2.3 1.1

Part of 2021 CMIMC

Problems(2)

2021 Alg/NT Div 1 P1 (Div 2 P3)

Source:

How many multiples of

12

12!

and have exactly

12

divisors?
Proposed by Adam Bertelli

2021 Combo Div 1 P1 (Div 2 P3)

Source:

Adam has a box with

15

pool balls in it, numbered from

1

15

, and picks out

5

of them. He then sorts them in increasing order, takes the four differences between each pair of adjacent balls, and finds exactly two of these differences are equal to

1

. How many selections of

5

balls could he have drawn from the box?
Proposed by Adam Bertelli