1) Find $\displaystyle \sum_{n=2}^{n=\infty }\frac{n^22n4}{n^4+4n^2+16}$ 2) Evaluate $\displaystyle \int_{0}^{2}\frac{\left ( 16x^2 \right )x}{16x^2+\sqrt{\left ( 4x \right )\left ( 4+x \right )\left ( 12+x^2 \right )}}\: dx$ 3) Find the least positive integer $n$ such that $2^{2014}$ divides $19^n1$ 4) Suppose we are given a 19 × 19 chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^21=360$ squares with 4 × 1 and 1 × 4 rectangles? (So each of the 360 squares is covered by exactly one rectangle.) Justify your answer. 5) Let $n\geq 1$ and $r\geq 2$ be positive integers. Prove that there is no integer $m$ such that $n\left ( n+1 \right )\left ( n+2 \right )=m^r$. 6) Let $S$ denote the set of $2$ by $2$ matrices with integer entries and determinant $1$, and let $T$ denote those matrices of $S$ which are congruent to the identity matrix $I$ mod $3$ (so $\displaystyle \left ( \begin{matrix} a & b\\ c & d \end{matrix} \right )\in T$ means that $a,b,c,d\in \mathbb{Z}$, $adbc=1$, and $3$ divides $b,c,a1,d1$; "$\in$" means "is in"). (a) Let $f:T\rightarrow \mathbb{R}$ (the real numbers) be a function such that for every $X,Y\in T$ with $Y\neq I$, either $f\left ( XY \right )>f\left ( X \right )$ or $f\left ( XY^{1} \right )>f\left ( X \right )$ (or both). Show that given two finite nonempty subsets $A, B\; \text{of}\; T$, there are matrices $a\in A$ and $b\in B$ such that if $a'\in A$, $b'\in B$ and $a'b'=ab$, then $a'=a$ and $b'=b$. (b) Show that there is no $f:S\rightarrow \mathbb{R}$ such that for every $X,Y\in S$ with $Y\neq \pm I$, either $f\left ( XY \right )>f\left ( X \right )$ or $f\left ( XY^{1} \right )>f\left ( X \right )$. 7) Let $A$, $B$ be two points in the plane with integer coordinates $\displaystyle A=\left ( x_1,y_1 \right )$ and $\displaystyle B=\left ( x_2,y_2 \right )$. (Thus $\displaystyle x_i,y_i\in\mathbb{Z}$, for $i=1,2$.) A path $\pi$: $A\rightarrow B$ is a sequence of down and right steps, where each step has an integer length, and the initial step starts from $A$, the last step ending at $B$. In the figure below, we indicated a path from $\displaystyle A_1=\left ( 4,9 \right )$ to $\displaystyle B_1=\left ( 10,3 \right )$. The distance $d\left ( A,B \right )$ between $A$ and $B$ is the number of such paths. For example, the distance between $A=\left ( 0,2 \right )$ and $B=\left ( 2,0 \right )$ equals 6. Consider now two pairs of points in the plane $\displaystyle A_i=\left ( x_i,y_i \right )$ and $\displaystyle B_i=\left ( u_i,z_i \right )$ for $i=1,2$ with integer coordinates, and in the configuration shown in the picture (but with arbitrary coordinates):
(a) Find the distance between two points $A$ and $B$ as before, as a function of the coordinates of $A$ and $B$. Assume that $A$ is NorthWest of $B$.
(b) Consider the 2 × 2 matrix $\displaystyle M=\left ( \begin{matrix} d\left ( A_1,B_1 \right ) & d\left ( A_1,B_2 \right ) \\ d\left ( A_2,B_1 \right ) & d\left ( A_2,B_2 \right ) \end{matrix} \right )$. Prove that for any configuration of points $A_1,A_2,B_1,B_2$ as described before, $\det M>0$. Here are some interesting facts about math! Proof is left as an exercise for the reader. Part I contains numbers 1 through 10. Stay tuned for Part II which will feature numbers 11 through 23! 1. If you write out pi to two decimal places, backwards it spells “pie”. 3.14=PIE 2. A French word for pie chart is “camembert”. Because of course it is. 3. The spiral shapes of sunflowers follow a Fibonacci sequence. That’s where you add the two preceding numbers in the sequence to give you the next one. So it starts 1, 1, 2, 3, 5, 8, 13, 21, etc. The Fibonacci sequence shows up in nature a fair bit. 4. The Fibonacci sequence is encoded in the number 1/89. $\frac{1}{89}$=0.01+0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + 0.000000021 + 0.0000000034, ect 5. A pizza that has radius “z” and height “a” has volume Pi × z × z × a. Because the area of a circle is Pi multiplied by the radius squared (which can be written out as Pi × z × z). Then you multiply by the height to get the total volume. 6. The word hundred is derived from the word “hundrath”, which actually means 120 and not 100. Hundrath is Old Norse. 7. 111,111,111 × 111,111,111 = 12,345,678,987,654,321. It also works for smaller numbers: $111\times111=12,321$ 8. In a room of just 23 people there’s a 50% chance that two people have the same birthday It’s called the Birthday Problem. In a room of 75 there’s a 99% chance of two people matching. 9. Zero is the only number that can’t be represented in Roman numerals.10. (6 × 9) + (6 + 9) = 69. Stay Tuned for Part II featuring numbers 11 through 23!
Back by popular demand, we present to you Funny Little Calculus Text by Robert W. Ghrist. You can buy a copy of this in Google Play at https://play.google.com/store/books/details/Robert_W_Ghrist_FLCT_Funny_Little_Calculus_Text! Need a new wallpaper for your computer? If so, why not make it "mathy" by using some of the ones provided below! To save a copy, click the image you want and select the Download! link.
