3

Part of 2020 JBMO Shortlist

Problems(2)

JBMO Shortlist 2020 A3

Source: JBMO Shortlist 2020

Find all triples of positive real numbers

(a, b, c)

so that the expression

M = \frac{(a + b)(b + c)(a + b + c)}{abc}

gets its least value.

JuniorBalkanshortlistalgebraInequalityminimum

JBMO Shortlist 2020 N3

Source: JBMO Shortlist 2020

Find the largest integer

k

k \ge 2

), for which there exists an integer

n

n \ge k

) such that from any collection of

n

consecutive positive integers one can always choose

k

numbers, which verify the following conditions:
1. each chosen number is not divisible by

6

7

8

; 2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the numbers

6

7

8

JuniorBalkanshortlist2020number theory