 Problem of the Month | Department of Mathematics

Problem of the Month The Competition. The UNT Math Department and the UNT Math Club invite all undergraduate students currently enrolled at UNT to take part in the newly redesigned Problem of the Month Competition. The competition, which runs during the regular semesters, consists in solving and submitting a solution to one proposed math problem each month.

The Rules. Problems and subsequent solutions will be emailed to the UNT undergraduate math major e-mail distribution list (if you would like to be added to the distribution list, contact Rita at <rhsears@unt.edu>), and also featured on the Math Club Bulletin Board, located on the fourth floor of the GAB Building. Work on problems individually and submit your solution to the Math Department Office, GAB 435, by the specified deadline. Please include your name, student ID number, and your UNT email address. The entries will be graded promptly by a panel of judges, on correctness, completeness, and style. The ruling of the judges will be final. No awards will be given to solutions which are not correct and complete. Identical entries will be disqualified. If two or more correct solutions are turned in then the winner and runner-up will be the first and second entries received chronologically.

The Awards. There will be two types of awards associated with the competition: Winner and Winner/Runner Up. All awardees will be prominently featured on the Math Dept. website (www. math.unt.edu). Also, they will be given awards to be used for tuition, as follows:

• Winner -- $\$ $50 -- if there is only one correct answer. • Winner --$ \30 -- Runner-Up -- $\$ $20. Problem for October 2019 Let$0<x<1$. Simplify as much as possible: $$\ln(x) \ln(1-x) + \sum_{n=1}^{\infty} \frac{x^{n}}{n^2} +\sum_{n=1}^{\infty} \frac{(1-x)^{n}}{n^2}.$$ Deadline: Thursday October 31, 2019${\bf Past \ Problems \ and \ Winners}$ Sept. 2019 Let$a$and$b$be real numbers with$b\neq 0$. Determine: $$\int_{0}^{2\pi} \frac{d\theta}{\cos^{2}(\theta) + 2a\cos(\theta)\sin(\theta) + (a^{2}+b^{2})\sin^{2}(\theta)}.$$ Winner: Rhythm Garg 2018-2019 Academic Year Apr. 2019 Determine all positive integer solutions to the system of equations: $$xy=z+w$$ $$zw=x+y.$$ Winner: Rhythm Garg Runner-Up: Jingyi Dai Mar. 2019 Let$a0$and$b>0$. Determine: $$\int_{0}^{2\pi} \frac{ab}{a^2 \cos^{2}(t) + b^2\sin^{2}(t)} \, dt.$$ Winner: None Apr. 2018 Determine: $$\sin\left(\frac{\pi}{n}\right) \sin\left(\frac{2\pi}{n}\right) \sin\left(\frac{3\pi}{n}\right) \cdots \sin\left(\frac{(n-1)\pi}{n}\right).$$ Hint: Note that: $$z^{n}-1 = (z-1)(z - e^{\frac{2\pi i}{n}}) (z - e^{\frac{4\pi i}{n}}) (z - e^{\frac{6\pi i}{n}}) \cdots (z - e^{\frac{2(n-1)\pi i}{n}}).$$ Winner: Riya Danait Mar. 2018 Let$x>1$and define$f(e)=e.$For$x>1$and$x\neq e$let$f(x)>0$be the unique number such that$f(x)\neq x$and$x^{f(x)} = f(x)^{x}.$Sketch the graph of$f(x)$and calculate$f'(x)$(you may leave this answer in terms of$x$and$f(x)$). Finally calculate$f'(e)$. Winner: Tien Le Feb. 2018 Determine the values of$\theta \in [0, \pi] $for which: $$\sum_{n=1}^{\infty} \frac{\sin^{2}(n\theta)}{n} \textrm{ converges}.$$ Winner: None Jan. 2018 Let$a, b, c$be real numbers such that$ax^{2} + 2bxy +cy^2>0$for all$(x,y)\neq (0,0)$. Determine: $$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} \left( e^{-(ax^2 + 2bxy + cy^2)} \right) \, dx \, dy.$$ You may use the well-known fact that for$H>0$: $$\int_{-\infty}^{\infty} e^{-Ht^2} \, dt = \sqrt\frac{\pi}{H}.$$ Winner: None Nov. 2017 Let: $$x_{n+1} = \frac{1}{n+1} x_{n-1} + \frac{n}{n+1}x_{n}$$ with: $$x_{0} = a, x_{1} = b.$$ Determine if: $$\lim\limits_{n \to \infty} x_{n} \textrm{exists}$$ and if so then find the limit. Winner: Yuqing Liu Oct. 2017 Determine: $$\sum_{n=2}^{\infty} (-1)^{n}\frac{\ln(n)}{n}.$$ Express your answer in terms of Euler's constant,$\gamma$.Note that:$\gamma = \lim\limits_{n \to \infty} \left( 1 + \frac{1}{2} + \cdots \frac{1}{n} - \ln(n) \right). $Hint: You may assume there exists a constant$A$such that: $$\lim\limits_{n \to \infty} (\frac{\ln(2)}{2} + \frac{\ln(3)}{3} + \cdots + \frac{\ln(n)}{n} - \frac{1}{2}\ln^{2}(n) ) = A.$$ Winner: Yuqing Liu Sept. 2017 Place a circle,$C_1$, with radius 1 centered at (-1,1) in the xy plane and place a second circle,$C_2$, with radius 1 centered at (1,1) in the xy plane. Next place a circle that goes through (0,0) and that is tangent to both$C_1$and$C_2$. Call this circle$D_1$and denote its diameter as$d_1$. Next place a circle,$D_2$, directly above$D_1$that is tangent to$D_1$,$C_1$, and$C_2$. Call its diameter$d_2$. Continue this process to obtain circles$D_3$,$D_4$, . . . with diameters$d_3$,$d_4$, . . . . Find a formula for$d_n$and also determine$\sum\limits_{n=1}^{\infty} d_n$. Winner: Linda Yu Runner-Up: Austin McGregor 2016-2017 Academic Year April 2017 Determine the area of the largest equilateral triangle that can be inscribed inside a square with side of length 1. Winner: Brandon Ohl Runner-Up: Ethan Seal March 2017 Consider the sum:$$\begin{array}{1 2 3 4 5 6 7 } \ & \ & \textrm{M} & \textrm{A} & \textrm{J} & \textrm{O} & \textrm{R} \\ \ & + & \textrm{M} & \textrm{I} & \textrm{N} & \textrm{O} & \textrm{R} \\ = & \textrm{R} & \textrm{E} & \textrm{S} & \textrm{U} & \textrm{M} & \textrm{E} \end{array}$$ where each letter represents a different nonnegative integer 0, 1, .... 9. What is the largest number that RESUME can represent and still have the equation be true? Note: There are 10 different letters in this sum so all of the nonnegative integers from 0 to 9 will be used exactly once. Winner: Ethan Seal Runner-Up: Brandon Ohl January 2017 Let$A=(0,2)$,$B = (3,0)$. Find a point$C$on the unit circle centered at the origin so that triangle$ABC$is of largest area. Winner: William Liu Runner-Up: Xiangyu Kong December 2016 Find all real solutions of: $$\root 3 \of{x+9} - \root 3 \of{x-9} = 3.$$ Co-winners: Xiangyu Kong, William Liu November 2016 Prove that the polynomial$p(x) = x^3 - 12x^2 + ax - 64$has all of its roots real and nonnegative for exactly one real number$a$. Determine$a$. Winner: William Liu Runner-Up: Aoxue Chen October 2016 a. Let$A$and$B$be two linear transformations from${\mathbb R}^N \to {\mathbb R}^N.$Show that it is impossible for$AB-BA=I$where$I$is the identity map. b. On the other hand show that it is possible to find two linear transformations (defined on infinite dimensional spaces) with$AB-BA=I$. Winner: William Liu September 2016 Determine $$\lim_{n \to \infty} \frac{1}{n^4} \prod_{i=1}^{2n} (n^2 + i^2)^{\frac{1}{n}}.$$ Note: The$\prod$sign is a short-hand for product. That is:$\prod\limits_{i=1}^{2n} b_i = b_{1}b_{2} \cdots b_{2n-1} b_{2n}.$Winner: William Liu 2015-2016 Academic Year April 2016 Let$ f(x,y) = \int_{0}^{\infty} \frac{1}{(1+x^2t^2)(1+y^2t^2)} \, dt. $Prove that$f(x,y) = \frac{\pi}{2(x+y)}$and then calculate$ \int_{0}^{1} \int_{0}^{1} f(x,y) \, dx \, dy$and determine$ \int_{0}^{\infty} \frac{\tan^{-1}(t^2)}{t^2} \, dt. $Winner: William Liu Runner-Up: Christopher Lee March 2016 Find all functions which satisfy: $$f(x) + 2f\left(\frac{1}{1-x}\right) = x.$$ Winner: William Liu Runner-Up: Jagath Vytheeswaran February 2016 Let$a,b,x,p$be real numbers with$0 < a \leq x \leq b$and$p>0$. Determine: $$\min_{p} \max_{x \in [a,b]} \frac{|x-p|}{x}.$$ Winner: William Liu January 2016 Show that the only solution of: $$m^2 + n^2 + p^2 = 2mnp$$ where$m,n$, and$p$are integers is:$m=n=p=0$. Winner: William Liu December 2015 Determine whether the following series converges: $$1 + \frac{1}{2}\left(\frac{19}{7}\right) + \frac{2!}{3^2}\left(\frac{19}{7}\right)^2 + \frac{3!}{4^3}\left(\frac{19}{7}\right)^3 + \cdots.$$ Winner: William Liu Runner-Up Brandon Ohl November 2015 Find all real solutions of: $$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^{2n}}{(2n)!} = 0.$$ Winner: William Liu October 2015 Let T be an equilateral triangle and let P be a point of T. Let$d_{1}, d_{2}$, and$d_{3}$be the distance of P to each of the sides of T. Show that$d_{1} + d_{2} + d_{3}$is independent of P. Winners: Xiangyu Kong, Shuhui Jiang September 2015 Let$f_{n}$be the Fibonacci sequence. Determine $$\sum_{n=1}^{\infty} \tan^{-1}\left(\frac{1}{f_{2n+1}} \right).$$ Hint: Use a trig identity and the following identity which holds for the Fibonacci sequence:$f_{n+1}f_{n+2} -f_{n}f_{n+3} = (-1)^n. $Winners: William Liu, Tamaki Ueno 2014-2015 Academic Year April 2015 Let$x\geq 0, y \geq 0, z \geq 0$. Find all solutions of: $$x^{1/3} - y^{1/3} - z^{1/3} = 16$$ $$x^{1/4} - y^{1/4} -z^{1/4} = 8$$ $$x^{1/6} - y^{1/6} - z^{1/6} = 4.$$ None March 2015 This is an approximate angle trisection method due to d'Ocagne. Consider the unit semicircle. Let A,P,B,D lie along the diameter where B is the center of the corresponding circle, A,D are the endpoints of the diameter, and P is the midpoint of the segment AB. Let C lie on the arc of the semicircle so that angle CBD is$\theta,$and let Q be the midpoint of the arc CD.Show that angle$\alpha=$QPC$\approx\theta/3$. More precisely show that:$$\lim_{\theta \to 0^{+}} \frac{\tan(\alpha)}{\theta} = \frac{1}{3}.$$ Winners: Zachary Gardner, Tamaki Ueno February 2015 Denote$p = \sum\limits_{k=1}^{\infty} \frac{1}{k^2}$and$q = \sum\limits_{k=1}^{\infty} \frac{1}{k^3}.$Express: $$\sum_{i=1}^{\infty}\sum_{j=1}^{\infty} \frac{1}{(i+j)^3}$$in terms of$p$and$q$. Winner: Zachary Gardner January 2015 Determine all nonnegative continuous functions which satisfy: $$f(x+t) = f(x) + f(t) + 2 \sqrt{f(x)}\sqrt{f(t)} \textrm{ for } x \geq 0, t \geq 0.$$ Winner: Mark Fincher December 2014 Determine: $$\sum_{k=1}^{n} {n \choose k} k^3.$$ Winner: Mark Fincher November 2014 Determine: $$\int_{0}^{\infty} \frac{\tan^{-1}(ax) - \tan^{-1}(x)}{x} \, dx \textrm{ when } a > 0.$$ Winner: Murray Lee Runner Up: Steven Grigsby October 2014 Determine: $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{3n-2} = 1 - \frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \cdots$$ Winner: Steven Grigsby Runner Up: Mark Fincher September 2014 Simplify: $$\frac{1}{\sqrt{1} + \sqrt{2} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{6}+\sqrt{9}} + \frac{1}{\sqrt{9} + \sqrt{12} + \sqrt{16}}$$ Winner: Tamaki Ueno Runner Up: Murray Lee 2013-2014 Academic Year April 2014 Squared Matrices and Characteristic Polynomials. Let$A$be a real$4\times4$-matrix with characteristic polynomial$p(\lambda)=\det(\lambda{I}-A)=\lambda^4-s_1\lambda^3+s_2\lambda^2-s_3\lambda+s_4.$Show that if$A$admits real square roots, in the sense that there is a real$4\times4$-matrix$S$such that$S^2=A,$then$s_1+s_2+s_3+s_4\ge-1.$Winner: Mark Fincher Runner-Up: Chia-Ting Han March 2014 Shortest Path in Triangle. In$\bigtriangleup BAC,\angle BAC = 40^\circ,AB = 5,$and$AC = 3.$Points$D$and$E$lie on$AB$and$AC$respectively, Whatis the minimum possible value of$BE + DE + CD?$Winner: Kevin Lin Runner-Up: Chia-Ting Han February 2014 Do It Without Fermat or Technology. By using a TI-83 calculator it appears that $$\sqrt[{}^{12}\;]{3987^{12} + 4365^{12}} = 4472.$$ Show that this is not true, without appealing to Fermat's Last Theorem or to computer technology. Winner: Chia-Ting Han Runner-Up: Kevin Lin January 2014 Largest Area Triangle. What is the largest area of a triangle inscribed in the ellipse$x^2 + xy + y^2 = 1?$Kevin Lin November 2013 Irrational? If$x$is a real number such that$x^3 + x$and$x^5 + x$are rational numbers, can$x$be an irrational number? Winner: Chia-Ting Han Runner-Up: Kevin Lin October 2013 Integer Part. Consider the sequence:$\displaystyle a_1=4; a_{n+1}=\frac{a_n^2}{a_n^2-3a_n+3},n\ge1.$Find the integer part of the 2013-th term of the sequence, that is the greatest integer smaller than or equal to$a_{2013}.$Winner: Kevin Lin Runner-Up: Chris James September 2013 A Geometric Product. Let$P_k$for$k = 1, 2,\ldots, n,$be the vertices of a regular polygon inscribed in a circle of radius$r.$Let$d_k$be the distance between$P_k$and$P_1.$Calculate$\displaystyle \prod_{k=2}^nd_k.$None 2012-2013 Academic Year April 2013 Integration Bee Warm-Up. Find an anti-derivative of the function $$f(x)=\sqrt{\sqrt{\sqrt{x}+1}-\sqrt{\sqrt{x}-1}},\;x>1.$$ Kevin Lin March 2013 A Determinant. Show that the determinant of the$3\times 3$matrix $$\left[\begin{array}{lll}1 + a^2 + a^4& 1 + ab + a^2b^2& 1 + ac + a^2c^2\\1 + ab + a^2b^2 &1 + b^2 + b^4 &1 + bc + b^2c^2\\1 + ac + a^2c^2& 1 + bc + b^2c^2& 1 + c^2 + c^4\end{array}\right]$$is a product of linear factors in$a$,$b$, and$c$. Winner: Heather Olney Runner-Up: Alyssa Sylvester February 2013 Pizza, Anyone? While discussing math$n$people seated at a round table eat a combined total of$n-1$slices of pizza. Show that there is a unique way of counting the people around the table so the fi rst person eats no pizza, the first two people eat no more than one slice, the fi rst three people eat no more than two slices, etc. None January 2013 A Geometric Inequality. Let$T$be a given (non-degenerate) triangle in a plane. Prove there is a constant$c(T)>0$with the following property: if a collection of$n$disks whose areas sum to$S$entirely contains the sides of$T,$then$\displaystyle n>\frac{c(T)}{S}.$None November 2012 Evaluate the improper integral$\displaystyle\int_0^{\pi/2}\ln\sin x\,dx.$Colin Campbell October 2012 Compute$\displaystyle\lim\limits_{n\to\infty}\frac{1}{n^4}\prod_{j=1}^{2n}(n^2+j^2)^{1/n}.$Colin Campbell September 2012 Find all integers$x$such that$x^4+x^3+x^2+x+1\$ is a perfect square. None 