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.

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., 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 -- Runner-Up $ \$ $20.

Current Problem

The next problem will appear in September 2017.

Past Problems and Winners

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

February 2017

Let $P= (-3/2, 9/4)$ and let $Q = (3,9)$. Note that $P$ and $Q$ both lie on $y=x^2$.
Let $y=f(x)$ be a continuous function on the interval $[-3/2, 3]$ and suppose the function $f(x)$ lies above the line $PQ$. Find a point $R$ on the curve $y=x^2$ between $P$ and $Q$ so that the area bounded by $y=f(x)$ and the straight line segments $PR$ and $QR$ is as large as possible.

Winner: William Liu, Runner-Up: 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, Runner-Up: Xiangyu Kong
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December 2016 Find all real solutions of: $$ \root 3 \of{x+9} - \root 3 \of{x-9} = 3. $$ Co-winners: 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

Runner-Up: 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 $AB-BA=I$ 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

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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 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
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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 and Runner-Up Christopher Lee
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March 2016 Find all functions which satisfy: $$ f(x) + 2f\left(\frac{1}{1-x}\right) = x. $$ Winner: William Liu and Runner-Up: 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:

$$ \min_{p} \max_{x \in [a,b]} \frac{|x-p|}{x}. $$

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 Runner-Up 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
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 and 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 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
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 and 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 and 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 and 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 and 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 and 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 and 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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