MathDB

9

Part of 2017 QEDMO 15th

Problems(2)

n integers in a circle, a_1 + a_2 +...+ a_k <= 2k -1

Source: 15th -a QEDMO problem 9 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/30/2021
Iskandar arranged n∈Nn \in N integer numbers in a circle, the sum of which is 2n−12n-1. Crescentia now selects one of these numbers and name the given numbers in clockwise direction with a1,a2,....,ana_1,a_2,...., a_n. Show that she can choose the starting number such that for all k∈{1,2,...,n}k \in \{1, 2,..., n\} the inequality a1+a2+...+ak≤2k−1a_1 + a_2 +...+ a_k \le 2k -1 holds.
combinatorics
n is prime if 2^{n-1}-1 and not 2^h-1 is divisible by n, where n = ph + 1

Source: 15th -b QEDMO problem 9 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

5/30/2021
Let pp be a prime number and hh be a natural number smaller than pp. We set n=ph+1n = ph + 1. Prove that if 2n−1−12^{n-1}-1, but not 2h−12^h-1, is divisible by nn, then nn is a prime number.
number theorydivisible