2017 Definition and 169 Threads

Here is this week's POTW: ----- Let $PQR$ be a triangle such that $PQ=3,\,QR=4$ and $PR=5$. Let $X$ be a point in the triangle. Find the minimal possible values of $PX^2+QX^2+RX^2$. ----- Remember to read the...

Here is this week's POTW: ----- Prove that the expression \[ \frac{\gcd(m,n)}{n}\binom{n}{m} \] is an integer for all pairs of integers $n\geq m\geq 1$. ----- 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: ----- Let $a_j,b_j,c_j$ be integers for $1\leq j\leq N$. Assume for each $j$, at least one of $a_j,b_j,c_j$ is odd. Show that there exist integers $r$, $s$, $t$ such that $ra_j+sb_j+tc_j$ is odd for at least $4N/7$ values of $j$, $1\leq j\leq N$. ----- Remember to...

Here is this week's POTW: ----- Find a triple of rational numbers $(x,\,y,\,z)$ such that $$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{x}+\sqrt[3]{y}+\sqrt[3]{z}$$. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to...

Here is this week's POTW: ----- Consider a strictly increasing sequence of natural numbers $(n_k)_{k = 1}^\infty$, and suppose $X$ is the subset of $[0,2\pi]$ consisting of all $x$ such that the sequence $(\sin(n_k x))_{k = 1}^\infty$ is convergent. Prove $X$ has Lebesgue measure zero.-----...

"Stable beams" has been declared 30 minutes ago. Similar to 2016, the initial collision rate is low (0.2% the design rate). The machine operators have to check that everything works and nothing presents a danger to the machine before more protons can be filled in. It will probably take a few...

Here is this week's POTW: ----- Let $f(x)$ be a polynomial with integer coefficients. Define a sequence $a_0,a_1,\ldots$ of integers such that $a_0=0$ and $a_{n+1}=f(a_n)$ for all $n\geq 0$. Prove that if there exists a positive integer $m$ for which $a_m=0$ then either $a_1=0$ or $a_2=0$...

Here is this week's POTW: ----- Without using a calculator, evaluate $${2000 \choose 2}+{2000 \choose 5}+{2000 \choose 8}+\cdots+{2000 \choose 2000}.$$ ----- 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: ----- Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$. ----- Remember to read the...

Here is this week's POTW: ----- Compute $$\int_{0}^{\pi} \dfrac{2\sin x+3\cos x-3}{13\cos x-5} \,dx$$. ----- 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: ----- Suppose $Z$ is a standard Gaussian random variable. Prove $\Bbb P(\lvert Z\rvert \ge z) = O[\exp(-z^2/2)]$ as $z\to \infty$.----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to...

Here is this week's POTW: ----- Show that the improper integral \[ \lim_{B\to\infty}\int_{0}^B \sin(x) \sin\left(x^2\right) \, dx\] converges. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...

Here is this week's POTW: ----- Acute triangle $ABC$ has $\angle BAC <45^\circ$. Point $D$ lies in the interior of triangle $ABC$ such that $BD=CD$ and $\angle BDC=4\angle BAC$. Point $E$ is the reflection of $C$ across line $AB$, and point $F$ is the reflection of $B$ across line $AC$...

Here is this week's POTW: ----- The octagon $P_1P_2P_3P_4P_5P_6P_7P_8$ is inscribed in a circle, with the vertices around the circumference in the given order. Given that the polygon $P_1P_3P_5P_7$ is a square of area 5, and the polygon $P_2P_4P_6P_8$ is a rectangle of area 4, find the...

Here is this week's POTW: ----- Prove that $$\int_{-\pi}^{\pi}\ln\lvert 1 - e^{it}\rvert\, dt = 0$$ ----- 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 exist infinitely many integers $n$ such that $n,n+1,n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.] ----- Remember to read the...

Hi MHB, sorry for missing one week of high school's POTW, I guess I can make it up by posting two POTWs this week.(Blush) Here is this week's another POTW: ----- Find all values of $k$ such that the equation $\left(\dfrac{1}{x+k}+\dfrac{k}{x-k}-\dfrac{2k}{k^2-x^2}\right)(|x-k|-k)=0$ has...

Here is this week's POTW: ----- In a triangle $ABC$, $\tan A:\tan B: \tan C=1:2:3$. Find $\dfrac{AC}{AB}$. ----- 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...

Time to think about what you can personally do to be a good steward of our home Earth. What activities are you doing today? I was hoping I could attend a sister science march in my city, but alas I have a wedding to attend. Instead this morning I will pick up trash I find in my community during...

Is there a way to do tables in the forum implementation of Latex? ( I asked this question in 2012 and the "tabular" environment wasn't implemented.)

There is a neat site called Kaggle that is home to lots of data science info and the place of very featured competitions with large cash prizes. The goal of this year's competition was to create a model to detect lung cancer. It just wrapped up last week and the results are being verified right...

Here is this week's POTW: ----- Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^\infty x_j^2$, given that $x_0,x_1,\ldots$ are positive numbers for which $\displaystyle\sum_{j=0}^\infty x_j=A$? ----- Remember to read the...

$ ( 2 * 4 * 6 * 8 \cdots * 2016) - ( 1 * 3 * 5 * 7 \cdots * 2015)$ is divided by 2017 what is the remainder

Here is this week's POTW: ----- Let $f : \Bbb S^1\subset \Bbb C \to \Bbb C$ be a continuous map. Show that if $f$ is continuously differentiable on $\Bbb S^1$, then its Fourier coefficient sequence $\{\hat{f}_n\}_{n\in \Bbb Z}$ belongs to $\ell^1(\Bbb Z)$.----- Remember to read the...

Here is this week's POTW, shamefully late. I can only say I will promise to do better in the next few weeks, and even try to catch up with the missed week: ----- For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times...

Here is this week's POTW: ----- Suppose $\tan A$ and $\tan B$ are the roots of $x^2+\pi x+\sqrt{2}=0$. Evaluate $\sin^2 (A+B) +\pi\sin (A+B)\cos (A+B) +\sqrt{2}\cos^2 (A+B)$ ----- Remember to read the...

Sorry for being late in coming to do the POTW, I just didn't feel well last week due to the prolonged fever, flu, sore throat and cough. To help make up for being late, I will present to you an intriguing problem which I truly hope you are going to have lots of fun solving the problem! Without...

Here is this week's POTW: ----- Prove that if $\Bbb R$ is homeomorphic to a cartesian product $A\times B$, then either $A$ or $B$ is a singleton. ----- Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how...

Here is this week's POTW: ----- Let $f$ be a real function with a continuous third derivative such that $f(x),f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$. ----- Remember to read the...

52nd Rencontres de Moriond EW 2017 présidé par Lydia Iconomidou-Fayard (LAL), Jean Marie Frere (ULB Brussels) has released results based on 36 fb-1 data @ 13 TEV they compare predictions from SUSY i.e squarks gluinos stops etc, with SM all results consistent with SM 95% No supersymmetry...

Here is this week's POTW: ----- Which number, $X$ or $Y$, is larger? $X=\dfrac{1}{2016}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2016}\right)$ $Y=\dfrac{1}{2017}\left(1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{2017}\right)$ ----- Remember to read the...

Here is this week's POTW: ----- Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let \[S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,\] where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate \[\lim_{(x,y)\rightarrow (1,1)...

Today is the bi...g day. Happy pi day everyone.

Here is this week's POTW: ----- Find all triples $(a,\,b,\,c)$ of real numbers that satisfy $a^2+b^2+c^2+1=ab+bc+ca+|a-2b+c|$. ----- 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 $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$, where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots. -----...

It seems that the March for Science is getting more traction and endorsements. The APS just released a statement endorsing this event. @Greg Bernhardt : Is PhysicsForums participating? :) Zz.

Here is this week's POTW: ----- Consider the Lebesgue space $L^1(\Bbb R)$ as an algebra with product given by convolution. Prove that $L^1(\Bbb R)$ is isomorphic as an algebra to an ideal in the algebra $M(\Bbb R)$ of complex Borel measures on $\Bbb R$, and identify the ideal. Note the product...

Here is this week's POTW: ----- Let $ABC$ be a triangle with centroid $G$ and circumcenter $O$. Prove that if $BC$ is its largest side, then $G$ lies in the interior of the circle with diameter $OA$. ----- Remember to read the...

Here is this week's POTW: ----- Right triangle $ABC$ has right angle at $C$ and $\angle BAC =\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE = \theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate...

Here is this week's POTW: ----- Let $(\mathcal{C}, \partial)$ be a chain complex of abelian groups. Suppose $f, g : \mathcal{C} \to \mathcal{C}$ are homotopic chain maps. Construct an explicit chain homotopy between the $n$-fold compositions $f^n$ and $g^n$.----- Remember to read the...

The AAPT Winter Meeting in Atlanta ( https://www.aapt.org/Conferences/wm2017/ ) is going on now. Unfortunately, this meeting was scheduled for now [in the middle of the semester], rather than its usual time in January [during my break]. So, I wasn't able to make it. However, the website for the...

Here is this week's POTW: ----- Find the exact value for the real root of the equation $x^3+3x-2=0$. ----- 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...

Please post results or questions on this Math Reu season. See last years discussionhttps://www.physicsforums.com/threads/math-reus-2016.858490/

I feel like it's about time we started this thread once again. Has anyone heard back from the schools they applied to? Furthermore, does anyone have questions about the REUs they have been accepted into?

Here is this week's POTW: ----- Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree $2017,$ then \[|p(0)|\leq C \int_{-1}^1 |p(x)|\,dx.\] ----- Remember to read the...

Here is this week's POTW: ----- Given that $\sqrt{x^3+1648}-\sqrt{4949-x^3}=75$ for $x\in\Bbb{N}$. Find $x$. ----- 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: ----- Consider the normed space $\mathcal{M}(X)$ of all complex regular Borel measures on a locally compact Hausdorff space $X$, with total variation norm $\|\mu\| := \lvert \mu\rvert (X)$, for all $\mu\in \mathcal{M}(X)$. Prove that $\mathcal{M}(X)$ is a Banach...

Here is this week's POTW: ----- Let $G$ be a group. If $\theta$ is an automorphism of $G$ and $N \vartriangleleft G$, prove that $\theta(N) \vartriangleleft G$. ----- Remember to read the...

Here is this week's POTW: ----- Sum the series \[\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.\] ----- 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: ----- Maximize $\sqrt{x^4-3x^2-6x+13}-\sqrt{x^4-x^2+1}$ for all $x\in \Bbb{R}$. ----- 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...