123 persons in a cafe, and pigeons and boxes

Question

$123$ persons are in a cafe. The sum of their ages is $3813$. Is it always possible to find $100$ among them so that their total age is greater or equal to $3100$?

Looks like Dirichlet-Principle type of problem, but I cant' see the solution.

score 2 · Accepted Answer · answered Jul 20 '14 at 17:10

2

is it possible to find 23 people so their age is less than or equal to 713? Suppose the group of the 23 youngest people has age greater than 713. Then the total age of the group is greater than $\frac{123\cdot 713}{23}=3813$ a contradiction.

answered Jul 20 '14 at 17:10

Asinomás

105,651

score 2 · Answer 2 · answered Jul 20 '14 at 17:10

2

Hint: what is the average age?

answered Jul 20 '14 at 17:10

Ross Millikan

374,822

score 1 · Answer 3 · answered Jul 20 '14 at 17:15

1

The worst case is that everyone is the same age, in which case picking the oldest 100 would give you only $\frac{100}{123}3813$ years in total. Does this suffice?

answered Jul 20 '14 at 17:15

Charles

32,122

123 persons in a cafe, and pigeons and boxes

3 Answers3