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
Let 
and 
. From 
, 
.
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 | ||
