2016 AIME II Problems/Problem 5
Problem
Triangle has a right angle at
. Its side lengths are pairwise relatively prime positive integers, and its perimeter is
. Let
be the foot of the altitude to
, and for
, let
be the foot of the altitude to
in
. The sum
. Find
.
Solution 1
Note that by counting the area in 2 ways, the first altitude is . By similar triangles, the common ratio is
for reach height, so by the geometric series formula, we have
. Multiplying by the denominator and expanding, the equation becomes
. Cancelling
and multiplying by
yields
, so
and
. Checking for Pythagorean triples gives
and
, so
Solution modified/fixed from Shaddoll's solution.
Solution 2
We start by splitting the sum of all into two parts: those where
is odd and those where
is even.
First consider the sum of the lengths of the segments for which is odd for each
. The perimeters of these triangles can be expressed using
and ratios that result because of similar triangles. Considering triangles where
is odd, we find that the perimeter for each such
is
. Thus,
.
Simplifying,
. (1)
Continuing with a similar process for the sum of the lengths of the segments for which is even,
.
Simplifying,
. (2)
Adding (1) and (2) together, we find that
.
Setting ,
, and
, we can now proceed as in Shaddoll's solution, and our answer is
.
Solution by brightaz
Solution 3
Let ,
, and
.
Note that the total length of the red segments in the figure above is equal to the length of the blue segment times
.
The desired sum is equal to the total length of the infinite path , shown in red in the figure below.
Since each of the triangles
on the left are similar, it follows that the total length of the red segments in the figure below is equal to the length of the blue segment times
.
In other words, we have that
.
Guessing and checking Pythagorean triples reveals that ,
,
, and
satisfies this equation.
Solution 4
This solution proceeds from . Note the general from for a primitive pythagorean triple,
and after substitution, letting
into the previous equation simplifies down very nicely into
. Thus
. Since we know all three side lengths are relatively prime, we must divide each by 2 and let n = 1 giving
yielding
.
Alternate way to find a, b, and c (from VarunGotem) : implies that
. Because
,
. Plugging in gives
. Simplifying gives
, and since
and
are relatively prime,
and
. This means
and
.
See also
2016 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |