1993 AIME Problems/Problem 4
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Problem
How many ordered four-tuples of integers
with
satisfy
and
?
Contents |
Solution
Solution 1
Let
so
. It follows that
. Hence
.
Solve them in tems of
to get
. The last two solution doesnt follow
, so we only need to consider the first two solutions.
The first solution gives us
and
, and the second one gives us
.
So the total number of such four-tuples is
.
Solution 2
Substituting
,
, and
into
,
Hence,
or
.
For
, we know that
, so there are
four-tuples. For
,
, and there are
four-tuples. In total, we have
four-tuples.
See also
| 1993 AIME (Problems • Resources) | ||
| Preceded by Problem 3 | Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||






