[u]Round 1[/u]


[b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!)


[b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started?


[b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips?


[u]Round 2[/u]


[b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine?


[b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved?


[b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$.


[u]Round 3 [/u]


[b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$.


[b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that
$\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$.
$\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over.
Find the smallest possible number of boxes of food.


[b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$.


[u]Round 4 [/u]


[b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$.


[b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$?


[b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$?


PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
### Problem 1
1. There are 7 rounds, and each round has 3 problems.
2. To find the total number of problems, we multiply the number of rounds by the number of problems per round:
   \[
   7 \times 3 = 21
   \]

The final answer is \(\boxed{1365}\)