For positive integers $n$ and $k$, let $f(n, k)$ be the remainder when $n$ is divided by $k$, and for $n > 1$ let $F(n) =\displaystyle\max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)$. Find the remainder when $\sum\limits_{n=20}^{100} F(n)$ is divided by $1000$.

What is the sum of the digits of the square of $111,111,111$?

Call a positive integer squarish if it contains the digits of the squares of its digits in order but not necessarily contiguous. For example, $14263$ contains $1^2 = 1$, $4^2 = 16$ and $2^2 = 4$. However, it is not squarish because it does not contain $3^2 = 9$, and $6^2 = 36$ is not in order. What is the smallest squarish number that includes at least one digit greater than $1$?

Find all primes $p$ for which there exist positive integers $x$, $y$, and $n$ such that $$p^n = x^3+y^3$$

A sequence satisfies $a_1 = 3, a_2 = 5$, and $a_{n+2} = a_{n+1} - a_n$ for $n \ge 1$. What is the value of $a_{2018}$?

Take a list of positive integers $1$, $2$, $3$, $\cdots$, $2017$. At each step, pick up two of the numbers on the list, say $a$ and $b$, cross them out and replace them by the single number $(ab+a+b)$. Keep doing this until only a single number is left. What is (are) the possible value(s) of this last number?

There are $100$ lights lined up in a long room. Each light has its own switch and is currently off. The room has an entry door and an exit door. There are $100$ people lined up outside the entry door. Each light is numbered consecutively from $1$ to $100$. So is each person.

Person No. $1$ enters the room, switches on every light, and exits. Person No. $2$ enters and flips the switch on every second light (i.e. turn off lights $2$, $4$, $6$...). Person No. $3$ enters and flips the switch on every third light (i.e. toggle lights $3$, $6$, $9$...). This continues until all $100$ people have passed through the room. How many of the lights are on at the end?

$\textbf{Cheating Husbands}$

A remote town comprises of $100$ married couples. Everyone in the town lives by the following rule: If a husband cheats on his wife, the husband is executed at the night as soon as his wife finds out about it. All the women in the town only gossip about husbands of other women. No woman ever tells another woman if that woman's husband is cheating on her. So every woman in the town knows about all the cheating husbands in the town except her own. It can also be assumed that a husband remains silent about his infidelity. One day, the mayor of the town announces to the whole town that there is at least $1$ cheating husband in the town. What will happen afterwards?