2

Part of 2024 Saint Petersburg Mathematical Olympiad

Problems(3)

A sequence 1,2,2,3,3,3,4,4,4,4,5,5,5,5,5

Source: Saint Petersburg olympiad 2024, 11.2

Given a sequence

a_n

1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots

(one '1', two '2' and so on) and another sequence

b_n

a_{b_n}=b_{a_n}

for all positive integers

n

. It is known that

b_k=1

k>100

b_m=1

m>k

algebrasequenses

7 coins are determined in 6 weighings; 32 real and 32 fake coins

Source: Saint Petersburg olympiad 2024, 10.2

32

32

fake coins are given the same appearance. All fake coins weigh equally and less than the real ones, which also all weigh the same. How to determine the type of at least seven coins in six weighings on a scale with two bowls?

Bambula carries weights

Source: Saint Petersburg olympiad 2024, 9.2

A strongman Bambula can carry several weights at the same time, if their total weight does not exceed

200

kg, and these weights are no more than three. On the way to work, he injured his finger and found that he could now carry no more than two weights (and still no more than

200

kg). At what minimum

k

is the statement true: any set of

100

weights that Bambula could previously carry in

50

runs, with a sore finger, he will be able to carry in no more than

k

algebracombinatorics