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Suppose $K$ is a normal subgroup of a group $G$ with $|G|$ odd. Prove that if $|K| = 5$, then $K$ is contained in the center of $G$.

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Let $X$ be a nonempty set of $n$ elements. Suppose $\emptyset \neq S_0 \subset S_1 \subset S_2\subset \cdots$ be a chain of subsets of $X$. Prove that to every positive integer $k \ge 2\log n$, there corresponds an index $j\in \{1,\ldots, k\}$ such that $$\operatorname{card}(S_j - S_{j-1}) \le \frac{2\operatorname{card}(S_{j-1})}{k}\log n$$

[Note: The logarithm here is the natural logarithm.]

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Remember to read the...

Problem Of The Week # 357 - Apr 22, 2020]]>

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Let $\phi_1$ and $\phi_2$ be harmonic functions on a bounded domain $\Omega \subset \mathbb R^3$ such that \[\phi_1 \frac{\partial \phi_1}{\partial n} + \phi_2 \frac{\partial \phi_2}{\partial n} = \phi_2 \frac{\partial \phi_1}{\partial n} + \phi_1 \frac{\partial \phi_2}{\partial n}\quad \text{on}\quad \partial \Omega\]

Prove that $\phi_1 = \phi_2$ everywhere in $\Omega$. [The operator $\frac{\partial}{\partial n}$ denotes the normal derivative on $\partial...

Problem Of The Week # 356 - Jan 21, 2020]]>

I'm sorry I haven't been around. For several months I've been very sick. I wish you all a Happy New Year! In respect of the MHB equations above, here's a good problem to start the new year:

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Prove the famous result \[\sum_{k = 1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}\] Use any method(s) you like.

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Problem Of The Week # 355 - Dec 31, 2019]]>

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Evaluate the integral $$\int_0^\infty \cos\!\big(x^5\big)\, dx$$

You may express your answer in terms of the Gamma function.

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Show that if $G$ is a finite group, then there are an odd number of elements $g\in G$ for which $g^3 = 1$.

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Suppose $X$ is compact Hausdorff. If $S$ is a subset of $X$ and $O$ is an open set in $X$ with $\overline{S} \subset O$, prove that there is another open set $V$ in $X$ with $\overline{S} \subset V \subset \overline{V} \subset O$.

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Problem Of The Week # 352 - Jul 23, 2019]]>

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Derive a formula for the volume of the solid unit $n$-sphere in $\Bbb R^{n+1}$.

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Prove that the tensor product of finitely many flat modules over a commuative ring is flat.

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Solve the linear system of ODE

\[\begin{align}

\frac{dx}{dt} = 3x + 4y\\

\frac{dy}{dt} = 4x - 3y

\end{align}\]

with initial conditions $x(0) = 1$, $y(0) = 0$.

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$\newcommand{\CC}{\mathbb{C}}$ Let the function $f$ be holomorphic in the open set $G\subset\CC$. Prove that the function $g(z)=\overline{f(\overline{z})}$ is holomorphic in the set $G^{\ast}=\{\overline{z}:z\in G\}$.

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Problem Of The Week # 348 - Jun 05, 2019]]>

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Prove that if $M$ is a metric space, then $M$ is finite if and only if the set $BC(M)$ of bounded continuous functions $f : M \to \Bbb R$ is a finite dimensional real vector space.

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Prove that if $z_1,z_2,z_3$ are noncollinear points in the complex plane, then the medians of the triangle with vertices $z_1,z_2,z_3$ intersect at the point $\frac{1}{3}(z_1+z_2+z_3)$.

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Problem Of The Week # 346 - May 09, 2019]]>

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Solve the PDE $$3\frac{\partial u}{\partial x} + 4 \frac{\partial u}{\partial y} = f(x,y)$$

where $f$ is a smooth function of $x$ and $y$.

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Prove that there is no simple group of order $2n$ where $n$ is an odd number $\ge 3$.

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Evaluate the sum of the series

$$\sum_{n = 0}^\infty \frac{(-1)^n}{2n+1} \sech\left[\frac{(2n+1)\pi}{2}\right]$$

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I was sick for some time, so I had not posted any new problems for either the uni POTW or the grad POTW for a couple weeks. Just this time, there will be a special of two problems posted today for both the university and graduate levels! Here is this week's two POTW:

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1. Suppose $f$ is a continuous, complex-valued function on the complex plane $\Bbb C$ such that $\lim\limits_{\lvert z\rvert \to \infty} \lvert f(z)\rvert = 0$. Prove that $f$ has maximum modulus in $\Bbb C$.

2...

Problem Of The Week # 342 - Mar 05, 2019]]>

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Suppose $f : M \to N$ is a local diffeomorphism between two smooth manifolds. Show that orientability of $N$ implies orientability of $M$.

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Given a nonempty set $A$ of positive integers, let $B$ be a subset of $A$ such that $\dfrac{m}{2}\notin A$ whenever $m\in B$. If $n$ is a positive number, prove that the partitions of $n$ into distinct parts selected from $A$ is equinumerous with the partitions of $n$ into parts selected from $B$.

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Problem Of The Week # 340 - Feb 13, 2019]]>

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A chain of uniform mass is on a table with negligible friction. Length $b$ hangs off the table and length $a$ is on the table. Find the velocity as the last link leaves the table.

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Problem Of The Week # 339 - Feb 05, 2019]]>

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Let $f$ be an analytic function on the half plane $\Omega := \{z\in \Bbb C : \operatorname{Im}(z) \ge 0\}$ such that for some $\alpha > 0$ and $M > 0$, $\lvert z^\alpha f(z)\rvert < M$ for all $z\in \Omega$. Prove that $f$ has integral representation

$$f(z) = \frac{1}{2\pi i} \int_{-\infty}^\infty \frac{f(t)}{t - z}\, dt\quad (\operatorname{Im}(z) > 0)$$

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Remember to read the...

Problem Of The Week # 338 - Feb 01, 2019]]>

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Suppose $q_1,\ldots, q_r$ are the primes in the interval $[1, n]$ where $n$ is an integer $> 1$. Prove

$$\prod_{j = 1}^r \left(1 - \frac{1}{q_j}\right)\sum_{k = 1}^n \frac{1}{k} < 1$$

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Suppose $X$ is a compact Hausdorff space. Let $S$ be a closed subspace of $X$. Show that the one-point compactification of $X - S$ is homeomorphic to the quotient space $X/S$.

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Prove that every commutative ring $A$ with unity in which every proper ideal is prime, is a field.

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Let $f,g : \Bbb R \to \Bbb R$ such that $$f(x) = \sum_{n =1}^{\lfloor x\rfloor} g\left(\frac{x}{n}\right)$$ Show that $$g(x) = \sum_{n = 1}^{\lfloor x\rfloor} \mu(n)\, f\left(\frac{x}{n}\right)$$ where $\mu(n)$ is the MÃ¶bius function.

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Problem Of The Week # 334 - Dec 11, 2018]]>

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If $X$ and $Y$ are independent, standard Cauchy random variables, find the density of the sum $X + Y$.

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Show that, for every compactly supported, smooth, real valued function $f : \Bbb R^3 \to \Bbb R$,

$$\iiint_{\Bbb R^3} \nabla^2\left(\frac{1}{\| \mathbf{x} - \mathbf{y}\|}\right) f(\mathbf{x})\, d\mathbf{x} = -4\pi f(\mathbf{y})$$

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Problem Of The Week # 332 - Nov 27, 2018]]>

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Show that

$$\int_0^\infty \frac{x^\alpha \log x}{x^2 + 1}\, dx = \frac{\pi^2}{4} \frac{\sin(\pi \alpha/2)}{\cos^2(\pi \alpha/2)}\quad (0 < \alpha < 1)$$

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For $n$ a positive integer, let $\phi(n)$ be the number of positive integers not exceeding $n$ and coprime to $n$. Show that every composite number $n \equiv 1\pmod{\phi(n)}$ has at least three distinct prime divisors. (In fact, $n$ would have at least four distinct prime divisors. You can prove this harder result if you like.)

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Remember to read the...

Problem Of The Week # 330 - Oct 30, 2018]]>

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If $p$ is the smallest prime divisor of the order of a finite group $G$, prove that any subgroup of $G$ of index $p$ is normal.

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Let $X$ be a compact Hausdorff space. If $X$ contains a dense, locally compact subspace $S$, show that $S$ is open in $X$.

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Let $\Omega$ be a bounded domain in $\Bbb R^2$. Suppose $u$ is a nonconstant, nonnegative solution of the PDE $\Delta u = mu$ in $\Omega$ where $m : \Omega \to (0,\infty)$ is continuous. Prove that $u$ cannot achieve its maximum in the interior of $\Omega$.

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Remember to read the...

Problem Of The Week # 327 - Oct 09, 2018]]>

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Consider a sequence $f_n : M\to M'$ between two metric spaces $M$ and $M'$, where $n = 1,2,3,\ldots$. Prove that if each $f_n$ is bounded and $f_n$ converges uniformly to $f$, then $f$ is bounded.

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Problem Of The Week # 326 - Oct 03, 2018]]>

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Find a formula for the number of subspaces of a vector space of dimension $n$ over a finite field with $p$ elements.

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Prove that the quotient of a Banach space $X$ by a closed linear subspace $M$ is Banach with respect to the norm $$\|x + M\| := \inf\{\|x + y\|_X : y\in M\}\quad (x\in X)$$

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Show that, for any basis $v_1, v_2 \in \Bbb R^2$, the sum $v_1 \otimes v_2 + v_2 \otimes v_1$ in $\Bbb R^2 \otimes_{\Bbb R} \Bbb R^2$ cannot be reduced to a simple tensor.

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Show that there are infinitely many primes of the form 4k + 1 where $k$ is an integer.

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Prove that in a compact topological space, any decreasing sequence of nonempty closed sets has non-empty intersection.

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Prove the vector identity

$$\nabla(\mathbf{A}\cdot \mathbf{B}) = (\mathbf{A}\cdot \nabla)\mathbf{B} + (\mathbf{B}\cdot \nabla)\mathbf{A} + \mathbf{A}\times (\nabla \times \mathbf{B}) + \mathbf{B}\times (\nabla \times \mathbf{A})$$

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Problem Of The Week # 320 - Aug 25, 2018]]>

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For $0 < x < \pi$, find the sum of the series

$$\sum_{n = 1}^\infty \frac{\sin^2(nx)}{n^2}$$

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Problem Of The Week # 319 - Aug 17, 2018]]>

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Give a proof of the number-theoretic equation $$\sum_{d|n} \phi(d) = n$$ where $\phi(d)$ is the number of positive integers $\le d$ and relatively prime to $d$.

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Find the sum of the series

$$\sum_{n = 1}^\infty \frac{(-1)^{n-1}}{n^4}$$

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Let $K : [0,1]\times [0,1] \to \Bbb R$ be a continuous function such that $\sup\limits_{x\in [0,1]} \int_0^1 |K(x,y)|\, dy \le 1$ and $\sup\limits_{y\in [0,1]} \int_0^1|K(x,y)|\, dx \le 1$. Prove that

$$\int_0^1 \left(\int_0^1 K(x,y)f(y)\, dy\right)^2\, dx \le 1$$ for all continuous functions $f : [0,1]\to \Bbb R$ such that $\int_0^1 f(y)^2\, dy \le 1$.

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Remember to read the...

Problem Of The Week # 316 - Jul 25, 2018]]>

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If $f$ is a function from $\Bbb R$ into a metric space $(X,d)$ such that for some $\gamma > 1$, $d(f(x),f(y)) \le |x - y|^{\gamma}$ for all $x,y\in \Bbb R$, show that $f$ must be constant.

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If $M$ is a complex $m \times n$ matrix of rank $1$, show that $M$ can be written as $\bf{uv^T}$ where $\bf{u}$ is an $m\times 1$ matrix and $\bf{v}$ is an $n\times 1$ matrix.

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If $R$ is a nonzero commutative ring such that the direct sums $R^m$ and $R^n$ are isomorphic, show that $m = n$.

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Suppose $D$ is a compact domain of $\Bbb R^3,$ $F : D \to \Bbb R^3$ is continuous, and $\phi_1, \phi_2 : D \to \Bbb R$ are $C^2$-solutions of the PDE $\nabla^2 \phi = F$. If $\dfrac{\partial \phi_1}{\partial n} = \dfrac{\partial \phi_2}{\partial n}$ on the boundary $\partial D$ and $\phi_1(x_0) = \phi_2(x_0)$ for some $x_0\in \partial D,$ show that $\phi_1 = \phi_2$.

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Remember to read...

Problem Of The Week # 312 - Jun 20, 2018]]>

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Determine the volume of a tetrahedron $ABCD$ if

$$\overline{AB}=\overline{AC}=\overline{AD}=5$$

and

$$\overline{BC}=3, \; \overline{CD}=4,\;\overline{DB}=5.$$

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Prove that, for any two bounded functions $g_1, g_2: \mathbb{R} \to [1, \infty)$, there exist functions $h_1, h_2: \mathbb{R} \to \mathbb{R}$ such that, for every $x \in \mathbb{R},$

$$

\sup_{s \in \mathbb{R}} (g_1(s)^x g_2(s)) = \max_{t \in \mathbb{R}} (x h_1(t) + h_2(t)).

$$

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Problem Of The Week # 310 - May 17, 2018]]>

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Given $n$ points in the plane, any listing (permutation) $p_1, p_2,\dots,p_n$ of them determines the path, along straight segments, from $p_1$ to $p_2$, then from $p_2$ to $p_3,\dots,$ ending with the segment from $p_{n-1}$ to $p_n$. Show that the shortest such broken-line path does not cross itself.

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Problem Of The Week # 309 - May 08, 2018]]>

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Find all values of $\alpha$ for which the curves $y = \alpha x^2 +\alpha x + \dfrac{1}{24}$ and $x = \alpha y^2 + \alpha y + \dfrac{1}{24}$ are tangent to each other.

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John tosses $6$ fair coins, and Mary tosses $5$ fair coins. What is the probability that John gets more "heads" than Mary?

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What is the largest possible radius of a circle contained in a $4$-dimensional hypercube of side length $1?$

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Solve for $x, y, z$ (in terms of $a, r, s, t$):

\begin{align*}

yz&=a(y+z)+r\\

zx&=a(z+x)+s\\

xy&=a(x+y)+t.

\end{align*}

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A round-robin tournament of $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?

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Remember to read the...

Problem Of The Week # 304 - Mar 30, 2018]]>

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Four distinct lines $L_1, \, L_2,\,L_3,\,L_4$ are given in the plane, with $L_1$ and $L_2$ respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.

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Problem Of The Week # 303 - Mar 23, 2018]]>

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Let $k$ be a positive integer. Suppose that the integers $1, 2, 3,\dots, 3k+1$ are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by $3?$ Your answer should be in closed form, but may include factorials.

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Remember to read the...

Problem Of The Week # 302 - Mar 16, 2018]]>

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Let $T_1$ and $T_2$ be two acute-angled triangles with respective side lengths $a_1, b_1, c_1$ and $a_2, b_2, c_2$, areas $\Delta_1$ and $\Delta_2$, circumradii $R_1$ and $R_2$ and inradii $r_1$ and $r_2$. Show that, if $a_1\ge a_2, \; b_1\ge b_2, \; c_1\ge c_2,$ then $\Delta_1\ge\Delta_2$ and $R_1\ge R_2$, but it is not necessarily true that $r_1\ge r_2$.

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Remember to read the...

Problem Of The Week # 301 - Mar 06, 2018]]>

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Find a real number $c$ and a positive number $L$ for which

$$\lim_{r\to\infty} \frac{\displaystyle r^c \int_0^{\pi/2} x^r \sin(x) \,dx}{\displaystyle \int_0^{\pi/2} x^r \cos(x) \,dx} = L.$$

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Problem Of The Week # 300 - Feb 26, 2018]]>

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Seventy-five coplanar points are given, no three collinear. Prove that, of all the triangles which can be drawn with these points as vertices, not more than seventy percent are acute-angled.

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Problem Of The Week # 299 - Feb 03, 2018]]>

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Let $a_0=1$, $a_1=2$, and $a_n=4a_{n-1}-a_{n-2}$ for $n\geq 2$. Find an odd prime factor of $a_{2015}$.

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For which nonnegative integers $n$ and $k$ is

$$ (k+1)^n+(k+2)^n+(k+3)^n+(k+4)^n+(k+5)^n $$

divisible by $5?$

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Let $p$ be a prime number. Let $h(x)$ be a polynomial with integer coefficients such that $h(0), \, h(1), \, \dots, \, h(p^2-1)$ are distinct modulo $p^2$. Show that $h(0), \, h(1), \, \dots, \, h(p^3-1)$ are distinct modulo $p^3$.

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Problem Of The Week # 296 - Jan 12, 2018]]>

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Tetrahedron $OABC$ is such that lines $OA, OB,$ and $OC$ are mutually perpendicular. Prove that triangle $ABC$ is not a right-angled triangle.

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For any continuous real-valued function $f$ defined on the interval $[0,1]$, let

\begin{gather*}

\mu(f) = \int_0^1 f(x)\,dx, \,

\mathrm{Var}(f) = \int_0^1 (f(x) - \mu(f))^2\,dx, \\

M(f) = \max_{0 \leq x \leq 1} \left| f(x) \right|.

\end{gather*}

Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1]$, then

\[

\mathrm{Var}(fg) \leq 2 \mathrm{Var}(f) M(g)^2 + 2 \mathrm{Var}(g) M(f)^2.

\]

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Remember to read the...

Problem Of The Week # 294 - Dec 27, 2017]]>

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What is the greatest common divisor of the set of numbers $\left\{16^n+10n-1 \;|\; n=1, 2, 3, \dots\right\}?$

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Let $a_0 = \dfrac52$ and $a_k = a_{k-1}^2 - 2$ for $k \geq 1$. Compute $\displaystyle\prod_{k=0}^\infty \left(1 - \frac{1}{a_k} \right)$ in closed form.

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A novel has 6 chapters. As usual, starting from the first page of the first chapter, the pages of the novel are numbered 1, 2, 3, 4, . . . . Also, each chapter begins on a new page. The last chapter is the longest and the page numbers of its pages add up to 2010. How many pages are there in the first 5 chapters ?

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Problem Of The Week # 291 - Dec 01, 2017]]>

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If $a_0\ge a_1 \ge a_2\ge \cdots\ge a_n\ge 0,$ prove that any root $r$ of the polynomial

$$P(z)\equiv a_0 z^n+a_1 z^{n-1}+\cdots+a_n$$

satisfies $|r|\le 1$; i.e., all the roots lie inside or on the unit circle centered at the origin in the complex plane.

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Remember to read the...

Problem Of The Week # 290 - Nov 24, 2017]]>

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Prove that every nonzero coefficient of the Taylor series of

\[

\left(1 - x + x^2\right) e^x

\]

about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number.

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Problem Of The Week # 289 - Nov 16, 2017]]>

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Let $f$ be a polynomial with positive integer coefficients. Prove that if $n$ is a positive integer, then $f(n)$ divides $f(f(n)+1)$ if and only if $n=1$. Assume $f$ is non-constant.

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For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k}\,)$. Evaluate

\[

\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.

\]

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Show that for each positive integer $n$, all the roots of the polynomial

\[

\sum_{k=0}^n 2^{k(n-k)} x^k

\]

are real numbers.

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For any positive integer $n$, let $\langle n\rangle$ denote the closest integer to $\sqrt{n}$. Evaluate

\[\sum_{n=1}^\infty \frac{2^{\langle n\rangle}+2^{-\langle n\rangle}}{2^n}.\]

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Let $X$ be a topological group; let $A$ be a subgroup of $X$ such that $A$ and $X/A$ are connected. Show that $X$ is connected.

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Evaluate the sum of the alternating series $$\sum\limits_{n = 1}^\infty \frac{(-1)^{n-1}}{n^4}$$

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Let $G$ be a group. If $\theta$ is an automorphism of $G$ and $N \vartriangleleft G$, prove that $\theta(N) \vartriangleleft G$.

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Evaluate the integral $\displaystyle \int_0^{\infty}\frac{dx}{(1+x^2)^{\alpha/2}}$ for $\alpha>1$. Express your answer using Gamma functions, where

$$\Gamma(x) :=\int_{0}^{\infty}t^{x-1}e^{-t} \, dt.$$

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Problem of the Week # 282 - Sep 25, 2017]]>

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Solve the ODE $(y^3+xy^2+y) \, dx + (x^3+x^2y+x) \, dy=0$.

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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 $\displaystyle \int_a^\infty f(x)\, dx \le -A/B$.

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Problem of the Week # 280 - Sep 12, 2017]]>

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If the commutator, $Z$, of two complex $n\times n$ matrices commutes with one of those matrices, must $Z$ be nilpotent?

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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*}

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Problem of the Week # 278 - Aug 29, 2017]]>

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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 each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum...

Problem of the Week # 277 - Aug 22, 2017]]>

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Can an arc of a parabola inside a circle of radius 1 have a length greater than 4?

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Prove that there are unique positive integers $a$, $n$ such that $a^{n+1}-(a+1)^n=2001$.

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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 $RST$.

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Problem of the Week # 274 - Aug 01, 2017]]>

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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?

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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 answer as a rational function of $n$.

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Problem of the Week # 272 - Jul 18, 2017]]>

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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 $a,b\in S$.

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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.

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Problem of the Week # 270 - Jul 03, 2017]]>

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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 $N$ for which $S_N=S_0\cup\{N+a: a\in S_0\}$.

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Problem of the Week # 269 - Jun 28, 2017]]>

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Let $f(x)$ be a continuous function such that $f(2x^2-1)=2xf(x)$ for all $x$. Show that $f(x)=0$ for $-1\leq x\leq 1$.

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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

\[N_0\leq N_1\leq N_2\leq \cdots \mbox{ and } \lim_{k\to\infty} N_k = 2N.\]

[Editorial clarification: only zeroes in $[0, 1)$ should be counted.]

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Problem of the Week # 267 - Jun 12, 2017]]>

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Prove that the expression

\[ \frac{\gcd(m,n)}{n}\binom{n}{m} \]

is an integer for all pairs of integers $n\geq m\geq 1$.

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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 $j$, $1\leq j\leq N$.

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Problem of the Week # 265 - May 30, 2017]]>

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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 $a_2=0$.

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Problem of the Week # 264 - May 23, 2017]]>

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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 $r^{1/3}$.

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Show that the improper integral

\[ \lim_{B\to\infty}\int_{0}^B \sin(x) \sin\left(x^2\right) \, dx\]

converges.

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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 maximum possible area of the octagon.

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Problem of the Week # 261 - May 01, 2017]]>

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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: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]

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