Found problems: 6
2014 ASDAN Math Tournament, 1
Compute the number of three digit numbers such that all three digits are distinct and in descending order, and one of the digits is a $5$.
2014 ASDAN Math Tournament, 3
A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0\leq a,b\leq5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
2015 ASDAN Math Tournament, 3
You have a circular necklace with $10$ beads on it, all of which are initially unpainted. You randomly select $5$ of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?
2015 ASDAN Math Tournament, 2
Heesu plays a game where he starts with $1$ piece of candy. Every turn, he flips a fair coin. On heads, he gains another piece of candy, unless he already has $5$ pieces of candy, in which case he loses $4$ pieces of candy and goes back to having $1$ piece of candy. On tails, the game ends. What is the expected number of pieces of candy that Heesu will have when the game ends?
2014 ASDAN Math Tournament, 2
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.
2015 ASDAN Math Tournament, 1
Rachel has $3$ children, all of which are at least $2$ years old. The ages of the children are all pairwise relatively prime, but Rachel’s age is a multiple of each of her children’s ages. What is Rachel’s minimum possible age?