This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 30

2017 ASDAN Math Tournament, 8
Tags: 2017 , Discrete Math Test
Let $S=\{1,2,3,4,5,6\}$. Compute the number of functions $f:S\rightarrow S$ such that $f(f(f(s)))=2$ if $s$ is odd and $f(f(f(s)))=1$ if $s$ is even.

2017 ASDAN Math Tournament, 9
Tags: 2017 , Discrete Math Test
Compute the number of positive integers $n\leq1330$ for which $\tbinom{2n}{n}$ is not divisible by $11$.

2017 ASDAN Math Tournament, 7
Tags: 2017 , Discrete Math Test
Alice and Bob play a game where on each turn, Alice rolls a die and Bob flips a coin. Bob wins the game if he flips $3$ heads before Alice rolls a $6$. What is the probability that Bob wins? Note that Bob does not in if he flips his third head the same turn Alice rolls her first $6$.

2018 ASDAN Math Tournament, 10
Tags: 2018 , Discrete Math Test
Let $p$ be an odd prime. A degree $d$ polynomial $f$ with non-negative integer coefficients less than $p$ is called $p-floppy$ if the coefficients of $f(x)f(-x) - f(x^2)$ are all divisible by $p$ and if exactly $d$ entries in the sequence $(f(0), f(1), f(2), \dots , f(p-1))$ are divisible by $p$. How many non-constant $61$-floppy polynomials are there?

2018 ASDAN Math Tournament, 8
Tags: 2018 , Discrete Math Test
Compute the remainder when $$\sum_{n=1}^{2018} n^4$$ is divided by $53$.

2016 ASDAN Math Tournament, 8
Tags: 2016 , Discrete Math Test
Consider all fractions $\tfrac{a}{b}$ where $1\leq b\leq100$ and $0\leq a\leq b$. Of these fractions, let $\tfrac{m}{n}$ be the smallest fraction such that $\tfrac{m}{n}>\tfrac{2}{7}$. What is $\tfrac{m}{n}$?

2016 ASDAN Math Tournament, 4
Tags: 2016 , Discrete Math Test
At a festival, Jing Jing plays a game where she must knock down ten targets with as few balls as possible. Every time Jing Jing knocks down a target, she can reuse the ball she just threw and does not have to pick up a new ball. Suppose that Jing Jing knocks down each target with a probability of $\tfrac{3}{4}$. Compute the expected number of balls that Jing Jing needs to knock down all ten targets.

2018 ASDAN Math Tournament, 4
Tags: Discrete Math Test , 2018
What is the remainder when $13^{16} + 17^{12}$ is divided by $221$?

2018 ASDAN Math Tournament, 1
Tags: 2018 , Discrete Math Test
Moor has $3$ different shirts, labeled $T, E,$ and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days?

2018 ASDAN Math Tournament, 2
Tags: 2018 , Discrete Math Test
Aurick throws $2$ fair $6$-sided dice labeled with the integers from $1$ through $6$. What is the probability that the sum of the rolls is a multiple of $3$?

2018 ASDAN Math Tournament, 6
Tags: 2018 , Discrete Math Test
Sam and Ben are each flipping fair coins. If Sam flips a single coin until he gets a tails, and Ben flips $10$ coins in total, what is the probability Sam and Ben get the same number of heads?

2017 ASDAN Math Tournament, 1
Tags: 2017 , Discrete Math Test
Clara and Nick each randomly and independently pick an integer between $0$ and $2017$, inclusive. What is the probability that the two integers they pick sum to an even number?

2016 ASDAN Math Tournament, 6
Tags: 2016 , Discrete Math Test
A container is filled with a total of $51$ red and white balls and has at least $1$ red ball and $1$ white ball. The probability of picking up $3$ red balls and $1$ white ball, without replacement, is equivalent to the probability of picking up $1$ red ball and $2$ white balls, without replacement. Compute the original number of red balls in the container.

2018 ASDAN Math Tournament, 9
Tags: 2018 , Discrete Math Test
Alice starts at the top of Pascal’s triangle. Every move, she moves one layer below, choosing either the left or the right with equal probability. After making $6$ moves, what is the expected sum of the values she visited, including the starting and ending values? For example, in the path shown below, the sum of the values Alice visited is $1 + 1 + 1 + 3 + 6 + 10 + 20 = 42$. [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMC84L2JjZDhiYjkzNjgyMTczMGQ0ZWIzZjE4NDVkOWIxODQxYzQxODdlLnBuZw==&rn=QS5wbmc=[/img][/center]

2016 ASDAN Math Tournament, 10
Tags: 2016 , Discrete Math Test
Let $\mathcal{S}$ be the set of all possible $9$-digit numbers that use $1,2,3,\dots,9$ each exactly once as a digit. What is the probability that a randomly selected number $n$ from $\mathcal{S}$ is divisible by $27$?

2017 ASDAN Math Tournament, 4
Tags: 2017 , Discrete Math Test
How many $6$-digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit.

2017 ASDAN Math Tournament, 6
Tags: 2017 , Discrete Math Test
The sum $$\sum_{n=0}^{2016\cdot2017^2}2018^n$$ can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$.

2016 ASDAN Math Tournament, 5
Tags: 2016 , Discrete Math Test
$ABCD$ is a four digit number ($A\neq0$) such that both $ABC$ and $BCD$ are divisible by $9$ ($ABCD$ is not necessarily divisible by $9$, and $B,C,D$ may be $0$). Compute the number of four digit numbers satisfying this property.

2016 ASDAN Math Tournament, 1
Tags: 2016 , Discrete Math Test
Moor owns $3$ shirts, one each of black, red, and green. Moor also owns $3$ pairs of pants, one each of white, red, and green. Being stylish, he decides to wear an outfit consisting of one shirt and one pair of pants that are different colors. How many combinations of shirts and pants can Moor choose?

2018 ASDAN Math Tournament, 7
Tags: 2018 , Discrete Math Test
Nathan starts with the number $0$, and randomly adds either $1$ or $2$ with equal probability until his number reaches or exceeds $2018$. What is the probability his number ends up being exactly $2018$?

2017 ASDAN Math Tournament, 10
Tags: 2017 , Discrete Math Test
Alice lives on a continent with $6$ countries labeled $1$ through $6$. Each country randomly chooses one other country to allow entry from. Alice can travel to any country that allows entry from the country she is currently in, and can travel along a path through multiple countries in this manner. If Alice starts in county $1$, what is the expected number of countries that she can reach (including country $1$)?

2018 ASDAN Math Tournament, 3
Tags: 2018 , Discrete Math Test
In a bag are all natural numbers less than or equal to $999$ whose digits sum to $6$. What is the probability of drawing a number from the bag that is divisible by $11$?

2016 ASDAN Math Tournament, 2
Tags: 2016 , Discrete Math Test
The largest factor of $n$ not equal to $n$ is $35$. Compute the largest possible value of $n$.

2016 ASDAN Math Tournament, 9
Tags: 2016 , Discrete Math Test
Define $\phi_n(x)$ to be the number of integers $y$ less than or equal to $n$ such that $\gcd(x,y)=1$. Also, define $m=\text{lcm}(2016,6102)$. Compute $$\frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}.$$

2018 ASDAN Math Tournament, 5
Tags: 2018 , Discrete Math Test
An ant traverses between vertices on a unit cube such that at each vertex, it uniformly at random chooses an adjacent vertex to travel to. What is the expected distance travelled by the ant until it returns to its starting vertex?