2007 AMC 12A Problems/Problem 22
- The following problem is from both the 2007 AMC 12A #22 and 2007 AMC 10A #25, so both problems redirect to this page.
Problem
For each positive integer , let
denote the sum of the digits of
For how many values of
is
Solution
Solution 1
For the sake of notation let . Obviously
. Then the maximum value of
is when
, and the sum becomes
. So the minimum bound is
. We do casework upon the tens digit:
Case 1: . Easy to directly disprove.
Case 2: .
, and
if
and
otherwise.
- Subcase a:
. This exceeds our bounds, so no solution here.
- Subcase b:
. First solution.
Case 3: .
, and
if
and
otherwise.
- Subcase a:
. Second solution.
- Subcase b:
. Third solution.
Case 4: . But
, and the these clearly sum to
.
Case 5: . So
and
, and
. Fourth solution.
In total we have solutions, which are
and
.
Solution 2
Clearly, . We can break this up into three cases:
Case 1:
- Inspection gives
.
Case 2: ,
,
- If you set up an equation, it reduces to
- which has as its only solution satisfying the constraints
,
.
Case 3: ,
,
- This reduces to
. The only two solutions satisfying the constraints for this equation are
,
and
,
.
The solutions are thus and the answer is
.
See also
2007 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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All AMC 12 Problems and Solutions |