Found problems: 30
2017 ASDAN Math Tournament, 2
Two distinct positive factors of $144$ are selected at random. What is the probability that their product is greater than $144$?
2017 ASDAN Math Tournament, 3
What is the remainder when $2^{1023}$ is divided by $1023$?
2017 ASDAN Math Tournament, 5
A $\textit{shuffle}$ is a permutation of the integers $1,2,3,4,5$. More formally, a shuffle is a function $f:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ such that if $i\neq j$ then $f(i)\neq f(j)$. For example, $12345\mapsto23154$ denotes a shuffle $f$ so that $f(1)=2$, $f(2)=3$, $f(3)=1$, $f(4)=5$, and $f(5)=4$. A shuffle can be repeated some number of times to obtain another shuffle. For example, if $f$ is the shuffle $12345\mapsto23154$ from above, then repeating $f$ twice gives the shuffle $g(x)=f(f(x))$ which is $12345\mapsto31245$. How many shuffles are there that, when repeated $6$ times, give the shuffle $12345\mapsto12345$?
2016 ASDAN Math Tournament, 3
Julia adds up the numbers from $1$ to $2016$ in a calculator. However, every time she inputs a $2$, the calculator malfunctions and inputs a $3$ instead (for example, when Julia inputs $202$, the calculator inputs $303$ instead). How much larger is the total sum returned by the broken calculator? (No $2$s are replaced by $3$s in the output, and the calculator only malfunctions while Julia is inputting numbers.)
2016 ASDAN Math Tournament, 7
Heesu, Xingyou, and Bill are in a class with $9$ other children. The teacher randomly arranges the children in a circle for story time. However, Heesu, Xingyou, and Bill want to sit near each other. Compute the probability that all $3$ children are seated within a consecutive group of $5$ seats.