous, Lebesgue integrable, real-valued function on $[0,\infty)$ has limit $0$ at infinity. ----- 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: ----- You're required to color the squares of a $2\times 2016$ grid in 3 colors, namely yellow, purple and green. How many ways can you color the squares such that no two squares of the same color share an edge? ----- Remember to read the...

I know it's really early in the year, but I want to have some opinion of my POTENTIAL fall 2016 schedule. No harsh comments.Taking calc 1 over the summer. I've been preparing since of December as I've been practicing my algebra skills. I'm taking Precalc this semester! There will be two...

How the people of Flint, Mich., ended up with contaminated drinking water http://news.yahoo.com/how-the-people-of-flint--michigan-ended-up-with-contaminated-drinking-water-043602916.html As far as I can tell, it was a local/municipal decision, so why it's a problem, or crisis, for the...

Does anyone have any idea of participating in Google science fair 2016? Is anyone interested?

Here is this week's POTW: ----- Three numbers from the first 99 positive integers are chosen at random (with repetitions allowed). What is the probability that the sum is divisible by 3? ----- Remember to read the...

Here is this week's POTW: ----- Let $a,\,b,\,c,\,d$ be real numbers such that the equation $x^5-20x^4+ax^3+bx^2+cx+d=0$ has real roots only. Find the biggest possible value of $a$. ----- Remember to read the...

Prove the inequality \sqrt{\frac{1\cdot 2}{3^2}}+\sqrt{\frac{2\cdot 3}{5^2}}+\sqrt{\frac{3\cdot 4}{7^2}}+\cdots+\sqrt{\frac{4032\cdot 4033}{8065^2}}<2016

Here is this week's POTW: ----- Let $M$ be a closed, connected Riemannian manifold. Prove that every $C^\infty(M;\Bbb R)$-solution of the PDE $$f\Delta f = -\alpha |\nabla f|^2\quad (\alpha\in \Bbb R)$$ is constant. ----- Remember to read the...

Here is this week's POTW: ----- Let \begin{align*} \sigma_1&=\begin{bmatrix}0&1\\1&0\end{bmatrix} \\ \sigma_2&=\begin{bmatrix}0&-i\\i&0\end{bmatrix} \\ \sigma_3&=\begin{bmatrix}1&0\\0&-1\end{bmatrix} \end{align*} be the three Pauli spin matrices. Let $\vec{v}$ be a real, three-dimensional...

Here is this week's POTW: ----- Evaluate the following limit: \large \lim_{x \rightarrow 2} \left(\sqrt[6]{\frac{6x^4-12x^3-x+2}{x+2}} \times \frac{\sqrt[3]{x^3-\sqrt {x^2+60}}}{\sqrt{x^2-\sqrt[3] {x^2+60}}} \right) ----- Remember to read the...

The 2016 question: WHAT DO YOU CONSIDER THE MOST INTERESTING RECENT [SCIENTIFIC] NEWS? WHAT MAKES IT IMPORTANT? Includes some interesting physics related essays. A couple of them are on the theme that the big news is what was not found: Paul J. Steinhardt The Big Bang Cannot Be What We Thought...

Here is this week's POTW: ----- A set $S$ has two binary operations $\#$ and $*$ on it, and the following axioms hold: There is an element $z$ in $S$ such that $z\# s=s$ for all $s\in S$. For all $s,t,u\in S$ if $s\# u=t\#u$ then $s=t$. For all $s,t\in S$ if $z*s=z*t$ then $s=t$. For all...

Here is this week's POTW: -----Let $l$ be a circle with center $O$ and let $AB$ be a chord of $l$ whose midpoint $M$ is distinct from $O$. The ray from $O$ through $M$ meets $l$ again at $R$. Let $P$ be a point on the minor arc $AR$ of $l$, let $PM$ meet $l$ again at $Q$, and let $AB$ meets...

Lindsey Graham is a Senator who ran for president. Recently, however, he has decided to drop out of the race. The problem I see with this video is that he doesn't explain why he is leaving the race. He only says that he's leaving the race and talks about the military. I personally think that...