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: 20

2015 ASDAN Math Tournament, 5
Tags: 2015 , Advanced Topics Test
Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.

2014 ASDAN Math Tournament, 9
Tags: 2014 , Advanced Topics Test
Compute how many permutations of the numbers $1,2,\dots,8$ have no adjacent numbers that sum to $9$.

2015 ASDAN Math Tournament, 7
Tags: 2015 , Advanced Topics Test
What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$?

2014 ASDAN Math Tournament, 5
Tags: 2014 , Advanced Topics Test
Compute the smallest $9$-digit number containing all the digits $1$ to $9$ that is divisible by $99$.

2014 ASDAN Math Tournament, 2
Tags: 2014 , Advanced Topics Test
Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers.

2015 ASDAN Math Tournament, 9
Tags: 2015 , Advanced Topics Test
You play a game with a biased coin, which has probability $\tfrac{3}{4}$ of landing heads. Each time you toss heads, you score $1$ point, while tossing tails earns no points. After any turn, you can stop playing the game and keep the points you currently have. However, if you are still playing when you toss tails for the second time, you lose all of your points. If you play to maximize your expected score, what is your expected score from playing this game?

2015 ASDAN Math Tournament, 8
Tags: 2015 , Advanced Topics Test
You have $8$ friends, each of whom lives at a different vertex of a cube. You want to chart a path along the cube’s edges that will visit each of your friends exactly once. You can start at any vertex, but you must end at the vertex you started at, and you cannot travel on any edge more than once. How many different paths can you take?

2015 ASDAN Math Tournament, 1
Tags: 2015 , Advanced Topics Test
How many integers between $2$ and $100$ have only odd numbers in their prime factorizations?

2015 ASDAN Math Tournament, 6
Tags: 2015 , Advanced Topics Test
You, your friend, and two strangers are sitting at a table. A standard $52$-card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces.

2014 ASDAN Math Tournament, 8
Tags: 2014 , Advanced Topics Test
Nick has a $3\times3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create.

2015 ASDAN Math Tournament, 3
Tags: 2015 , Advanced Topics Test
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.

2014 ASDAN Math Tournament, 4
Tags: 2014 , Advanced Topics Test
Cynthia and Lynnelle are collaborating on a problem set. Over a $24$-hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$-hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.

2014 ASDAN Math Tournament, 1
Tags: 2014 , Advanced Topics Test
Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.

2015 ASDAN Math Tournament, 2
Tags: 2015 , Advanced Topics Test
Nick is taking a $10$ question test where each answer is either true or false with equal probability. Nick forgot to study, so he guesses randomly on each of the $10$ problems. What is the probability that Nick answers exactly half of the questions correctly?

2014 ASDAN Math Tournament, 6
Tags: 2014 , Advanced Topics Test
Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords.

2014 ASDAN Math Tournament, 10
Tags: 2014 , Advanced Topics Test
Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$. Your answer should be an integer between $0$ and $42$.

2015 ASDAN Math Tournament, 4
Tags: 2015 , Advanced Topics Test
Given a positive integer $x>1$ with $n$ divisors, define $f(x)$ to be the product of the smallest $\lceil\tfrac{n}{2}\rceil$ divisors of $x$. Let $a$ be the least value of $x$ such that $f(x)$ is a multiple of $X$, and $b$ be the least value of $n$ such that $f(y)$ is a multiple of $y$ for some $y$ that has exactly $n$ factors. Compute $a+b$.

2015 ASDAN Math Tournament, 10
Tags: 2015 , Advanced Topics Test
Let $\sigma(n)$ be the sum of all the positive divisors of $n$. Let $a$ be the smallest positive integer greater than or equal to $2015$ for which there exists some positive integer $n$ satisfying $\sigma(n)=a$. Finally, let $b$ be the largest such value of $n$. Compute $a+b$.

2014 ASDAN Math Tournament, 7
Tags: 2014 , Advanced Topics Test
Two math students play a game with $k$ sticks. Alternating turns, each one chooses a number from the set $\{1,3,4\}$ and removes exactly that number of sticks from the pile (so if the pile only has $2$ sticks remaining the next player must take $1$). The winner is the player who takes the last stick. For $1\leq k\leq100$, determine the number of cases in which the first player can guarantee that he will win.

2014 ASDAN Math Tournament, 3
Tags: 2014 , Advanced Topics Test
A mouse is playing a game of mouse hopscotch. In mouse hopscotch there is a straight line of $11$ squares, and starting on the first square the mouse must reach the last square by jumping forward $1$, $2$, or $3$ squares at a time (so in particular the mouse’s first jump can be to the second, third, or fourth square). The mouse cannot jump past the last square. Compute the number of ways there are to complete mouse hopscotch.