2

Part of 1991 China Team Selection Test

Problems(2)

f(n+2) = 23 * f(n+1) + f(n)

Source: China TST 1991, problem 5

f

f: \mathbb{N} \cup \{0\} \mapsto \mathbb{N},

and satisfies the following conditions: (1)

f(0) = 0, f(1) = 1,

f(n+2) = 23 \cdot f(n+1) + f(n), n = 0,1, \ldots.

Prove that for any

m \in \mathbb{N}

, there exist a

d \in \mathbb{N}

m | f(f(n)) \Leftrightarrow d | n.

functionnumber theory unsolvednumber theory

We write numbers on points

Source: China TST 1991, problem 2

i = 1,2, \ldots, 1991

n_i

points and write number

i

on them (each point has only written one number on it). A set of chords are drawn such that:
(i) They are pairwise non-intersecting. (ii) The endpoints of each chord have distinct numbers.
If for all possible assignments of numbers the operation can always be done, find the necessary and sufficient condition the numbers

n_1, n_2, \ldots, n_{1991}

must satisfy for this to be possible.

combinatorics unsolvedcombinatorics