2023 AMC 8
Complete problem set with solutions and individual problem pages
Fifteen integers are arranged in order on a number line. The integers are equally spaced and have the property that
What is the sum of digits of
- A.
- B.
- C.
- D.
- E.
Solution 1
We can find the possible values of the common difference by finding the numbers which satisfy the conditions. To do this, we find the minimum of the last two: , and the maximum–. There is a difference of between them, so only and work, as , so satisfies . The number is similarly found. , however, is too much.
Now, we check with the first and last equations using the same method. We know . Therefore, . We test both values we just got, and we can realize that is too large to satisfy this inequality. On the other hand, we can now find that the difference will be , which satisfies this inequality.
The last step is to find the first term. We know that the first term can only be from to since any larger value would render the second inequality invalid. Testing these three, we find that only will satisfy all the inequalities. Therefore, . The sum of the digits is therefore .
Solution 2
Let the common difference between consecutive be . Since , we find from the first and last inequalities that . As must be an integer, this means . Substituting this into all of the given inequalities so we may extract information about gives
The second inequality tells us that , while the last inequality tells us , so we must have . Finally, to solve for , we simply have , so our answer is .
