AMC 10B/12B 2025 (Problem 17/14)
Consider a decreasing sequence of positive integers that satisfies the following two conditions:
• The average (arithmetic mean) of the first terms in the sequence is .
• For all , the average of the first terms in the sequence is less than the average of the first terms in the sequence.
What is the greatest possible value of ?
• The average (arithmetic mean) of the first terms in the sequence is .
• For all , the average of the first terms in the sequence is less than the average of the first terms in the sequence.
What is the greatest possible value of ?
Find .
We can guess that
(At the end we will prove it.)
If we want to rigorously prove it:
We want to prove
by induction on .
Base Case :
By Hint 2,
Suppose the formula holds up to . For ,
and we can assume
(B)