MathDB

4

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/4: Many composites in 1, ..., n!-1

Source:

2/17/2019
Find all positive integers nn for which there do not exist nn consecutive composite positive integers less than n!n!.
HMMTnumber theory
HMMT Algebra/NT 2019/4: Binary recursive function

Source:

2/17/2019
Let N\mathbb{N} be the set of positive integers, and let f:N→Nf: \mathbb{N} \to \mathbb{N} be a function satisfying
[*] f(1)=1f(1) = 1, [*] for n∈Nn \in \mathbb{N}, f(2n)=2f(n)f(2n) = 2f(n) and f(2n+1)=2f(n)−1f(2n+1) = 2f(n) - 1.
Determine the sum of all positive integer solutions to f(x)=19f(x) = 19 that do not exceed 2019.
HMMTalgebrafunction
HMMT Combinatorics 2019/4: Yannick is Also Playing a Game

Source:

2/17/2019
Yannick is playing a game with 100100 rounds, starting with 11 coin. During each round, there is an n%n\% chance that he gains an extra coin, where nn is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?
HMMTprobabilityexpected value
HMMT Geometry 2019/4: Hexagon with three areas

Source:

2/17/2019
Convex hexagon ABCDEFABCDEF is drawn in the plane such that ACDFACDF and ABDEABDE are parallelograms with area 168. ACAC and BDBD intersect at GG. Given that the area of AGBAGB is 10 more than the area of CGBCGB, find the smallest possible area of hexagon ABCDEFABCDEF.
HMMTgeometry