Problem of the Month | Department of Mathematics

Problem of the Month

Problem of the Month Competition

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 <>), 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. 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 $a<b$ be positive integers. Find all solutions of: $$a^b = b^a. $$

Winner: Edoardo Luna

Runner-Up: Rhythm Garg

Feb. 2019

Take a square piece of paper with vertices labeled $A, B, C, D$ when proceeding in the counterclockwise direction with vertex $A$ in the upperleft corner. Now take the side $AB$ and fold it in such a way that $A$ gets folded to a point $A'$ and $B$ to $B'$ where $B'$ is on $CD$ and $A'$ is outside the square $ABCD$. Then $A'B'$ intersects $AD$ at $E$. Next draw a circle that is tangent to $CD$, tangent to $AD$ and tangent to $A'B'$. Denote the radius of this circle by $r$. Prove that $r= A'E.$

Hint: A well-known fact from geometry states that $r= \frac{2 \textrm{Area}(T)}{\textrm{Perimeter}(T)} $ where $T$ is the triangle $EB'D$.

Winner: Rhythm Garg
Jan. 2019 Consider the unit square with corners P,Q,R,S (in the counterclockwise direction). Using each corner as a center, draw a quarter circle of radius 1. Find the area, A, of the intersection of these 4 quarter circles.

Winner: Rhythm Garg

Runner-Up: Fernando Moreno

Dec. 2018 Consider a square ABCD with vertices labeled counterclockwise and with vertex A in the upperleft corner. There are three segments - one from corner A, one from corner B, and one from corner C. The segments intersect at a point P in the interior of the square and P lies below the diagonal AC. The length of the segment from P to A is 1, the length of the segment from P to B is 2, and the length of the segment from P to C is 3. Determine angle APB.

Winner: Rhythm Garg

Runner-Up: Trenton Hicks

Nov. 2018

Let $0<a<b$. Determine:

$$ \int_{a}^{b} \cos^{-1}\left( \frac{x}{\sqrt{(a+b)x - ab}} \right)\, dx. $$

Winner: Rhythm Garg
Oct. 2018 Let $n$ be a nonzero integer. Multiply this by 3 (write your answer in base 10). Now add the digits together and call this number $m$ (written in base 10). If $0 \leq m \leq 9$ then this process stops. Otherwise repeat the process - add the digits of $m$ together to get a new number $p$ (written in base 10). As above if $0\leq p \leq 9$ then stop but otherwise continue this process until obtaining just one digit. Make a conjecture about what this last number is and then prove it.

Winner: Tien Le

Runner-Up: Rhythm Garg

Sep 2018

Investigate the convergence of:

$$ \sum_{n=3}^{\infty} \frac{1}{\ln(\ln(n))^{\ln(n)}} $$


$$ \sum_{n=3}^{\infty} \frac{1}{\ln(\ln(n))^{\ln(\ln(n))}} $$

Winner: Rhythm Garg
2017-2018 Academic Year
May 2018

Let $a>0$ 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


$$ \sin\left(\frac{\pi}{n}\right) \sin\left(\frac{2\pi}{n}\right) \sin\left(\frac{3\pi}{n}\right) \cdots \sin\left(\frac{(n-1)\pi}{n}\right). $$

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: $$ \frac{1}{\sqrt[3]{1} + \sqrt[3]{2} + \sqrt[3]{4}} + \frac{1}{\sqrt[3]{4} + \sqrt[3]{6}+\sqrt[3]{9}} + \frac{1}{\sqrt[3]{9} + \sqrt[3]{12} + \sqrt[3]{16}} $$

Winner: Tamaki Ueno

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, What
is 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

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