MathDB

2.4

Part of 2021 CMIMC

Problems(3)

2021 Alg/NT Div 2 P4

Source:

3/2/2021
What is the 101101st smallest integer which can represented in the form 3a+3b+3c3^a+3^b+3^c, where a,b,a,b, and cc are integers?
Proposed by Dilhan Salgado
algebranumber theory
2021 Geo Div 2 P4

Source:

3/2/2021
A 252\sqrt5 by 454\sqrt5 rectangle is rotated by an angle θ\theta about one of its diagonals. If the total volume swept out by the rotating rectangle is 62π62\pi, find the measure of θ\theta in degrees.
Proposed by Connor Gordon
geometry
2021 Combo Div 2 P4

Source:

3/2/2021
Vijay has a stash of different size stones: in particular, he has 20212021 types of stones, with sizes from 00 through 20202020, and he has 2r+12r+1 stones of size rr.
Vijay starts randomly (and without replacement) taking out stones from his stash and laying them out in a line. Vijay notices that the first stone of size 20202020 comes before the first stone of size 20192019, the first stone of size 20192019 is before the first stone of size 20182018, and so on. What is the probability of this happening?
Express your answer in terms of only basic arithmetic operations (division, exponentiation, etc.) and the factorial function.
Proposed by Misha Ivkov
combinatorics