Olympiad problems

2018 ASDAN Math Tournament, 3

Each day, the city of Berkeley is either rainy or foggy, with a $\tfrac{3}{4}$ chance of the weather remaining the same as that of the previous day. If we only know that it is rainy today, what is the probability that it is rainy in $7$ days?

2019 ASDAN Math Tournament, 1

Tags: 2019 , Discrete Math Tiebreaker

What is the greatest positive integer $x$ for which $2^{2^x+1}+2$ is divisible by $17$?

2018 ASDAN Math Tournament, 2

Tags: 2018 , Discrete Math Tiebreaker

What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?

2019 ASDAN Math Tournament, 2

Tags: 2019 , Discrete Math Tiebreaker

Let $P_1,P_2,\dots,P_{720}$ denote the integers whose digits are a permutation of $123456$, arranged in ascending order (so $P_1=123456$, $P_2=123465$, and $P_{720}=654321$). What is $P_{144}$?

2019 ASDAN Math Tournament, 3

Tags: 2019 , Discrete Math Tiebreaker

$5$ monkeys, $5$ snakes, and $5$ tigers are standing in line at the local grocery store, with animals of the same species being indistinguishable. A monkey stands at the front of the line and a tiger stands at the end of the line. Unfortunately, monkeys and tigers are sworn enemies, so monkeys and tigers cannot stand in adjacent places in line. Compute the number of possible arrangements of the line.

2017 ASDAN Math Tournament, 1

Tags: 2017 , Discrete Math Tiebreaker

Two arbitrary distinct lattice points are selected on the coordinate plane within the square marked by the points $(0,0)$, $(3,0)$, $(0,3)$, and $(3,3)$ (the lattice points may lie on a side or a corner of the square). What is the probability that the distance between the two points is at most $\sqrt{2}$?

2016 ASDAN Math Tournament, 1

Tags: 2016 , Discrete Math Tiebreaker

You own two cats, Chocolate and Tea. Chocolate and Tea sleep for $C$ and $T$ hours a day respectively, where $C$ and $T$ are chosen independently and uniformly at random from the interval $[5,10]$. In a given day, what is the probability that Chocolate and Tea will together sleep for a total of at least $14$ hours?

2017 ASDAN Math Tournament, 2

Tags: 2017 , Discrete Math Tiebreaker

Find the remainder of $7^{1010}+8^{2017}$ when divided by $57$.

2016 ASDAN Math Tournament, 2

Tags: 2016 , Discrete Math Tiebreaker

Define a $\textit{subsequence}$ of a string $\mathcal{S}$ of letters to be a positive-lenght string using any number of the letters in $\mathcal{S}$ in order. For example, a subsequence of $HARRISON$ is $ARRON$. Compute the number of subsequences in $HARRISON$.

2016 ASDAN Math Tournament, 3

Tags: 2016 , Discrete Math Tiebreaker

Find the $2016$th smallest positive integer that is a solution to $x^x\equiv x\pmod{5}$.

2018 ASDAN Math Tournament, 1

Tags: 2018 , Discrete Math Tiebreaker

Each vertex on a cube is colored black or white independently at random with equal probability. What is the expected number of edges on the cube that connect vertices of different colors?

2017 ASDAN Math Tournament, 3

Tags: 2017 , Discrete Math Tiebreaker

Alex and Zev are two members of a group of $2017$ friends who all know each other. Alex is trying to send a package to Zev. The delivery process goes as follows: Alex sends the package randomly to one of the people in the group. If this person is Zev, the delivery is done. Otherwise, the person who received the package also randomly sends it to someone in the group who hasn't held the package before and this process repeats until Zev gets the package. What is the expected number of deliveries made?