Integration Bee
Problem of the Month Competition
William Lowell Putnam 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 posted here at the beginning of each month of the academic year (except September and January when the new problem will be posted on the first day of the semester), and also featured on the Math Club Bulletin Board, located on the fourth floor of the GAB Building. Work on problems individually and email your solution to iaia@unt.edu 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. Winners will no longer be determined chronologically. All who answer correctly will be listed on the Math Club Bulletin Board the following month.
The Awards. There will be two types of awards associated with the competition: Winner and 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 January 2021
Find an equation of a circle $(x-h)^2 + (y-k)^2 =r^2$ that has exactly one rational point on it. Also find the equation of a circle that exactly two rational points on it.
Note: It can be shown that if the equation of a circle has 3 rationals points on it then it has an infinite number of rational points on it and these points form a dense subset of the circle.
Deadline: Friday, January 29, 2021
SCROLL down to the bottom of this page to find SOLUTIONS.
${\bf Past \ Problems \ and \ Winners}$
Dec. 2020 Show that there are no rational solutions of $x^2 + y^2 = 3.$
Winner: Michalis Paizanis
Runner-Up: Subiksha Sankar
Nov. 2020 Find all Pythagorean triples where the area of the triangle is equal to its perimeter.
Runner-Up: Angela Yuan
other correct solutions by: Alejandro Castellanos, Subisksha, Sankar
Oct. 2020 Show that none of the numbers is a square: 11, 111, 1111, 11111, . . .
Winner: Subiksha Sankar
Runner-Up: Michalis Paizanis
Sept. 2020 Take a quarter circle with radius $R$ in quadrant 1 with one side on the x-axis and one side on the y-axis. Take a semicircle of radius $r<R$ and inscribe it in the larger quarter circle so that the arc intersects and is tangent to the vertical and horizontal part of the quarter circle. Determine how $r$ and $R$ are related. See the accompanying figure here: https://wizardofodds.com/wizfiles/img/568/problem231a.png
Winner: David Duhon
Runner Up: Peyton Thibodeaux
other correct solutions by: Michalis Paizanis, Subiksha Sankar
Show that: $$(\sin^{-1}(x))^2 = \sum_{n=0}^{\infty} \frac{2^{2n} (n!)^2 \, x^{2n+2}}{(2n+1)! (n+1)} $$ on $[0,1].$
Use this to determine: $$ \sum_{n=1}^{\infty} \frac{1}{ n^2 {2n \choose n} } . $$
Hint: Let: $$y = \sum_{n=0}^{\infty} \frac{2^{2n} (n!)^2 \, x^{2n+2}}{(2n+1)! (n+1)} $$ and show that $y$ satisfies a second order linear differential equation. Then solve the differential equation.
Winner: None
Mar. 2020 Let $m<n<p$ be positive integers and suppose $N$ is an integer such that: $$\frac{1}{m} + \frac{1}{n} + \frac{1}{p} = N.$$ Prove that there is only one solution and find the solution.
Winner: Jingyi Dai
Runner-Up: Hubert Yang
Other correct solutions by: Anush Beeram, Elson Darby, David Duchon, Rhythm Garg Harrison Rodi
Feb. 2020 Let $f: [0,1] \to {\mathbb R}$ be continuous with $f(0)=f(1).$ Show that there is an $x_2$ with $0<x_{2}<1$ such that $f(x_{2} + \frac{1}{2}) = f(x_{2}).$ Similarly show for each positive integer $n>2$ that there is an $x_n$ with $0< x_n < 1$ such that $f(x_{n} + \frac{1}{n}) = f(x_{n}).$
Runner-Up: Rhythm Garg
Find all functions $f: {\mathbb R} \to {\mathbb R}$ that satisfy: $f(x+y) = f(x) + f(y)$ and $f(xy)=f(x)f(y).$ (You may NOT assume that $f$ is continuous!) Hint: Try to show $f$ is nondecreasing.
Winner: Rhythm Garg
Prove the following:
$$ \frac{1}{3} = \frac{1+3}{5+7} = \frac{1+3+5}{7+9+11} = \cdots. $$
Winner: Chris Howard
Runner-Up: Andrew Sansom
Nov. 2019 Determine $a_{n}$ if: $$a_{n+1} = 2 a_{n} + n \textrm{ and } a_{0} = 1. $$
Winner: Arnav Iyer
Runner-Up: Aryan Agarwal
Let $0<x<1$. Simplify as much as possible:
$$ \ln(x) \ln(1-x) + \sum_{n=1}^{\infty} \frac{x^{n}}{n^2} +\sum_{n=1}^{\infty} \frac{(1-x)^{n}}{n^2}. $$
Winner: Weilun Sun
Let $a$ and $b$ be real numbers with $b\neq 0$. Determine:
$$ \int_{0}^{2\pi} \frac{d\theta}{\cos^{2}(\theta) + 2a\cos(\theta)\sin(\theta) + (a^{2}+b^{2})\sin^{2}(\theta)}. $$
Determine all positive integer solutions to the system of equations:
$$ xy=z+w$$
$$ zw=x+y.$$
Runner-Up: Jingyi Dai
Mar. 2019 Let $a<b$ be positive integers. Find all solutions of: $$a^b = b^a. $$
Winner: Edoardo Luna
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 $EB'D$.
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.
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.
Runner-Up: Trenton Hicks
Let $0<a<b$. Determine:
$$ \int_{a}^{b} \cos^{-1}\left( \frac{x}{\sqrt{(a+b)x - ab}} \right)\, dx. $$
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
Investigate the convergence of:
$$ \sum_{n=3}^{\infty} \frac{1}{\ln(\ln(n))^{\ln(n)}} $$
$$ \sum_{n=3}^{\infty} \frac{1}{\ln(\ln(n))^{\ln(\ln(n))}} $$
Let $a>0$ and $b>0$. Determine:
$$ \int_{0}^{2\pi} \frac{ab}{a^2 \cos^{2}(t) + b^2\sin^{2}(t)} \, dt. $$
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
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 $f'(e)$.
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
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 $H>0$: $$ \int_{-\infty}^{\infty} e^{-Ht^2} \, dt = \sqrt\frac{\pi}{H}. $$
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
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
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
Consider the sum:
$$ \begin{array}{1 2 3 4 5 6 7 }
\ & \ & \textrm{M} & \textrm{A} & \textrm{J} & \textrm{O} & \textrm{R} \\
\ & + & \textrm{M} & \textrm{I} & \textrm{N} & \textrm{O} & \textrm{R} \\
= & \textrm{R} & \textrm{E} & \textrm{S} & \textrm{U} & \textrm{M} & \textrm{E} \end{array}
where each letter represents a different nonnegative integer 0, 1, .... 9. What is the largest number that RESUME can represent and still have the equation be true?
Note: There are 10 different letters in this sum so all of the nonnegative integers from 0 to 9 will be used exactly once.
Winner: Ethan Seal
Runner-Up: Brandon Ohl
Let $A=(0,2)$, $B = (3,0)$. Find a point $C$ on the unit circle centered at the origin so that triangle $ABC$ is of largest area.
Winner: William Liu
Runner-Up: Xiangyu Kong
December 2016 Find all real solutions of: $$ \root 3 \of{x+9} - \root 3 \of{x-9} = 3. $$ Co-winners: Xiangyu Kong, William Liu
Prove that the polynomial $p(x) = x^3 - 12x^2 + ax - 64$ has all of its roots real and nonnegative for exactly one real number $a$. Determine $a$.
Runner-Up: Aoxue Chen
a. Let $A$ and $B$ be two linear transformations from ${\mathbb R}^N \to {\mathbb R}^N.$ Show that it is impossible for $AB-BA=I$ where $I$ is the identity map.
b. On the other hand show that it is possible to find two linear transformations (defined on infinite dimensional spaces) with $AB-BA=I$.
Determine $$ \lim_{n \to \infty} \frac{1}{n^4} \prod_{i=1}^{2n} (n^2 + i^2)^{\frac{1}{n}}. $$
Note: The $\prod$ sign is a short-hand for product. That is: $\prod\limits_{i=1}^{2n} b_i = b_{1}b_{2} \cdots b_{2n-1} b_{2n}.$
April 2016 Let $ f(x,y) = \int_{0}^{\infty} \frac{1}{(1+x^2t^2)(1+y^2t^2)} \, dt. $ Prove that $f(x,y) = \frac{\pi}{2(x+y)}$ and then calculate $ \int_{0}^{1} \int_{0}^{1} f(x,y) \, dx \, dy$ and determine $ \int_{0}^{\infty} \frac{\tan^{-1}(t^2)}{t^2} \, dt. $
Runner-Up: Christopher Lee
March 2016 Find all functions which satisfy: $$ f(x) + 2f\left(\frac{1}{1-x}\right) = x. $$
Runner-Up: Jagath Vytheeswaran
February 2016 Let $a,b,x,p$ be real numbers with $0 < a \leq x \leq b$ and $p>0$. Determine:
$$ \min_{p} \max_{x \in [a,b]} \frac{|x-p|}{x}. $$
January 2016 Show that the only solution of: $$ 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.$$
Runner-Up Brandon Ohl
November 2015 Find all real solutions of: $$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^{2n}}{(2n)!} = 0.$$ Winner: William Liu
October 2015 Let T be an equilateral triangle and let P be a point of T. Let $d_{1}, d_{2}$, and $d_{3}$ be the distance of P to each of the sides of T. Show that $d_{1} + d_{2} + d_{3}$ is independent of P. Winners: Xiangyu Kong, Shuhui Jiang
Let $f_{n}$ be the Fibonacci sequence. Determine $$\sum_{n=1}^{\infty} \tan^{-1}\left(\frac{1}{f_{2n+1}} \right). $$
Hint: Use a trig identity and the following identity which holds for the Fibonacci sequence: $f_{n+1}f_{n+2} -f_{n}f_{n+3} = (-1)^n. $
Winners: William Liu, Tamaki Ueno
April 2015 Let $x\geq 0, y \geq 0, z \geq 0$. Find all solutions of: $$ 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} $$
in terms of $p$ and $q$. Winner: Zachary Gardner
Determine all nonnegative continuous functions which satisfy:
$$ f(x+t) = f(x) + f(t) + 2 \sqrt{f(x)}\sqrt{f(t)} \textrm{ for } x \geq 0, t \geq 0. $$
Winner: Mark Fincher
Determine: $$ \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$$
Winner: Steven Grigsby
Runner Up: Mark Fincher
Simplify: $$ \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}} $$
Winner: Tamaki Ueno
Runner Up: Murray Lee
Squared Matrices and Characteristic Polynomials. Let $A$ be a real $4\times4$-matrix with characteristic polynomial $p(\lambda)=\det(\lambda{I}-A)=\lambda^4-s_1\lambda^3+s_2\lambda^2-s_3\lambda+s_4.$ Show that if $A$ admits real square roots, in the sense that there is a real $4\times4$-matrix $S$ such that $S^2=A,$ then $s_1+s_2+s_3+s_4\ge-1.$
Runner-Up: Chia-Ting Han
March 2014 Shortest Path in Triangle. In $\bigtriangleup BAC,$ $\angle BAC = 40^\circ,$ $AB = 5,$ and $AC = 3.$ Points $D$ and $E$ lie on $AB$ and $AC$ respectively, What
is the minimum possible value of $BE + DE + CD?$
Winner: Kevin Lin
February 2014 Do It Without Fermat or Technology. By using a TI-83 calculator it appears that $$\sqrt[{}^{12}\;]{3987^{12} + 4365^{12}} = 4472.$$ Show that this is not true, without appealing to Fermat's Last Theorem or to computer technology.
Winner: Chia-Ting Han
Runner-Up: Kevin Lin
January 2014 Largest Area Triangle. What is the largest area of a triangle inscribed in the ellipse $x^2 + xy + y^2 = 1?$ Kevin Lin
November 2013 Irrational? If $x$ is a real number such that $x^3 + x$ and $x^5 + x$ are rational numbers, can $x$ be an irrational number?
October 2013 Integer Part. Consider the sequence: $\displaystyle a_1=4; a_{n+1}=\frac{a_n^2}{a_n^2-3a_n+3},n\ge1.$ Find the integer part of the 2013-th term of the sequence, that is the greatest integer smaller than or equal to $a_{2013}.$
Runner-Up: Chris James
September 2013 A Geometric Product. Let $P_k$ for $k = 1, 2,\ldots, n,$ be the vertices of a regular polygon inscribed in a circle of radius $r.$ Let $d_k$ be the distance between $P_k$ and $P_1.$ Calculate $\displaystyle \prod_{k=2}^nd_k.$ None
Integration Bee Warm-Up. Find an anti-derivative of the function $$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
December 2020 Solution 59.56 KB
November 2020 Solution 72.48 KB
October 2020 Solution 46.55 KB
September 2020 Solution 65.56 KB
April 2020 Solution 85.95 KB
March 2020 Solution 66.84 KB
February 2020 Solution 67.43 KB
January 2020 Solution 76.92 KB
November 2019 Solution 109.37 KB
October 2019 Solution 101.7 KB
February 2019 Solution 100.15 KB
September 2018 Solution 101.39 KB
May 2018 Solution 87.33 KB
April 2018 Solution 104.23 KB
March 2018 Solution 108.03 KB
January 2018 Solution 106.04 KB
December 2017 Solution 106.23 KB
November 2017 Solution 79.3 KB
March 2017 Solution 57.4 KB