Problem:
In how many ways can be written as the sum of two or more consecutive odd positive integers that are arranged in increasing order?
Answer Choices:
A.
B.
C.
D.
E.
Solution 1:
The main idea of this problem is noting that an integer can be written as a sum of consecutive positive odd integers if and only if
Notice that condition is only possible if is also odd. Since is even, we can ignore this case. So we just need to check positive even integers that divide which are less than . Note that and , but , so only values out of the work. Therefore, the answer is just .
Note: the corresponding sequences for that sum to are and respectively.
Solution 2:
Note that since the numbers are odd, we must have an even number of numbers. Let's case on how many numbers we have summing up to :
Case 1: numbers
We get a solution from .
Case 2: numbers
We see that and . Since these two consecutive solutions skip , we will never get a solution with numbers.
Case 3: numbers
We see that , which gives us another solution.
Case 4: numbers or more
The sum of the first positive odd integers is , so if we ever try to sum or more numbers, we will get a sum too large.
Summing up cases gives a total of solutions.
The problems on this page are the property of the MAA's American Mathematics Competitions