[hide=D stands for Descartes, L stands for Leibniz]they had two problem sets under those two names[/hide] 

[u]Set 1[/u]


[b]D.1 / L.1[/b] Find the units digit of $3^{1^{3^{3^7}}}$.


[b]D.2[/b] Find the positive solution to the equation $x^3 - x^2 = x - 1$.


[b]D.3[/b] Points $A$ and $B$ lie on a unit circle centered at O and are distance $1$ apart. What is the degree measure of $\angle AOB$?


[b]D.4[/b] A number is a perfect square if it is equal to an integer multiplied by itself. How many perfect squares are there between $1$ and $2019$, inclusive?


[b]D.5[/b]  Ted has four children of ages $10$, $12$, $15$, and $17$. In fifteen years, the sum of the ages of his children will be twice Ted’s age in fifteen years. How old is Ted now?


[u]Set 2[/u]


[b]D.6[/b] Mr. Schwartz is on the show Wipeout, and is standing on the first of $5$ balls, all in a row. To reach the finish, he has to jump onto each of the balls and collect the prize on the final ball. The probability that he makes a jump from a ball to the next is $1/2$, and if he doesn’t make the jump he will wipe out and no longer be able to finish. Find the probability that he will finish.


[b]D.7 / L. 5[/b] Kevin has written $5$ MBMT questions. The shortest question is $5$ words long, and every other question has exactly twice as many words as a different question. Given that no two questions have the same number of words, how many words long is the longest question?


[b]D.8  / L. 3[/b] Square $ABCD$ with side length $1$ is rolled into a cylinder by attaching side $AD$ to side $BC$. What is the volume of that cylinder?


[b]D.9 / L.4[/b] Haydn is selling pies to Grace. He has $4$ pumpkin pies, $3$ apple pies, and $1$ blueberry pie. If Grace wants $3$ pies, how many different pie orders can she have?


[b]D.10[/b] Daniel has enough dough to make $8$ $12$-inch pizzas and $12$ $8$-inch pizzas. However, he only wants to make $10$-inch pizzas. At most how many $10$-inch pizzas can he make?


[u]Set 3[/u]


[b]D.11 / L.2[/b] A standard deck of cards contains $13$ cards of each suit (clubs, diamonds, hearts, and spades). After drawing $51$ cards from a randomly ordered deck, what is the probability that you have drawn an odd number of clubs?


[b]D.12 / L. 7[/b] Let $s(n)$ be the sum of the digits of $n$. Let $g(n)$ be the number of times s must be applied to n until it has only $1$ digit. Find the smallest n greater than $2019$ such that $g(n) \ne g(n + 1)$.


[b]D.13 / L. 8[/b] In the Montgomery Blair Meterology Tournament, individuals are ranked (without ties) in ten categories. Their overall score is their average rank, and the person with the lowest overall score wins. Alice, one of the $2019$ contestants, is secretly told that her score is $S$. Based on this information, she deduces that she has won first place, without tying with anyone. What is the maximum possible value of $S$?


[b]D.14 / L. 9[/b] Let $A$ and $B$ be opposite vertices on a cube with side length $1$, and let $X$ be a point on that cube. Given that the distance along the surface of the cube from $A$ to $X$ is $1$, find the maximum possible distance along the surface of the cube from $B$ to $X$.


[b]D.15[/b] A function $f$ with $f(2) > 0$ satisfies the identity $f(ab) = f(a) + f(b)$ for all $a, b > 0$. Compute $\frac{f(2^{2019})}{f(23)}$.



PS. You should use hide for answers.  D.1-15 / L1-9 problems have been collected [url=https://artofproblemsolving.com/community/c3h2790795p24541357]here [/url] and L10,16-30  [url=https://artofproblemsolving.com/community/c3h2790825p24541816]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the volume of the cylinder formed by rolling square \(ABCD\) with side length 1 into a cylinder by attaching side \(AD\) to side \(BC\), we need to follow these steps:

1. **Determine the radius of the cylinder:**
   - When the square is rolled into a cylinder, the side length of the square becomes the circumference of the base of the cylinder.
   - The circumference \(C\) of the base of the cylinder is equal to the side length of the square, which is 1.
   - Using the formula for the circumference of a circle, \(C = 2\pi r\), we can solve for the radius \(r\):
     \[
     2\pi r = 1 \implies r = \frac{1}{2\pi}
     \]

2. **Determine the height of the cylinder:**
   - The height \(h\) of the cylinder is equal to the other side length of the square, which is 1.

3. **Calculate the volume of the cylinder:**
   - The volume \(V\) of a cylinder is given by the formula \(V = \pi r^2 h\).
   - Substituting the values of \(r\) and \(h\) into the formula:
     \[
     V = \pi \left(\frac{1}{2\pi}\right)^2 \cdot 1 = \pi \cdot \frac{1}{4\pi^2} \cdot 1 = \frac{\pi}{4\pi^2} = \frac{1}{4\pi}
     \]

Thus, the volume of the cylinder is \(\boxed{\frac{1}{4\pi}}\).