3

Part of 2020 Francophone Mathematical Olympiad

Problems(2)

min k such among k reals, exist a,b in R of which |a -b|<1/n or |a-b| >n

Source: The Francophone MO Juniors P3

n

be an integer greater than or equal to

1

. Find, as a function of

n

, the smallest integer

k\ge 2

such that, among any

k

real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than

1 / n

, either strictly greater than

n

combinatoricsalgebraFrancophone

Find n for which k<=a_n<k+1

Source: The francophone mathematical olympiads P3

(a_i)_{i\in \mathbb{N}}

be a sequence with

a_1=\frac{3}2

a_{n+1}=1+\frac{n}{a_n}

n

2020\le a_n <2021

SequencealgebraFrancophone