Difference between revisions of "1996 AHSME Problems/Problem 12"
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Revision as of 14:07, 5 July 2013
Contents
Problem 12
A function from the integers to the integers is defined as follows:
Suppose is odd and
. What is the sum of the digits of
?
Solution
First iteration
To get , you could either have
and add
, or
and divide by
.
If you had the former, you would have , and the function's rule would have you divide. Thus,
is the only number for which
.
Second iteration
Going out one step, if you have , you would have to have
. For
, you would either have
and add
, or
and divide by
.
Both are possible: and
return values of
. Thus,
, and
.
Third iteration
Going out the final step, if you have , you would have to have
or
.
If you doubled either of these, would not be odd. So you must subtract
.
If you subtract from
, you would compute
, which would halve it, and not add the
back.
If you subtract from
, you would compute
, which would add the
back.
Thus, , and
is odd. The desired sum of the digits is
, and the answer is
.
See also
1996 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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