MathDB

1.5

Part of 2022 CMIMC

Problems(3)

2022 Alg/NT DIv 1 P5

Source:

2/28/2022
Grant is standing at the beginning of a hallway with infinitely many lockers, numbered, 1,2,3,…1, 2, 3, \ldots All of the lockers are initially closed. Initially, he has some set S={1,2,3,…}S = \{1, 2, 3, \ldots\}.
Every step, for each element ss of SS, Grant goes through the hallway and opens each locker divisible by ss that is closed, and closes each locker divisible by ss that is open. Once he does this for all ss, he then replaces SS with the set of labels of the currently open lockers, and then closes every door again.
After 20222022 steps, SS has nn integers that divide 102022{10}^{2022}. Find nn.
Proposed by Oliver Hayman
algebranumber theory
2022 Geo Div 1 P5

Source:

2/28/2022
In triangle ABCABC, let I,O,HI, O, H be the incenter, circumcenter and orthocenter, respectively. Suppose that AI=11AI = 11 and AO=AH=13AO = AH = 13. Find OHOH.
Proposed by Kevin You
geometry
2022 Combo Div 1 P5

Source:

2/28/2022
At CMIMC headquarters, there is a row of nn lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things:
[*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken). [*] Check if a given lightbulb is on or off. [*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on. Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most nn seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value nn such that he can complete his task on time?
Proposed by Adam Bertelli
combinatorics