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 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 on correctness, completeness, and style. No awards will be given to solutions which are not correct and complete. 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 -- $\$ $20. The PROBLEM OF THE MONTH is on summer break and will return on the first day of classes for the Fall of 2024, August 19. SCROLL down to the bottom of this page to find SOLUTIONS.${\bf Past \ Problems \ and \ Winners}$ May 2024 This problem was provided by UNT TAMS student Victor Lin. Determine which of the following numbers is a perfect square: 1, 14, 144, 1444, 14444, 144444, . . . . Winners: None Apr. 2024 Let$i = \sqrt{-1}$. Determine whether the following infinite products converge or diverge: $$\prod_{n=1}^{\infty} \left(1 + \frac{i}{n}\right) \textrm{ and } \prod_{n=1}^{\infty} \big\vert1 + \frac{i}{n}\big\vert.$$ Note: $$\prod_{n=1}^{\infty} \left(1 + \frac{i}{n}\right) = \lim_{N\to \infty} \left((1 + i)(1+\frac{i}{2})(1 + \frac{i}{3})\cdots(1 + \frac{i}{N})\right).$$ Winner: Victor Lin Runner-Up: Rishabh Mallidi Mar. 2024 Determine the values of$p>0$for which: $$\sum_{m=1}^{\infty}\left( \sum_{n=1}^{\infty} \frac{1}{(m+n)^p} \right)$$ converges and for which$p>0$it diverges. Winner: Rishabh Mallidi Runner-Up: Victor Lin Feb. 2024 Let$f:{\mathbb R} \to {\mathbb R}$be a function that satisfies: $$f(f(x)) = -x \textrm{ for every } x \in {\mathbb R}.$$ Prove that$f$is one-to-one and onto. In addition, prove that any such function$f$cannot be continuous. (In fact, it is known that any such$f(x)$has an infinite number of discontinuities!) Winner: Rishabh Mallidi Runner-Up: Victor Lin Correct Solution also submitted by Matthew Li Jan. 2024 Determine: $$\lim_{n \to \infty}( n!e - [n!e] )$$ Note:$[x] $is the largest integer$\leq x$so for example$[1.2] =1, [\pi]=3, [-1.2] = -2$. Hint:$e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots. $Winner: Orion Jordan Runner-Up: Victor Lin Dec. 2023 This problem was provided by UNT TAMS student Victor Lin. Let$n$be a positive integer and let$x>0$. Define: $$f(x) = \sum_{k=1}^{2n} \frac{x^\frac{2k}{2n+1}}{ x^\frac{2k}{2n+1} + x }.$$ Determine$f(2023).$Winner: Orion Jordan Nov. 2023 Prove: $$\sum_{k=1}^{n} \frac{k(k+1)(k+2)\cdots(k+N-1)}{N!} = \frac{n(n+1)(n+2)\cdots(n+N) }{(N+1)!}$$ Winner: Victor Lin Runner-Up: Orion Jordan Oct. 2023 Determine: $$\int_{0}^{\infty} \frac{(\ln x)^2}{1+x^2} \, dx.$$ Winners: None Sept. 2023 Let$00$,$b>0$.Determine: $$\int_{0}^{2\pi} \frac{ab}{a^2 \cos^2 t + b^2 \sin^2 t } \, dt.$$ Winner: Zachary Li Runner-Up: Eric Peng Correct solutions were also submitted by Ria Garg and Victor Lin Apr. 2023 Let$n$be a positive integer. Prove that:$$(\sqrt{2} -1)^n = \sqrt{m} - \sqrt{m-1} \textrm{ for some positive integer } m.$$ Winner: Victor Lin Runner-Up: Tyson Ramirez A correct solution was also submitted by Ria Garg Mar. 2023 The${\it Bell \ Numbers \ }$,$b_n$, are the number of ways of partitioning a set of$n$objects into subsets. We define$b_0=1$. It can be shown that$b_1=1, b_2=2, b_3=5, b_4 = 15, $and more generally: $$b_{n+1} = \sum\limits_{k=0}^{n} {n \choose k} b_k.$$ Prove that: $$e^{\left({ e^x-1}\right)} = \sum_{n=0}^{\infty} \frac{b_n}{n!} x^n \ \ \textrm{ and } \ \ b_n = \frac{1}{e} \sum_{k=1}^{\infty} \frac{k^n}{k!}.$$ Winner: Eric Peng Runner-Up: Victor Lin Feb. 2023 Prove that if all the coefficients in$ax^2 + bx+c=0$are odd integers then the roots of the equation cannot be rational. Winner: Victor Lin Runner-Up: Tyson Ramirez Correct solutions were also submitted by: Merdangeli Bayramov, Ria Garg, and Eric Peng. Jan. 2023 Determine the triangle of maximum area that can be inscribed in a circle of radius 1. Winner: Saisneha Ghatti Runner-Up: Zachary Li Correct solutions were also submitted by Ria Garg, Helen Li, and Victor Lin. Dec. 2022 Determine which function is larger:$\sin(\cos(x))$or$\cos(\sin(x)).$Winner: Victor Lin Runner-Up: Zachary Li A correct solution was also submitted by Saisneha Ghatti and Eric Peng. Nov. 2022 Of all triangles with base length$b_0$and perimeter$P_0$prove that the one that has maximum area must be isosceles (i.e. has two sides of equal length). Winner: Eric Peng Runner-Up: Victor Lin Correct Solutions were also submitted by Ria Garg, Saisneha Ghatti, Matthew Li Oct. 2022 Determine all continuous functions which satisfy:$$f\left(\frac{x+y}{2}\right) = \frac{ f(x) + f(y)}{2} \textrm{ for all real } x,y.$$ Winner: Victor Lin Runner-Up: Saisneha Ghatti A correct solutions was also submitted by Eric Peng Partial solutions were submitted by Atharv Chagi, Ria Garg, Zachary Li Sept. 2022 Let$x_i$be real numbers with$x_10$. HINT: It is known that if$A$is the area of the triangle and$P$the perimeter then$A= \frac{1}{2} rP.$Winner: Eric Peng Runner-Up: James Heath Dec. 2021 Determine: $$\prod_{n=2}^{\infty} \frac{n^3-1}{n^3 +1}.$$ Note:$\prod\limits_{n=2}^{\infty} a_n= \lim\limits_{n \to \infty} (a_2 a_3 \cdots a_n).$Winner: Michalis Paizanis Runner-Up: Eric Peng Nov. 2021 Let$00$and$d>0.$Let: $$A_{n} = \frac{1}{n}\sum_{k=0}^{n-1} (a + kd)$$and: $$G_{n} = \left( a(a+d)(a+2d)\cdots(a+(n-1)d) \right)^{1/n}.$$ Determine:$$\lim_{n \to \infty} \frac{G_n}{A_{n}}.$$ Winner: Eric Peng Runner-Up: Yugendra Uppalapati A correct solution was also submitted by Michalis Paizanis Sept. 2021 Determine: $$\lim_{n \to \infty} n \sin(2\pi e n!).$$ Hint:$ e =1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots. $Winner: Michael Holland, Runner-Up: Divya Darji 2020-2021 Academic Year Apr. 2021 Find a simple formula for: $$1{n \choose 1} + 2 { n \choose 2} + 3 {n \choose 3} + \cdots + n{ n \choose n}.$$ Winner: Michalis Paizanis Mar. 2021 Show that there is a continuous function$f: {\mathbb R} \to {\mathbb R}$that takes on every value exactly 3 times. (That is, for any given$y_0 \in {\mathbb R}$there are exactly three solutions$x_1 < x_2 < x_3$of$f(x)=y_0$). Note: It is known that there is no continuous function$f: {\mathbb R} \to {\mathbb R}$that takes on every value exactly 2 times. Winner: Michalis Paizanis Feb. 2021 Show that in any Pythagorean triangle one of the sides must be divisible by 3, one side must be divisible by 4, and one side must be divisible by 5.(A ''Pythagorean triangle'' is a right triangle with sides$a,b$and hypotenuse$c$such that$a,b,c$are positive integers and$c^2=a^2+b^2$). Note: A Pythagorean triple$(a,b,c) $is called ''primitive'' if$(a,b,c)$have no common factors. It is known that every primitive Pythagorean triple can be written in the form$a=m^2-n^2, b= 2mn, c = m^2 + n^2$where$m$,$n$are positive integers with$m>n.$Hint: Try to prove the result for primitive Pythagorean triples first. Winner: Michalis Paizanis Runner-Up: Angela Yuan a correct solution was also submitted by Neel Shanmugam Jan. 2021 Find an equation of a circle$(x-h)^2 + (y-k)^2 =r^2$that has exactly one rational point on it. Also find the equation of a circle that has exactly two rational points on it. Winner: Michalis Paizanis Dec. 2020 Show that there are no rational solutions of$x^2 + y^2 = 3.$Winner: Michalis Paizanis Runner-Up: Subiksha Sankar Nov. 2020 Find all Pythagorean triples where the area of the triangle is equal to its perimeter. Winner: Michalis Paizanis Runner-Up: Angela Yuan other correct solutions by: Alejandro Castellanos, Subisksha Sankar Oct. 2020 Show that none of the numbers is a square: 11, 111, 1111, 11111, . . . Winner: Subiksha Sankar Runner-Up: Michalis Paizanis Sept. 2020 Take a quarter circle with radius$R$in quadrant 1 with one side on the x-axis and one side on the y-axis. Take a semicircle of radius$r2$that there is an$x_n$with$0< x_n < 1$such that$f(x_{n} + \frac{1}{n}) = f(x_{n}).$Winner: David Duhon Runner-Up: Rhythm Garg Jan. 2020 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 Dec. 2019 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 Oct. 2019 Let$00$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: $$\frac{1}{\sqrt[3]{1} + \sqrt[3]{2} + \sqrt[3]{4}} + \frac{1}{\sqrt[3]{4} + \sqrt[3]{6}+\sqrt[3]{9}} + \frac{1}{\sqrt[3]{9} + \sqrt[3]{12} + \sqrt[3]{16}}$$ 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, 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$\displaystyle \prod_{k=2}^nd_k.$Winners: 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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