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 email 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.
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), and presented with Certificates of Excellence. Also, they will be given awards to be used for tuition, as follows:
 Winner  $50  if there is only one correct answer.
 Winner  $ \$ $30  RunnerUp $ \$ $20.
Current Problem  Due November 30, 2017
Let: $$ x_{n+1} = \frac{1}{n+1} x_{n1} + \frac{n}{n+1}x_{n} $$ 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.
Past Problems and Winners
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 RunnerUp: Austin McGregor 

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 RunnerUp: Ethan Seal 

March 2017 
Consider the sum: 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 RunnerUp: Brandon Ohl 

February 2017 
Let $P= (3/2, 9/4)$ and let $Q = (3,9)$. Note that $P$ and $Q$ both lie on $y=x^2$. 
Winner: William Liu, RunnerUp: Rohit Kopparthy 

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, RunnerUp: Xiangyu Kong  
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December 2016  Find all real solutions of: $$ \root 3 \of{x+9}  \root 3 \of{x9} = 3. $$  Cowinners: Xiangyu Kong, William Liu  
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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 RunnerUp: Aoxue Chen 

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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 $ABBA=I$ 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 $ABBA=I$. 
Winner: William Liu 

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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 shorthand for product. That is: $\prod\limits_{i=1}^{2n} b_i = b_{1}b_{2} \cdots b_{2n1} b_{2n}.$ 
Winner: William Liu  
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20152016 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 and RunnerUp Christopher Lee  
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March 2016  Find all functions which satisfy: $$ f(x) + 2f\left(\frac{1}{1x}\right) = x. $$  Winner: William Liu and RunnerUp: Jagath Vytheeswaran  
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February 2016  Let $a,b,x,p$ be real numbers with $0 < a \leq x \leq b$ and $p>0$. Determine:
$$ \min_{p} \max_{x \in [a,b]} \frac{xp}{x}. $$ 
Winner: William Liu  
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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 and RunnerUp Brandon Ohl  
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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  
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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  
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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  
20142015 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 and Runner Up: Steven Grigsby  
October 2014  Determine: $$ \sum_{n=1}^{\infty} \frac{(1)^{n+1}}{3n2} = 1  \frac{1}{4} + \frac{1}{7}  \frac{1}{10} + \cdots$$  Winner: Steven Grigsby and Runner Up: Mark Fincher  
September 2014 
Simplify: $$ \frac{1}{\sqrt[3]{1} + \sqrt[3]{2} + \sqrt[3]{4}} + \frac{1}{\sqrt[3]{4} + \sqrt[3]{6}+\sqrt[3]{9}} + \frac{1}{\sqrt[3]{9} + \sqrt[3]{12} + \sqrt[3]{16}} $$ 
Winner: Tamaki Ueno and Runner Up: Murray Lee  
20132014 Academic Year  
April 2014 
Squared Matrices and Characteristic Polynomials. Let $A$ be a real $4\times4$matrix with characteristic polynomial $p(\lambda)=\det(\lambda{I}A)=\lambda^4s_1\lambda^3+s_2\lambda^2s_3\lambda+s_4.$ 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 $s_1+s_2+s_3+s_4\ge1.$ 
Winner: Mark Fincher and RunnerUp: ChiaTing 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, What is the minimum possible value of $BE + DE + CD?$ 
Winner: Kevin Lin and RunnerUp: ChiaTing Han  
February 2014  Do It Without Fermat or Technology. By using a TI83 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: ChiaTing Han and RunnerUp: 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: ChiaTing Han and RunnerUp: Kevin Lin 

October 2013  Integer Part. Consider the sequence: $\displaystyle a_1=4; a_{n+1}=\frac{a_n^2}{a_n^23a_n+3},n\ge1.$ Find the integer part of the 2013th term of the sequence, that is the greatest integer smaller than or equal to $a_{2013}.$  Winner: Kevin Lin and RunnerUp: 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  
20122013 Academic Year  
April 2013 
Integration Bee WarmUp. Find an antiderivative 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 and RunnerUp: Alyssa Sylvester  
February 2013  Pizza, Anyone? While discussing math $n$ people seated at a round table eat a combined total of $n1$ 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 (nondegenerate) 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 