2017 Definition and 169 Threads

Here is this week's POTW: ----- Assume that $\alpha,\, \beta,\,\gamma$ satisfy $0<\alpha<\beta<\gamma<2\pi$. If $\cos (x+\alpha)+\cos(x+\beta)+\cos(x+\gamma)=0$ for arbitrary $x\in \Bbb{R}$, evaluate $\gamma-\alpha$. ----- Remember to read the...

Here is this week's POTW: ----- Using contour integration, prove $$\int_0^\infty \sin(x^\alpha)\, dx = \sin\left(\frac{\pi}{2\alpha}\right)\,\Gamma\!\left(1 + \frac{1}{\alpha}\right),\quad \alpha > 1.$$----- Remember to read the...

Here is this week's POTW: ----- Evaluate the integral $\displaystyle \int_0^{\infty}\frac{dx}{(1+x^2)^{\alpha/2}}$ for $\alpha>1$. Express your answer using Gamma functions, where $$\Gamma(x) :=\int_{0}^{\infty}t^{x-1}e^{-t} \, dt.$$ ----- Remember to read the...

Here is this week's POTW: ----- The sum of the angles $A$ and $C$ of a convex quadrilateral $ABCD$ is less than $180^\circ$. Prove that $AB\cdot CD+AD \cdot BC <AC(AB+AD)$. ----- Remember to read the...

Hello everyone, I'm currently doing a CSE bachelor (europe) in which I'll focus on physics. Now, I'd like to also put in some extra work to strengthen my focus on physics. This will put me ahead of others but it might also give me the chance of doing a masters in physics. Anyway, currently...

The goal is to create the most beautiful or interesting equation aesthetically (pleasing to the eye). This is not about it's mathematical significance. Get your inner designer on! Each member is allowed to post one equation The equation can be completely made up Must use LaTeX Be creative...

Here is this week's POTW (I will identify the problem source when I post the solution): ----- Solve the ODE $(y^3+xy^2+y) \, dx + (x^3+x^2y+x) \, dy=0$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find...

Hello, The Nobel Prize in Physics is upcoming soon and I wanted to hear your opinions and expectations about some important discoveries or inventions which might end up receiving this most major prize in science. I look forward to hearing your thoughts! Thanks.

Hello, I have heard that the 2017 Nobel Prize in Chemistry is going to be announced in the next few months. Do you know about any interesting and/or useful discoveries that chemistry achieved worth the Nobel Prize in your opinion? Thanks.

Here is this week's POTW: ----- Given positive real numbers $a,\,b,\,c,\,d$ satisfy the equalities $a^2-ad+d^2=b^2+bc+c^2$ and $a^2+b^2=c^2+d^2$, find all possible values of the expression $\dfrac{ab+cd}{ad+bc}$. ----- Remember to read the...

Here is this week's POTW: ----- Let $f : [a,\infty)\to \Bbb R$ be a continuous function that satisfies the inequality $\displaystyle f(x) \le A + B\int_a^x f(t)\, dt$, where $A$ and $B$ are constants with $B < 0$. If $\displaystyle \int_a^\infty f(x)\, dx$ exists, show that $\displaystyle...

The Lasker Prizes (regarded as the US’s most prestigious biomedical research awards) were announced (Science article) (The Lasker Foundation's Awards Announcement) and are interesting to me for two reasons: 1) Michael Hall, 64, of the University of Basel’s Biozentrum in Switzerland...

Here is this week's POTW: ----- Prove that if $n > 0$, an even map between $n$-spheres has even homological degree.----- 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: ----- If the commutator, $Z$, of two complex $n\times n$ matrices commutes with one of those matrices, must $Z$ be nilpotent? ----- 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: ----- In how many ways can we paint 16 seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same color is always odd? ----- Remember to read the...

A quick shot of the sun this morning Nth lower left, East limb lower right 400mm telephoto = x2 teleconverter, 500th sec, f11

Here is this week's POTW: ----- Find all pairs of real numbers $(x,y)$ satisfying the system of equations \begin{align*} \frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2) \\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4). \end{align*} ----- Remember to read the...

Here is this week's POTW: ----- Find all quadruples $(a,\,b,\,c,\,d)$ of real numbers that simultaneously satisfy the following equations: $$\begin{array}{rcr}ab^2+cd^2\hspace{-10px} & = & \hspace{-10px}-6 \\ a^2b+c^2d\hspace{-12px} & = & 0 \\ a^3+c^3\hspace{-10px} & = & 2 \\...

What courses are you taking first semester?

Here is this week's POTW: ----- Let $n$ be an even positive integer. Write the numbers $1,2,\ldots,n^2$ in the squares of an $n\times n$ grid so that the $k$-th row, from left to right, is \[(k-1)n+1,(k-1)n+2,\ldots, (k-1)n+n.\] Color the squares of the grid so that half of the squares in...

corona chromosphere

Here is this week's POTW: ----- If a quadrilateral is circumscribed about a circle, prove that its diagonals and the two chords joining the points of contact of opposite sides are all concurrent. ----- Remember to read the...

Here is this week's POTW: ----- Let $\Bbb D$ be the open unit disc in the complex plane, and let $f$ be a continuous complex function on $\partial\Bbb D$. Consider the function $$F(re^{i\phi}) \,\dot{=}\, \frac1{2\pi}\int_0^{2\pi} f(e^{i\theta})\frac{1-r^2}{1-2r\cos(\theta-\phi) + r^2}\...

Here is this week's POTW: ----- Can an arc of a parabola inside a circle of radius 1 have a length greater than 4? ----- 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: ----- Suppose $$\frac{3}{2}\le x \le 5.$$ Prove that $$2\sqrt{x+1}+\sqrt{2x-3}+\sqrt{15-3x}<2\sqrt{19}$$. ----- 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 there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$. ----- 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 $f$ satisfy $x=f(x)e^{f(x)}$. Calculate $$\int_{0}^{e} f(x)\,dx$$. ----- 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: ----- Show that the tensor algebra of $\Bbb Z/n\Bbb Z$ is isomorphic to $\Bbb Z[x]/(nx)$. ----- 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: ----- Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$. ----- Remember to read...

Here is this week's POTW: ----- Minimize $|x|-|y|$, given $$\log_4{(x+2y)}+\log_4{(x-2y)}=1$$. ----- 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...

Hello MHB Community! :D As I'm sure most of you who live in the U.S. are aware, we will be witness to a solar eclipse on the 21st of August and the path of totality will cut across the continental U.S.: I live in NE Florida, and will experience almost 90% obscurity, but I have family in TN...

Here is this week's POTW: ----- For each integer $m$, consider the polynomial \[P_m(x)=x^4-(2m+4)x^2+(m-2)^2.\] For what values of $m$ is $P_m(x)$ the product of two non-constant polynomials with integer coefficients? ----- Remember to read the...

Here is this week's POTW: ----- Find the integral part of $$\large \sum_{n=1}^{10^9} n^{\small-\dfrac{2}{3}}$$. ----- 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's this week's POTW: ----- Determine the value of the definite integral $$\int_0^\infty \frac{dt}{(1+t^2)t^{1/2}}$$----- 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: ----- You have coins $C_1,C_2,\ldots,C_n$. For each $k$, $C_k$ is biased so that, when tossed, it has probability $\displaystyle \frac{1}{2k+1}$ of falling heads. If the $n$ coins are tossed, what is the probability that the number of heads is odd? Express the...

Here is this week's POTW: ----- On the sides $AC$ and $BC$ of an isosceles right-angled triangle $ABC$, points $D$ and $E$ are chosen such that $|CD|=|CE|$. The perpendiculars from $C$ and $D$ on $AE$ intersect the hypotenuse $AB$ at $L$ and $K$ respectively. Prove that $|LK|=|LB|$. -----...

Here is this week's POTW: ----- Consider a set $S$ and a binary operation $*$, i.e., for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b\in S$. ----- Remember to read the...

AR2665 ... largest spot group for some time Canon 6D, 800mm, f11, 125th, ISO100 ( the 800mm is a 100-400mm L lens with a x2 teleconverter) With my eyesight going downhill, I have really been struggling of late to be able to get sharp manual focus Dave

Here is this week's POTW: ----- Compute the smallest value $k$ such that for all $n>k$, the cubic function $x^3+x^2+nx+9$ has exactly one real root.----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out...

Here is this week's POTW: ----- Let $B$ be a set of more than $2^{n+1}/n$ distinct points with coordinates of the form $(\pm 1,\pm 1,\ldots,\pm 1)$ in $n$-dimensional space with $n\geq 3$. Show that there are three distinct points in $B$ which are the vertices of an equilateral triangle...

It's that time again! Summer Games Done Quick is starting tomorrow! What is SGDQ you ask? I'll let wiki do the talking: Games Done Quick is a semiannual video game speedrun charity marathon held in the United States, originally organized by the Speed Demos Archive and Speedruns Live...

Here is this week's POTW: ----- How many digits are there in (decimal representation of) the integer $19^{9^9}$? ----- 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 $S_0$ be a finite set of positive integers. We define finite sets $S_1,S_2,\ldots$ of positive integers as follows: the integer $a$ is in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$...

This new virus used multiple means of attack to infect machines on a network. The initial attack came from a legitimate software updater program: http://www.zdnet.com/article/microsoft-petya-ransomware-attacks-were-spread-by-hacked-software-updater/

Here is this week's POTW: ----- Let $(X_n)$ be a sequence of $L^1$ random variables on a probability space $(\Omega, \Bbb P)$. Let $f$ be a continuous, nondecreasing function from $[0,\infty)$ onto itself such that 1. $\Bbb E[f(|X_n|)]$ is uniformly bounded 2. $\dfrac{f(x)}{x}\to \infty$ as...

Here is this week's POTW: ----- Evaluate $$\sum_{i=1}^{1995}\dfrac{1}{f(i)}$$, given that $$f(k)$$ be the integer closest to $$\sqrt[4]{k}$$. ----- 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 $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\leq x\leq 1$. ----- 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: ----- Determine $a^2+b^2+c^2+d^2$ if...

Here is this week's POTW: ----- Let $f : M \to M$ be a self-map of a smooth manifold $M$. Prove that the graph of $f$ is transversal to the diagonal of $M$ if and only if the fixed points of $M$ are nondegenerate, i.e., for all fixed points $p$, $+1$ is not an eigenvalue of $df_p$. -----...

Here is this week's POTW: ----- Let $\displaystyle f(t)=\sum_{j=1}^N a_j \sin(2\pi jt)$, where each $a_j$ is real and $a_N$ is not equal to 0. Let $N_k$ denote the number of zeroes (including multiplicities) of $\dfrac{d^k f}{dt^k}$. Prove that \[N_0\leq N_1\leq N_2\leq \cdots \mbox{ and }...