MathDB

3

Part of ICMC 6

Problems(2)

Fibonacci hops in the Euclidean plane

Source: ICMC 2023 Round 1 P3

11/28/2022
Bugs Bunny plays a game in the Euclidean plane. At the nn-th minute (n≥1)(n \geq 1), Bugs Bunny hops a distance of FnF_n in the North, South, East, or West direction, where FnF_n is the nn-th Fibonacci number (defined by F1=F2=1F_1 = F_2 =1 and Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2} for n≥3n \geq 3). If the first two hops were perpendicular, prove that Bugs Bunny can never return to where he started.
Proposed by Dylan Toh
ICMCcombinatoricsFibonaccicollege contests
Erasing random numbers on a blackboard

Source: ICMC 2023 Round 2 P3

3/1/2023
The numbers 1,2,…,n1, 2, \dots , n are written on a blackboard and then erased via the following process: [*] Before any numbers are erased, a pair of numbers is chosen uniformly at random and circled. [*] Each minute for the next n−1n -1 minutes, a pair of numbers still on the blackboard is chosen uniformly at random and the smaller one is erased. [*] In minute nn, the last number is erased.
What is the probability that the smaller circled number is erased before the larger?
Proposed by Ethan Tan
ICMCcollege contestsprobabilitycombinatoricsrandom process