2016 AMC 10B Problems/Problem 24
Contents
Problem
How many four-digit integers , with
, have the property that the three two-digit integers
form an increasing arithmetic sequence? One such number is
, where
,
,
, and
.
Solution 1
The numbers are and
. Note that only
can be zero, the numbers
,
, and
cannot start with a zero, and
.
To form the sequence, we need . This can be rearranged as
. Notice that since the left-hand side is a multiple of
, the right-hand side can only be
or
. (A value of
would contradict
.) Therefore we have two cases:
and
.
Case 1
If , then
, so
. This gives
.
If
, then
, so
. This gives
.
If
, then
, so
, giving
.
There is no solution for
.
Added together, this gives us
answers for Case 1.
Case 2
This means that the digits themselves are in an arithmetic sequence. This gives us answers,
Adding the two cases together, we find the answer to be
.
Solution 2 (Brute Force, when you have lots of time)
Looking at the answer options, all the numbers are pretty small so it is easy to make a list.
Counting all the cases we get our answer of which is
-srisainandan6
Solution 3
Let be the difference between the numbers
,
, and
. We then have
and
Subtracting the second equation from the first and then simplifying, we are left with:
Notice that . Because the values of
and
are irrelevant compared to the other numbers, we can just find pairs of
that work. Trying out each value of
from
to
and summing the number of pairs yields
- cappucher
Video Solution
https://www.youtube.com/watch?v=UhPxvZ6V4Zs
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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