MathDB

n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}

Source: 2010 Indonesia TST stage 2 test 4 p4

December 16, 2020

inequalitiesalgebra

Problem Statement

Given a positive integer

n

and

I = \{1, 2,..., k\}

with

k

is a positive integer. Given positive integers

a_1, a_2, ..., a_k

such that for all

i \in I

1 \le a_i \le n

and

\sum_{i=1}^k a_i \ge 2(n!).

Show that there exists

J \subseteq I

such that

n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}