1997 CEMC Pascal Problems/Problem 10

Problem

The sum of nine consecutive positive integers is . The largest of these integers is

Solution 1

Let be the starting number. The full list of numbers is then:

We can now set up an equation involving the sum:

Using the fact that for some non-negative integer , and the fact that appears times, we have:

We can either subtract from both sides or factor out a :

or

The problem asks us for the highest number in the list, which is .

~anabel.disher

Solution 2

Let be the highest number in the list of consecutive integers. The full list is then:

Using a similar process to solution 1, we get:

~anabel.disher

Solution 3

Let be the middle term in the list. The full list is then:

We can then write an equation for the sum:

We can see that all of the constant terms cancel out, and appears times, so we have:

The largest term in the list is .

~anabel.disher