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INMO 2022

Source: INMO 2022 P3

March 6, 2022

INMO 2022Arrangements

Problem Statement

For a positive integer

N

, let

T(N)

denote the number of arrangements of the integers

1, 2, \cdots N

into a sequence

a_1, a_2, \cdots a_N

such that

a_i > a_{2i}

for all

i

,

1 \le i < 2i \le N

and

a_i > a_{2i+1}

for all

321

,

312

. For example,

T(3)

is

2

, since the possible arrangements are

321

and

312

(a) Find

T(7)

(b) If

K

is the largest non-negative integer so that

2^K

divides

T(2^n - 1)

, show that

K = 2^n - n - 1

. (c) Find the largest non-negative integer

K

so that

2^K

divides

T(2^n + 1)

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