MathDB

5

Part of ICMC 5

Problems(2)

Tanned vectors

Source: ICMC 2022 Round 1 P5

12/4/2021
A tanned vector is a nonzero vector in R3\mathbb R^3 with integer entries. Prove that any tanned vector of length at most 20212021 is perpendicular to a tanned vector of length at most 100100.
Proposed by Ethan Tan
3dVectorsdot productvectorlinear algebracollege contestsICMC
A robot's random walk

Source: ICMC 2022 Round 2 P5

3/1/2022
A robot on the number line starts at 11. During the first minute, the robot writes down the number 11. Each minute thereafter, it moves by one, either left or right, with equal probability. It then multiplies the last number it wrote by n/tn/t, where nn is the number it just moved to, and tt is the number of minutes elapsed. It then writes this number down. For example, if the robot moves right during the second minute, it would write down 2/2=12/2=1.
Find the expected sum of all numbers it writes down, given that it is finite.
Proposed by Ethan Tan
probabilityexpected valueICMCcollege contests