2015 AMC 10A Problems/Problem 24
- The following problem is from both the 2015 AMC 12A #19 and 2015 AMC 10A #24, so both problems redirect to this page.
Contents
Problem
For some positive integers , there is a quadrilateral
with positive integer side lengths, perimeter
, right angles at
and
,
, and
. How many different values of
are possible?
Solution 1
Let and
be positive integers. Drop a perpendicular from
to
to show that, using the Pythagorean Theorem, that
Simplifying yields
, so
. Thus,
is one more than a perfect square.
The perimeter must be less than 2015. Simple calculations demonstrate that
is valid, but
is not. On the lower side,
does not work (because
), but
does work. Hence, there are 31 valid
(all
such that
for
), and so our answer is
Solution 2
If , then
. Thus,
and thus
. Since
must be an integer, we note that
is a perfect square and by simple computations it is seen that
works but
doesn't. Therefore
to
work, giving an answer of
.
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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 | ||
All AMC 10 Problems and Solutions |
2015 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 | |
All AMC 12 Problems and Solutions |
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