American Mathematical Monthly Problem - Volume 128, 2021

Proposed by Michael Elgersma, Plymouth, MN, and James R. Roche, Ellicott City, MD. Two weighted $m$-sided dice have faces labeled with the integers 1 to $m$. The first die shows the integer $i$ with probability $p_{i}$, while the second die shows the integer $i$ with probability $r_{i}$. Alice rolls the two dice and sums the resulting integers; Bob then independently does the same.
a) For each $m$ with $m \geq 2$, find the probability vectors $\left(p_{1}, \ldots, p_{m}\right)$ and $\left(r_{1}, \ldots, r_{m}\right)$ that minimize the probability that Alice's sum equals Bob's sum.
b) Generalize to $n$ dice, with $n \geq 3$.
Proposed by Cherng-tiao Perng, Norfolk State University, Norfolk, VA. Let $A B C$ be a triangle, with $D$ and $E$ on $A B$ and $A C$, respectively. For a point $F$ in the plane, let $D F$ intersect $B C$ at $G$ and let $E F$ intersect $B C$ at $H$. Furthermore, let $A F$ intersect $B C$ at $I$, let $D H$ intersect $E G$ at $J$, and let $B E$ intersect $C D$ at $K$. Prove that $I, J$, and $K$ are collinear.
Proposed by Pakawut Jiradilok, Massachusetts Institute of Technology, Cambridge, MA, and Wijit Yangjit, University of Michigan, Ann Arbor, MI. Let $\Gamma$ denote the gamma function, defined by $\Gamma(x)=\int_{0}^{\infty} e^{-t} t^{x-1} d t$ for $x>0$.
a) Prove that $\lceil\Gamma(1 / n)\rceil=n$ for every positive integer $n$, where $\lceil y\rceil$ denotes the smallest integer greater than or equal to $y$.
b) Find the smallest constant $c$ such that $\Gamma(1 / n) \geq n-c$ for every positive integer $n$.
Proposed by Jovan Vukmirovic, Belgrade, Serbia. Let $x_{1}, x_{2}$, and $x_{3}$ be real numbers, and define $x_{n}$ for $n \geq 4$ recursively by $x_{n}=\max \left\{x_{n-3}, x_{n-1}\right\}-x_{n-2}$. Show that the sequence $x_{1}, x_{2}, \ldots$ is either convergent or eventually periodic, and find all triples $\left(x_{1}, x_{2}, x_{3}\right)$ for which it is convergent.
Proposed by Gregory Galperin, Eastern Illinois University, Charleston, IL, and Yury J. Ionin, Central Michigan University, Mount Pleasant, MI. Prove that for any integer $n$ with $n \geq 3$ there exist infinitely many pairs ( $A, B)$ such that $A$ is a set of $n$ consecutive positive integers, $B$ is a set of fewer than $n$ positive integers, $A$ and $B$ are disjoint, and $\sum_{k \in A} 1 / k=\sum_{k \in B} 1 / k$.
Proposed by Hervé Grandmontagne, Paris, France. Prove $$\int_{0}^{1} \frac{(\ln x)^{2} \ln \left(2 \sqrt{x} /\left(x^{2}+1\right)\right)}{x^{2}-1} d x=2 G^{2},$$ where $G$ is Catalan's constant $\sum_{n=0}^{\infty}(-1)^{n} /(2 n+1)^{2}$.
Proposed by Moubinool Omarjee, Lycée Henri IV, Paris, France. Let $f:[0,1] \rightarrow \mathbb{R}$ be a function that has a continuous second derivative and that satisfies $f(0)=f(1)$ and $\int_{0}^{1} f(x) d x=0$. Prove $$30240\left(\int_{0}^{1} x f(x) d x\right)^{2} \leq \int_{0}^{1}\left(f^{\prime \prime}(x)\right)^{2} d x .$$
Proposed by David Callan, University of Wisconsin, Madison, WI. Let $[n]=$ $\{1, \ldots, n\}$. Given a permutation $\left(\pi_{1}, \ldots, \pi_{n}\right)$ of $[n]$, a right-left minimum occurs at position $i$ if $\pi_{j}>\pi_{i}$ whenever $j>i$, and a small ascent occurs at position $i$ if $\pi_{i+1}=\pi_{i}+1$. Let $A_{n, k}$ denote the set of permutations $\pi$ of $[n]$ with $\pi_{1}=k$ that do not have right-left minima at consecutive positions, and let $B_{n, k}$ denote the set of permutations $\pi$ of $[n]$ with $\pi_{1}=k$ that have no small ascents.
a) Prove $\left|A_{n, k}\right|=\left|B_{n, k}\right|$ for $1 \leq k \leq n$.
b) Prove $\left|A_{n, j}\right|=\left|A_{n, k}\right|$ for $2 \leq j<k \leq n$.
Proposed by George Apostolopoulos, Messolonghi, Greece. For an acute triangle $A B C$ with circumradius $R$ and inradius $r$, prove $$\sec \left(\frac{A-B}{2}\right)+\sec \left(\frac{B-C}{2}\right)+\sec \left(\frac{C-A}{2}\right) \leq \frac{R}{r}+1 .$$
Proposed by Seán Stewart, Bomaderry, Australia. Prove $$\int_{0}^{1} \int_{0}^{1} \frac{1}{\sqrt{x(1-x)} \sqrt{y(1-y)} \sqrt{1-x y}} d x d y=\frac{1}{4 \pi}\left(\int_{0}^{\infty} e^{-t} t^{-3 / 4} d t\right)^{4} .$$
Proposed by C. R. Pranesachar, Indian Institute of Science, Bengaluru, India. Let $n$ and $k$ be positive integers with $1 \leq k \leq(n+1) / 2$. For $1 \leq r \leq n$, let $h(r)$ be the number of $k$-element subsets of $\{1, \ldots, n\}$ that do not contain consecutive elements but that do contain $r$. For example, with $n=7$ and $k=3$, the string $h(1), \ldots, h(7)$ is $6,3,4,4,4,3,6$. Prove
a) $h(r)=h(r+1)$ when $r \in\{k, \ldots, n-k\}$.
b) $h(k-1)=h(k) \pm 1$.
c) $h(r)>h(r+2)$ when $r \in\{1, \ldots, k-2\}$ and $r$ is odd.
d) $h(r)<h(r+2)$ when $r \in\{1, \ldots, k-2\}$ and $r$ is even.
Proposed by Nicolai Osipov, Siberian Federal University, Krasnoyarsk, Russia. Let $p$ be an odd prime, and let $A x^{2}+B x y+C y^{2}$ be a quadratic form with $A, B$, and $C$ in $\mathbb{Z}$ such that $B^{2}-4 A C$ is neither a multiple of $p$ nor a perfect square modulo $p$. Prove that $$\prod_{0<x<y<p}\left(A x^{2}+B x y+C y^{2}\right)$$ is $1$ modulo $p$ if exactly one or all three of $A$, $C$, and $A+B+C$ are perfect squares modulo $p$ and is $-1$ modulo $p$ otherwise.
Proposed by George Stoica, Saint John, $N B$, Canada. Let $a_{0}, a_{1}, \ldots$ be a sequence of real numbers tending to infinity, and let $f: \mathbb{C} \rightarrow \mathbb{C}$ be an entire function satisfying $$\left|f^{(n)}\left(a_{k}\right)\right| \leq e^{-a_{k}}$$ for all nonnegative integers $k$ and $n$. Prove $f(z)=c e^{-z}$ for some constant $c \in \mathbb{C}$ with $|c| \leq 1$.
Proposed by Navid Safaei, Sharif University of Technology, Tehran, Iran. Let $p_{k}$ be the $k$ th prime number, and let $a_{n}=\prod_{k=1}^{n} p_{k}$. Prove that for $n \in \mathbb{N}$ every positive integer less than $a_{n}$ can be expressed as a sum of at most $2 k$ distinct divisors of $a_{n}$.
Proposed by Donald E. Knuth, Stanford University, Stanford, CA. Let $x_{0}=1$ and $x_{n+1}=x_{n}+\left\lfloor x_{n}^{3 / 10}\right\rfloor$ for $n \geq 0$. What are the first 40 decimal digits of $x_{n}$ when $n=10^{100} ?$.
Proposed by Tran Quang Hung, Hanoi, Vietnam. Let $A B C D$ be a convex quadrilateral with $A D=B C$. Let $P$ be the intersection of the diagonals $A C$ and $B D$, and let $K$ and $L$ be the circumcenters of triangles $P A D$ and $P B C$, respectively. Show that the midpoints of segments $A B, C D$, and $K L$ are collinear.
Proposed by David Altizio, University of Illinois, Urbana, IL. Determine all positive integers $r$ such that there exist at least two pairs of positive integers $(m, n)$ satisfying the equation $2^{m}=n !+r$.
Proposed by Yue Liu, Fuzhou University, Fuzhou, China, and Fuzhen Zhang, Nova Southeastern University, Fort Lauderdale, FL. We denote by $A^{*}$ the conjugate transpose of the matrix $A$.
a) Let $x \in \mathbb{C}^{m}$ be a unit column vector. Find the eigenvalues of the $(m+1)$-by- $(m+1)$ matrices $$\left[\begin{array}{cc} x^{*} x & x^{*} \\ x & 0 \end{array}\right] \quad \text { and } \quad\left[\begin{array}{cc} x x^{*} & x \\ x^{*} & 0\end{array}\right]$$ b) More generally, let $X$ be an $m$-by-n complex matrix, and let $\rho$ be any real number. Find the eigenvalues of the $(m+n)$-by- $(m+n)$ matrices $$\left[\begin{array}{cc} X^{*} X & X^{*} \\ X & \rho I_{m} \end{array}\right] \quad \text { and } \quad\left[\begin{array}{cc}X X^{*} & X \\ X^{*} & \rho I_{n}\end{array}\right]$$
Proposed by Ovidiu Furdui and Alina Sîntămărian, Technical University of ClujNapoca, Cluj-Napoca, Romania. Prove $$\sum_{n=1}^{\infty}(-1)^{n} n\left(\frac{1}{4 n}-\ln 2+\sum_{k=n+1}^{2 n} \frac{1}{k}\right)=\frac{\ln 2-1}{8}$$
Proposed by Elena Corobea, Technical College Carol I, Constanţa, Romania. For $n \geq 1$, let $$I_{n}=\int_{0}^{1} \frac{\left(\sum_{k=0}^{n} x^{k} /(2 k+1)\right)^{2022}}{\left(\sum_{k=0}^{n+1} x^{k} /(2 k+1)\right)^{2021}} d x .$$ Let $L=\lim _{n \rightarrow \infty} I_{n}$. Compute $L$ and $\lim _{n \rightarrow \infty} n\left(I_{n}-L\right)$.
Proposed by $M$. L. Glasser, Clarkson University, Potsdam, NY. For $a>0$, evaluate $$\int_{0}^{a} \frac{t}{\sinh t \sqrt{1-\operatorname{csch}^{2} a \cdot \sinh ^{2} t}} d t .$$
Proposed by Rob Pratt, SAS Institute Inc., Cary, NC, Stan Wagon, Macalester College, St. Paul, MN, Douglas B. West, University of Illinois, Urbana, IL, and Piotr Zielinski, Cambridge, MA. A polyomino is a region in the plane with connected interior that is the union of a finite number of squares from a grid of unit squares. For which integers $k$ and $n$ with $4 \leq k \leq n$ does there exist a polyomino $P$ contained entirely within an $n$-by- $n$ grid such that $P$ contains exactly $k$ unit squares in every row and every column of the grid? Clearly such polyominos do not exist when $k=1$ and $n \geq 2$. Nikolai Beluhov noticed that they do not exist when $k=2$ and $n \geq 3$, and his Problem 12137 [2019, 756; 2021, 381], whose solution appears at the end of this column, shows that they do not exist when $k=3$ and $n \geq 5$.
Proposed by Jiahao Chen, Tsinghua University, Beijing, China. Suppose that two circles $\alpha$ and $\beta$, with centers $P$ and $Q$, respectively, intersect orthogonally at $A$ and $B$. Let $C D$ be a diameter of $\beta$ that is exterior to $\alpha$. Let $E$ and $F$ be points on $\alpha$ such that $C E$ and $D F$ are tangent to $\alpha$, with $C$ and $E$ on one side of $P Q$ and $D$ and $F$ on the other side of $P Q$. Let $S$ be the intersection of $C F$ and $Q A$, and let $T$ be the intersection of $D E$ and $Q B$. Prove that $S T$ is parallel to $C D$.
Proposed by Seán Stewart, Bomaderry, Australia. Let $\zeta$ be the Riemann zeta function, defined for $n \geq 2$ by $\zeta(n)=\sum_{k=1}^{\infty} 1 / k^{n}$. Let $H_{n}$ be the $n$th harmonic number, defined by $H_{n}=\sum_{k=1}^{n} 1 / k$. Prove $$\sum_{n=2}^{\infty} \frac{\zeta(n)}{n^{2}}+\sum_{n=2}^{\infty}(-1)^{n} \frac{\zeta(n) H_{n}}{n}=\frac{\pi^{2}}{6} .$$
Proposed by Prathap Kasina Reddy, Bhabha Atomic Research Centre, Mumbai, India. For positive real constants $a, b$, and $c$, prove $$\int_{0}^{\pi} \int_{0}^{\infty} \frac{a}{\pi\left(x^{2}+a^{2}\right)^{3 / 2}} \frac{x}{\sqrt{x^{2}+b^{2}+c^{2}-2 c x \cos \theta}} d x d \theta=\frac{1}{\sqrt{(a+b)^{2}+c^{2}}} .$$
Proposed by Askar Dzhumadil'daev, Almaty, Kazakhstan. Let $n$ be a positive integer, and let $x_{k}$ be a real number for $1 \leq k \leq 2 n$. Let $C$ be the $2 n$-by-2n skew-symmetric matrix with $i, j$-entry $\cos \left(x_{i}-x_{j}\right)$ when $1 \leq i<j \leq 2 n$. Prove $$\operatorname{det}(C)=\cos ^{2}\left(x_{1}-x_{2}+x_{3}-x_{4}+\cdots+x_{2 n-1}-x_{2 n}\right) .$$
Proposed by Florin Stanescu, Serban Cioculescu School, Gaesti, Romania. Prove $$\sum_{k=\lfloor n / 2\rfloor}^{n-1} \sum_{m=1}^{n-k}(-1)^{m-1} \frac{k+m}{k+1}\left(\begin{array}{c}k+1 \\m-1 \end{array}\right) 2^{k-m}=\frac{n}{2}$$ for any positive integer $n$.
Proposed by Dorin Mărghidanu, Colegiul National A. I. Cuza, Corabia, Romania. With $n \geq 4$, let $a_{1}, \ldots, a_{n}$ be the lengths of the sides of a polygon. Prove $$\sqrt{\frac{a_{1}}{-a_{1}+a_{2}+\cdots+a_{n}}}+\sqrt{\frac{a_{2}}{a_{1}-a_{2}+\cdots+a_{n}}}+\cdots+\sqrt{\frac{a_{n}}{a_{1}+a_{2}+\cdots-a_{n}}}>\frac{2 n}{n-1} .$$
Proposed by Roberto Tauraso, Università di Roma "Tor Vergata," Rome, Italy. Each point in the plane is colored either red or blue. Show that for any positive real number $S$, there is a proper convex pentagon of area $S$ all five of whose vertices have the same color. (By a proper convex pentagon we mean a convex pentagon whose internal angles are less than $\pi$.)
Proposed by Nguyen Quang Minh, Saint Joseph's Institution, Singapore. Let $k$, $q$, and $n$ be positive integers with $k \geq 2$, and let $P$ be the set of positive integers less than $k^{n}$ that are not divisible by $k$. Prove $$\sum_{p \in P}\left[\frac{\left\lfloor n-\log _{k} p\right\rfloor}{q}\right\rceil=\left\lfloor\frac{k^{q-1}\left(k^{n-1}-1\right)(k-1)}{k^{q}-1}\right\rfloor+1 .$$
Proposed by Alexandru Gîrban, Constanţa, Romania, and Bogdan D. Suceavă, Fullerton, CA. Let $A B C$ be a triangle, and let $D$ and $E$ be the contact points of the incircle of $A B C$ with the segments $B C$ and $C A$, respectively. Let $M$ be the intersection of the line $D E$ and the line through $A$ parallel to $B C$. Prove that the bisector of $\angle A B C$ passes through the midpoint of $D M$.
Proposed by Cezar Lupu, Texas Tech University, Lubbock, TX, and Tudorel Lupu, Constanta, Romania. Prove $$\sum_{n=0}^{\infty}\left(\frac{(-1)^{n}}{2 n+1} \sum_{k=1}^{n} \frac{1}{n+k}\right)=\frac{3 \pi}{8} \log 2-G$$ where $G$ is Catalan's constant $\sum_{k=0}^{\infty}(-1)^{k} /(2 k+1)^{2}$.
Proposed by Besfort Shala, student, University of Primorska, Koper, Slovenia. Given a positive integer $a_{0}$, define $a_{1}, \ldots, a_{n}$ recursively by $a_{i}=1^{2}+2^{2}+\cdots+a_{i-1}^{2}$ for $i \geq 1$. Is it true that, given any subset $A$ of $\{1, \ldots, n\}$, there is a positive integer $a_{0}$ such that, for $1 \leq i \leq n, 6$ divides $a_{i}$ if and only if $i \in A$ ?
Proposed by Paul Bracken, University of Texas, Edinburg, TX. Prove $$\int_{0}^{1} \frac{\log (1+x) \log (1-x)}{x} d x=-\frac{5}{8} \zeta(3),$$ where $\zeta$ (3) is Apéry's constant $\sum_{n=1}^{\infty} 1 / n^{3}$.
Proposed by Erich Friedman, Stetson University, DeLand, FL, and James Tilley, Bedford Corners, NY. An arrangement of equilateral triangles in the plane is called saturated if the intersection of any two is either empty or is a common vertex and every vertex is shared by exactly two triangles. What is the smallest positive integer $n$ such that there exists a saturated arrangement of $n$ equilateral triangles with integer length sides?
Proposed by Jeffrey $C$. Lagarias, University of Michigan, Ann Arbor, MI. Let $S$ be the set of positive integers $n$ such that $n$ ! is not the sum of three squares. Show that $S$ has bounded gaps, i.e., there is a positive constant $C$ such that for every positive integer $n$, there is an element of $S$ between $n$ and $n+C$.
Proposed by Giuseppe Fera, Vicenza, Italy. A triangle is Heronian if it has integer sides and integer area. A pair of noncongruent Heronian triangles is called a supplementary pair if the triangles have the same perimeter and the same area and some interior angle of one is the supplement of some interior angle of the other. Prove that there are infinitely many supplementary pairs of Heronian triangles.
Proposed by Seán M. Stewart, Bomaderry, Australia. Prove $$\int_{0}^{\infty} \frac{\sin ^{2} x-x \sin x}{x^{3}} d x=\frac{1}{2}-\log 2$$
Proposed by Albert Stadler, Herrliberg, Switzerland. Let $a_{n}$ be the number of equilateral triangles whose vertices are chosen from the vertices of the $n$-dimensional cube. Compute $\lim _{n \rightarrow \infty} n a_{n} / 8^{n}$.
Proposed by Li Zhou, Polk State College, Winter Haven, FL. For a nonnegative integer $m$, let $$A_{m}=\sum_{k=0}^{\infty}\left(\frac{1}{(6 k+1)^{2 m+1}}-\frac{1}{(6 k+5)^{2 m+1}}\right)$$ Prove $A_{0}=\pi \sqrt{3} / 6$ and, for $m \geq 1$, $$2 A_{m}+\sum_{n=1}^{m} \frac{(-1)^{n} \pi^{2 n}}{(2 n) !} A_{m-n}=\frac{(-1)^{m}\left(4^{m}+1\right) \sqrt{3}}{2(2 m) !}\left(\frac{\pi}{3}\right)^{2 m+1}$$
Proposed by Dong Luu, Hanoi National University of Education, Hanoi, Vietnam. In triangle $A B C$, let $D, E$, and $F$ be the points at which the incircle of $A B C$ touches the sides $B C, C A$, and $A B$, respectively. Let $D^{\prime}, E^{\prime}$, and $F^{\prime}$ be three other points on the incircle with $E^{\prime}$ and $F^{\prime}$ on the minor arc $E F$ and $D^{\prime}$ on the major arc $E F$ and such that $A D^{\prime}, B E^{\prime}$, and $C F^{\prime}$ are concurrent. Let $X, Y$, and $Z$ be the intersections of lines $E F$ and $E^{\prime} F^{\prime}$, lines $F D$ and $F^{\prime} D^{\prime}$, and lines $D E$ and $D^{\prime} E^{\prime}$, respectively. Prove that $A X, B Y$, and $C Z$ are either concurrent or parallel.
Proposed by Navid Safaei, Sharif University of Technology, Tehran, Iran. Let $P_{d}$ be the set of all polynomials of the form $\sum_{0 \leq i, j \leq d} a_{i, j} x^{i} y^{j}$ with $a_{i, j} \in\{1,-1\}$ for all $i$ and $j$. Prove that there is a positive integer $d$ such that more than 99 percent of the elements of $P_{d}$ are irreducible in the ring of polynomials with integer coefficients.
Proposed by Ross Dempsey, student, Princeton University, Princeton, NJ. For a fixed positive integer $k$, let $a_{0}=a_{1}=1$ and $a_{n}=a_{n-1}+(k-n)^{2} a_{n-2}$ for $n \geq 2$. Show that $a_{k}=(k-1)$ !.
Proposed by Haoran Chen, Xi'an Jiaotong-Liverpool University, Suzhou, China. A union of a finite number of squares from a grid is called a polyomino if its interior is simply connected. Given a polyomino $P$ and a subpolyomino $Q$, we write $c(P, Q)$ for the number of components that remain when $Q$ is removed from $P$. Let $f(k)=\max _{P} \min _{Q} c(P, Q)$, where the maximum is taken over all polyominoes and the minimum is taken over all subpolyominoes $Q$ of $P$ of size $k$. For example, $f(2) \geq 3$, because any domino removed from the pentomino at right breaks the pentomino into 3 pieces. Is $f$ bounded?
Proposed by Michel Bataille, Rouen, France. Let $x, y$, and $z$ be nonnegative real numbers such that $x+y+z=1$. Prove $$\begin{aligned}&(1-x) \sqrt{x(1-y)(1-z)}+(1-y) \sqrt{y(1-z)(1-x)}+(1-z) \sqrt{z(1-x)(1-y)} \\ &\geq 4 \sqrt{x y z} .\end{aligned}$$
Proposed by Samina Boxwala Kale, Nowrosjee Wadia College, Pune, India, Vašek Chvátal, Concordia University, Montreal, Canada, Donald E. Knuth, Stanford University, Stanford, CA, and Douglas B. West, University of Illinois, Urbana, IL.
a) Show that there is an easy way to decide whether the edges of a graph can each be colored red or green so that at each vertex the number of incident edges with one color differs from the number having the other color by at most $1$.
b) Show that it is NP-hard to decide whether the vertices of a graph can each be colored red or green so that at each vertex the number of neighboring vertices with one color differs from the number having the other color by at most 1 .
Proposed by Mehmet Şahin and Ali Can Güllü, Ankara, Turkey. Let $A B C$ be an acute triangle. Suppose that $D$, $E$, and $F$ are points on sides $B C, C A$, and $A B$, respectively, such that $F D$ is perpendicular to $B C, D E$ is perpendicular to $C A$, and $E F$ is perpendicular to $A B$. Prove $$\frac{A F}{A B}+\frac{B D}{B C}+\frac{C E}{C A}=1$$
Proposed by Moubinool Omarjee, Lycée Henri IV, Paris, France. Let $a_{0}=1$, and let $a_{n+1}=a_{n}+e^{-a_{n}}$ for $n \geq 0$. Show that the sequence whose $n$th term is $e^{a_{n}}-n-(1 / 2) \ln n$ converges.
Proposed by Steven Deckelman, University of Wisconsin-Stout, Menomonie, WI. Let $n$ be a positive integer. Evaluate $$\int_{0}^{2 \pi}\left|\sin \left((n-1) \theta-\frac{\pi}{2 n}\right) \cos (n \theta)\right| d \theta .$$
Proposed by H. A. ShahAli, Tehran, Iran, and Stan Wagon, Macalester College, St. Paul, MN.
a) For which integers $n$ with $n \geq 3$ do there exist distinct positive integers $a_{1}, \ldots, a_{n}$ such that $a_{i}+a_{i+1}$ is a power of 2 for all $i \in\{1, \ldots, n\}$ ? (Here subscripts are taken modulo $n$, so that $a_{n+1}=a_{1}$.)
b) What is the answer if the word "positive" is removed from part (a)?
Proposed by Hideyuki Ohtsuka, Saitama, Japan. Let $\zeta$ be the Riemann zeta function, defined by $\zeta(s)=\sum_{k=1}^{\infty} 1 / k^{s}$. For $s>1$, prove the following inequalities $$\sum_{\text {prime } p} \frac{1}{p^{s}-0.5}<\log \zeta(s), \quad \sum_{\text {prime } p} \frac{1}{p^{s}}<\log \frac{\zeta(s)}{\sqrt{\zeta(2 s)}}, \quad \sum_{\text {prime } p} \frac{1}{p^{s}+0.5}<\log \frac{\zeta(s)}{\zeta(2 s)}.$$
Proposed by Roberto Tauraso, Università di Roma "Tor Vergata," Rome, Italy. Evaluate $$\int_{0}^{1} \frac{\arctan x}{1+x^{2}}\left(\ln \left(\frac{2 x}{1-x^{2}}\right)\right)^{2} d x$$
Proposed by Yun Zhang, Xi'an, China. Let $x, y$, and $z$ be positive real numbers with $x+y+z=3$. Prove each of the following inequalities.
a) $x^{5} y^{5} z^{5}\left(x^{4}+y^{4}+z^{4}\right) \leq 3$.
b) $x^{8} y^{8} z^{8}\left(x^{5}+y^{5}+z^{5}\right) \leq 3$.
c) $x^{11} y^{11} z^{11}\left(x^{6}+y^{6}+z^{6}\right) \leq 3$.
d) $x^{16} y^{16} z^{16}\left(x^{7}+y^{7}+z^{7}\right) \leq 3$.
Proposed by Joe Santmyer, Las Cruces, NM. Prove $$\sum_{n=2}^{\infty} \frac{1}{n+1} \sum_{i=1}^{\lfloor n / 2\rfloor} \frac{1}{2^{i-1}(i-1) !(n-2 i) !}=1$$
Proposed by Cristian Chiser, Elena Cuza College, Craiova, Romania. Let $A, B$, and $C$ be three pairwise commuting 2 -by-2 real matrices. Show that if at least one of the matrices $A-B, B-C$, and $C-A$ is invertible, then the matrix $$A^{2}+B^{2}+C^{2}-A B-A C-B C$$ cannot have rank $1 .$
Proposed by Dao Thanh Oai, Thai Binh, Vietnam. Let $A B C$ be a scalene triangle, and let its external angle bisectors at $A, B$, and $C$ meet $B C$, $C A$, and $A B$ at $D$, $E$, and $F$, respectively. Let $l, m$, and $n$ be lines through $D$, $E$, and $F$ that (internally) trisect angles $\angle A D B, \angle B E C$, and $\angle C F A$, respectively, with the angle between $l$ and $A D$ equal to $1 / 3$ of $\angle A D B$, the angle between $m$ and $B E$ equal to $1 / 3$ of $\angle B E C$, and the angle between $n$ and $C F$ equal to $1 / 3$ of $\angle C F A$.
a) Show that $l, m$, and $n$ form an equilateral triangle.
b) The lines $l, m$, and $n$ each intersect $A D, B E$, and $C F$. Of these nine points of intersection, three are the points $D, E$, and $F$. Show that the other six lie on a circle.
Proposed by Brad Isaacson, Brooklyn, NY. Let $S(m, k)$ denote the number of partitions of a set with $m$ elements into $k$ nonempty blocks. (These are the Stirling numbers of the second kind.) Let $j$ and $n$ be positive integers of opposite parity with $j<n$. Prove $$ \sum_{r=j}^{n} \frac{(-1)^{r}(r-1) !\left(\begin{array}{c} r \\ j \end{array}\right) S(n, r)}{2^{r}}=0 .$$
Proposed by Nguyen Duc Toan, Da Nang, Vietnam. Let $A B C$ be an acute scalene triangle with circumcenter $O$ and orthocenter $H$. Let $M$ and $R$ be the midpoints of segments $B C$ and $O H$, respectively, let $S$ be the reflection across $B C$ of the circumcenter of triangle $B O C$, and let $T$ be the second point of intersection of the circumcircle of triangle $B H C$ and line $O H$. Prove that $M, R, S$, and $T$ are concyclic.
Proposed by Paolo Perfetti, Università di Roma "Tor Vergata," Rome, Italy. Evaluate $$\int_{0}^{\infty}\left(\frac{\cosh x}{\sinh ^{2} x}-\frac{1}{x^{2}}\right)(\ln x)^{2} d x .$$
Proposed by George Stoica, Saint John, NB, Canada. Prove that the multiplicative group generated by $\left\{\lfloor\sqrt{2} n\rfloor / n: n \in \mathbb{Z}^{+}\right\}$is the group of positive rational numbers.
Proposed by Yongge Tian, Shanghai Business School, Shanghai, China. Let $A$ and $B$ be two $n$-by- $n$ matrices that are orthogonal projections, that is, $A^{2}=A=A^{*}$ and $B^{2}=$ $B=B^{*}$. Let $\sqrt{A+B}$ denote the positive semidefinite square root of $A+B$. Prove $$\begin{aligned}\operatorname{trace}(A+B)-(2-\sqrt{2}) \operatorname{rank}(A B) & \leq \operatorname{trace} \sqrt{A+B} \\ & \leq (\sqrt{2}-1) \operatorname{trace}(A+B)+(2-\sqrt{2}) \operatorname{rank}(A+B),\end{aligned}$$ and show that equality holds simultaneously if and only if $A B=B A$.
Proposed by Zachary Franco, Houston, TX. Let $A B C$ be a triangle with circumcenter $O$, incenter $I$, orthocenter $H$, sides of integer length, and perimeter 2021. Suppose that the perpendicular bisector of $O H$ contains $A$ and $I$. Find the length of $B C$.
Proposed by Atul Dixit, Indian Institute of Technology, Gandhinagar, India. Prove $$\sum_{m=1}^{\infty} \int_{0}^{\infty} \frac{t \cos t}{t^{2}+m^{2} u^{2}} d t=\int_{0}^{\infty}\left(-\frac{\pi}{2 u} \cos t+\sum_{m=1}^{\infty} \frac{t \cos t}{t^{2}+m^{2} u^{2}}\right) d t$$for $u>0$.
Proposed by Ira Gessel, Brandeis University, Waltham, MA. Let $p$ be a prime number, and let $m$ be a positive integer not divisible by $p$. Show that the coefficients of $\left(1+x+\cdots+x^{m-1}\right)^{p-1}$ that are not divisible by $p$ are alternately 1 and $-1$ modulo $p$. For example, $\left(1+x+x^{2}+x^{3}\right)^{4} \equiv 1-x+x^{4}-x^{6}+x^{8}-x^{11}+x^{12}(\bmod 5)$.
Proposed by Ovidiu Furdui and Alina Sîntămărian, Technical University of ClujNapoca, Cluj-Napoca, Romania. Prove $$\sum_{n=1}^{\infty}\left(n\left(\sum_{k=n}^{\infty} \frac{1}{k^{2}}\right)^{2}-\frac{1}{n}\right)=\frac{3}{2}-\frac{1}{2} \zeta(2)+\frac{3}{2} \zeta(3),$$ where $\zeta$ is the Riemann zeta function, defined by $\zeta(s)=\sum_{k=1}^{\infty} 1 / k^{s}$.
Proposed by Seán Stewart, Bomaderry, Australia. Prove $$\int_{0}^{\infty}\left(1-x^{2} \sin ^{2}\left(\frac{1}{x}\right)\right)^{2} d x=\frac{\pi}{5}$$
Proposed by George E. Andrews, Pennsylvania State University, University Park, PA, and Mircea Merca, University of Craiova, Craiova, Romania. Prove $$\sum_{n=0}^{\infty} 2 \cos \left(\frac{(2 n+1) \pi}{3}\right) q^{n(n+1) / 2}=\prod_{n=1}^{\infty}\left(1-q^{n}\right)\left(1-q^{6 n-1}\right)\left(1-q^{6 n-5}\right),$$ when $|q|<1$.
Proposed by Walther Janous, Ursulinengymnasium, Innsbruck, Austria. Find all analytic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ that satisfy $$|f(x+i y)|^{2}=|f(x)|^{2}+|f(i y)|^{2}$$ for all real numbers $x$ and $y$.
Proposed by Leonard Giugiuc, Drobeta Turnu Severin, Romania, and Petru Braica, Satu Mare, Romania. The Nagel point of a triangle is the point common to the three segments that join a vertex of the triangle to the point at which an excircle touches the opposite side. Let $A B C$ be a triangle with incenter $I$ and Nagel point $J$. Prove that $A J$ is perpendicular to the line through the orthocenters of triangles $I A B$ and $I A C$.
Proposed by Nikolai Osipov, Siberian Federal University, Krasnoyarsk, Russia. Let $p$ be a prime number, and let $r=1 /(2 \cos (4 \pi / 7))$. Evaluate $\left\lfloor r^{p+2}\right\rfloor$ modulo $p$.