Difference between revisions of "1993 AIME Problems/Problem 4"
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== Solution == | == Solution == | ||
=== Solution 1 === | === Solution 1 === | ||
− | Let <math>k = a + d = b + c</math> so <math>d = k-a, b=k-c</math>. It follows that <math>(k-c)c - a(k-a) = (a-c)(a+c-k) = (c-a)(d-c) = 93</math>. Hence <math>(c - a,d - | + | Let <math>k = a + d = b + c</math> so <math>d = k-a, b=k-c</math>. It follows that <math>(k-c)c - a(k-a) = (a-c)(a+c-k) = (c-a)(d-c) = 93</math>. Hence <math>(c - a,d - c) = (1,93),(3,31),(31,3),(93,1)</math>. |
Solve them in tems of <math>c</math> to get | Solve them in tems of <math>c</math> to get |
Revision as of 17:08, 10 March 2010
Problem
How many ordered four-tuples of integers with satisfy and ?
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 • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |