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Difference between revisions of "2016 AIME II Problems/Problem 5"

(→‎Solution 2)
(Added another solution)
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Solution by brightaz
 
Solution by brightaz
 +
 +
== Solution 3 ==
 +
<asy>
 +
size(10cm);
 +
 +
// Setup
 +
pair A, B;
 +
pair C0, C1, C2, C3, C4, C5, C6, C7, C8;
 +
A = (5, 0);
 +
B = (0, 3);
 +
C0 = (0, 0);
 +
C1 = foot(C0, A, B);
 +
C2 = foot(C1, C0, B);
 +
C3 = foot(C2, C1, B);
 +
C4 = foot(C3, C2, B);
 +
C5 = foot(C4, C3, B);
 +
C6 = foot(C5, C4, B);
 +
C7 = foot(C6, C5, B);
 +
C8 = foot(C7, C6, B);
 +
 +
// Labels
 +
label("$A$", A, SE);
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label("$B$", B, NW);
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label("$C_0$", C0, SW);
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label("$C_1$", C1, NE);
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label("$C_2$", C2, W);
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label("$C_3$", C3, NE);
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label("$C_4$", C4, W);
 +
 +
// Drawings
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draw(A--B--C0--cycle);
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draw(C0--C1--C2, red);
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draw(C2--C3--C4--C5--C6--C7--C8);
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draw(C0--C2, green);
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</asy>
 +
 +
Let <math>a = BC_0</math>, <math>b = AC_0 </math>, and <math>c = AB</math>.
 +
Note that the total length of the red segments in the figure above is equal to the length of the green segment times <math>\frac{a+c}{b}</math>.
 +
 +
The desired sum is equal to the total length of the infinite path <math>C_0 C_1 C_2 C_3 \cdots</math>, shown in red in the figure below.
 +
Since each of the triangles <math>\triangle C_0 C_1 C_2, \triangle C_2 C_3 C_4, \dots</math> 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 green segment times <math>\frac{a+c}{b}</math>.
 +
In other words, we have that <math>a\left(\frac{a+c}{b}\right) = 6p</math>.
 +
 +
Guessing and checking Pythagorean triples reveals that <math>a = 84</math>, <math>b=13</math>, <math>c = 85</math>, and <math>p = a + b + c = \boxed{182}</math> satisfies this equation.
 +
 +
<asy>
 +
size(10cm);
 +
 +
// Setup
 +
pair A, B;
 +
pair C0, C1, C2, C3, C4, C5, C6, C7, C8;
 +
A = (5, 0);
 +
B = (0, 3);
 +
C0 = (0, 0);
 +
C1 = foot(C0, A, B);
 +
C2 = foot(C1, C0, B);
 +
C3 = foot(C2, C1, B);
 +
C4 = foot(C3, C2, B);
 +
C5 = foot(C4, C3, B);
 +
C6 = foot(C5, C4, B);
 +
C7 = foot(C6, C5, B);
 +
C8 = foot(C7, C6, B);
 +
 +
// Labels
 +
label("$A$", A, SE);
 +
label("$B$", B, NW);
 +
label("$C_0$", C0, SW);
 +
label("$a$", (B+C0)/2, W);
 +
label("$b$", (A+C0)/2, S);
 +
label("$c$", (A+B)/2, NE);
 +
 +
// Drawings
 +
draw(A--B--C0--cycle);
 +
draw(C0--C1--C2--C3--C4--C5--C6--C7--C8, red);
 +
draw(C0--B, green);
 +
</asy>
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2016|n=II|num-b=4|num-a=6}}
 
{{AIME box|year=2016|n=II|num-b=4|num-a=6}}

Revision as of 19:27, 26 May 2017

Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pariwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\overline{AB}$, and for $n \geq 2$, let $C_n$ be the foot of the altitude to $\overline{C_{n-2}B}$ in $\triangle C_{n-2}C_{n-1}B$. The sum $\sum_{i=1}^\infty C_{n-2}C_{n-1} = 6p$. Find $p$.

Solution 1

Note that by counting the area in 2 ways, the first altitude is $\dfrac{ac}{b}$. By similar triangles, the common ratio is $\dfrac{a}{c}$ for reach height, so by the geometric series formula, we have $6p=\dfrac{\dfrac{ab}{c}}{1-\dfrac{a}{c}}$. Multiplying by the denominator and expanding, the equation becomes $\dfrac{ab}{c}=6a+6b+6c-\dfrac{6a^2}{c}-\dfrac{6ab}{c}-6a$. Cancelling $6a$ and multiplying by $c$ yields $ab=6bc+6c^2-6a^2-6ab$, so $7ab = 6bc+6b^2$ and $7a=6b+6c$. Checking for Pythagorean triples gives $13,84,$ and $85$, so $p=13+84+85=\boxed{182}$

Solution modified/fixed from Shaddoll's solution.

Solution 2

We start by splitting the sum of all $C_{n-2}C_{n-1}$ into two parts: those where $n-2$ is odd and those where $n-2$ is even.


First consider the sum of the lengths of the segments for which $n-2$ is odd for each $n\geq2$. The perimeters of these triangles can be expressed using $p$ and ratios that result because of similar triangles. Considering triangles where $n-2$ is odd, we find that the perimeter for each such $n$ is $p(\frac{C_{n-1}C_{n}}{C_{0}B})$. Thus,

$p\sum_{n=1}^{\infty}\frac{C_{2n-1}C_{2n}}{C_{0}B}=6p+C_{0}B$.


Simplifying,

$\sum_{n=1}^{\infty}C_{2n-1}C_{2n}=6C_{0}B + \frac{(C_{0}B)^2}{p}=C_{0}B(6+\frac{C_{0}B}{p})$. (1)


Continuing with a similar process for the sum of the lengths of the segments for which $n-2$ is even,

$p\sum_{n=1}^{\infty}\frac{C_{2n-2}C_{2n-1}}{C_{0}B}=6p+C_{0}A+AB=7p-C_{0}B$.


Simplifying,

$\sum_{n=1}^{\infty}C_{2n-2}C_{2n-1}=C_{0}B(7-\frac{C_{0}B}{p})$. (2)


Adding (1) and (2) together, we find that

$6p=13C_{0}B \Rightarrow p=\frac{13C_{0}B}{6}=C_{0}B+C_{0}A+AB \Rightarrow \frac{7C_{0}B}{6}=C_{0}A+AB \Rightarrow 7C_{0}B=6C_{0}A + 6AB$.

Setting $a=C_{0}B$, $b=C_{0}A$, and $c=AB$, we can now proceed as in Shaddoll's solution, and our answer is $p=13+84+85=\boxed{182}$.

Solution by brightaz

Solution 3

[asy] size(10cm); // Setup pair A, B; pair C0, C1, C2, C3, C4, C5, C6, C7, C8; A = (5, 0); B = (0, 3); C0 = (0, 0); C1 = foot(C0, A, B); C2 = foot(C1, C0, B); C3 = foot(C2, C1, B); C4 = foot(C3, C2, B); C5 = foot(C4, C3, B); C6 = foot(C5, C4, B); C7 = foot(C6, C5, B); C8 = foot(C7, C6, B); // Labels label("$A$", A, SE); label("$B$", B, NW); label("$C_0$", C0, SW); label("$C_1$", C1, NE); label("$C_2$", C2, W); label("$C_3$", C3, NE); label("$C_4$", C4, W); // Drawings draw(A--B--C0--cycle); draw(C0--C1--C2, red); draw(C2--C3--C4--C5--C6--C7--C8); draw(C0--C2, green); [/asy]

Let $a = BC_0$, $b = AC_0$, and $c = AB$. Note that the total length of the red segments in the figure above is equal to the length of the green segment times $\frac{a+c}{b}$.

The desired sum is equal to the total length of the infinite path $C_0 C_1 C_2 C_3 \cdots$, shown in red in the figure below. Since each of the triangles $\triangle C_0 C_1 C_2, \triangle C_2 C_3 C_4, \dots$ 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 green segment times $\frac{a+c}{b}$. In other words, we have that $a\left(\frac{a+c}{b}\right) = 6p$.

Guessing and checking Pythagorean triples reveals that $a = 84$, $b=13$, $c = 85$, and $p = a + b + c = \boxed{182}$ satisfies this equation.

[asy] size(10cm); // Setup pair A, B; pair C0, C1, C2, C3, C4, C5, C6, C7, C8; A = (5, 0); B = (0, 3); C0 = (0, 0); C1 = foot(C0, A, B); C2 = foot(C1, C0, B); C3 = foot(C2, C1, B); C4 = foot(C3, C2, B); C5 = foot(C4, C3, B); C6 = foot(C5, C4, B); C7 = foot(C6, C5, B); C8 = foot(C7, C6, B); // Labels label("$A$", A, SE); label("$B$", B, NW); label("$C_0$", C0, SW); label("$a$", (B+C0)/2, W); label("$b$", (A+C0)/2, S); label("$c$", (A+B)/2, NE); // Drawings draw(A--B--C0--cycle); draw(C0--C1--C2--C3--C4--C5--C6--C7--C8, red); draw(C0--B, green); [/asy]

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
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