8

Part of 2023 LMT Fall

Problems(2)

2023 Fall LMT Team #8 JERRY = 1428

Source:

J

E

R

Y

be four positive integers chosen independently and uniformly at random from the set of factors of

1428

. What is the probability that

JERRY = 1428

? Express your answer in the form

\frac{a}{b\cdot 2^n}

n

is a nonnegative integer,

a

b

are odd, and gcd

(a,b) = 1

2023 Fall LMT Speed p8 - n bakers places 20 candles on cake

Source:

To celebrate the

20

th LMT, the LHSMath Team bakes a cake. Each of the

n

20

candles on the cake. When they count, they realize that there are

(n -1)!

total candles on the cake. Find

n

.
Proposed by Christopher Cheng