A piecewise linear function is defined by<img alt="\[f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \\ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}\]" height="53" src="https://latex.artofproblemsolving.com/2/1/b/21bf20dc5e2dae2f8d5368cec6e395912388f875.png" width="250" />and <img alt="$f(x + 4) = f(x)$" height="20" src="https://latex.artofproblemsolving.com/4/8/c/48ce760add14ed99bfe3ff972ea6b155ad82267f.png" width="128" /> for all real numbers <img alt="$x$" height="10" src="https://latex.artofproblemsolving.com/2/6/e/26eeb5258ca5099acf8fe96b2a1049c48c89a5e6.png" width="12" />. The graph of <img alt="$f(x)$" height="18" src="https://latex.artofproblemsolving.com/c/9/6/c96dd6ec1dc4ad7520fbdc78fcdbec9edd068d0c.png" width="34" /> has the sawtooth pattern depicted below. <img alt="[asy] import graph; size(300); Label f; f.p=fontsize(6); yaxis(-2,2,Ticks(f, 2.0)); xaxis(-6.5,6.5,Ticks(f, 2.0)); draw((0, 0)..(1/4,sqrt(1/136))..(1/2,sqrt(1/68))..(0.75,sqrt(0.75/34))..(1, sqrt(1/34))..(2, sqrt(2/34))..(3, sqrt(3/34))..(4, sqrt(4/34))..(5, sqrt(5/34))..(6, sqrt(6/34))..(7, sqrt(7/34))..(8, sqrt(8/34)), red); draw((0, 0)..(1/4,-sqrt(1/136))..(0.5,-sqrt(1/68))..(0.75,-sqrt(0.75/34))..(1, -sqrt(1/34))..(2, -sqrt(2/34))..(3, -sqrt(3/34))..(4, -sqrt(4/34))..(5, -sqrt(5/34))..(6, -sqrt(6/34))..(7, -sqrt(7/34))..(8, -sqrt(8/34)), red); draw((-7,0)--(7,0), black+0.8bp); draw((0,-2.2)--(0,2.2), black+0.8bp); draw((-6,-0.1)--(-6,0.1), black); draw((-4,-0.1)--(-4,0.1), black); draw((-2,-0.1)--(-2,0.1), black); draw((0,-0.1)--(0,0.1), black); draw((2,-0.1)--(2,0.1), black); draw((4,-0.1)--(4,0.1), black); draw((6,-0.1)--(6,0.1), black); draw((-7,1)..(-5,-1), blue); draw((-5,-1)--(-3,1), blue); draw((-3,1)--(-1,-1), blue); draw((-1,-1)--(1,1), blue); draw((1,1)--(3,-1), blue); draw((3,-1)--(5,1), blue); draw((5,1)--(7,-1), blue); [/asy]" height="148" src="https://latex.artofproblemsolving.com/1/2/e/12eaae0d7d3e6127a4d0af7dc27fd14c3197a740.png" width="502" /> The parabola <img alt="$x = 34y^{2}$" height="20" src="https://latex.artofproblemsolving.com/6/2/2/62204bac6b80a0be18395c7b40d6a24d48f52725.png" width="72" /> intersects the graph of <img alt="$f(x)$" height="18" src="https://latex.artofproblemsolving.com/c/9/6/c96dd6ec1dc4ad7520fbdc78fcdbec9edd068d0c.png" width="34" /> at finitely many points. The sum of the <img alt="$y$" height="13" src="https://latex.artofproblemsolving.com/0/9/2/092e364e1d9d19ad5fffb0b46ef4cc7f2da02c1c.png" width="11" />-coordinates of all these intersection points can be expressed in the form <img alt="$\tfrac{a + b\sqrt{c}}{d}$" height="26" src="https://latex.artofproblemsolving.com/c/1/7/c171ce0a357e21da980e46e7150120c82d71b2de.png" width="46" />, where <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" />, <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" />, <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" />, and <img alt="$d$" height="12" src="https://latex.artofproblemsolving.com/9/6/a/96ab646de7704969b91c76a214126b45f2b07b25.png" width="9" /> are positive integers such that <img alt="$a$" height="10" src="https://latex.artofproblemsolving.com/c/7/d/c7d457e388298246adb06c587bccd419ea67f7e8.png" width="11" />, <img alt="$b$" height="14" src="https://latex.artofproblemsolving.com/8/1/3/8136a7ef6a03334a7246df9097e5bcc31ba33fd2.png" width="10" />, <img alt="$d$" height="12" src="https://latex.artofproblemsolving.com/9/6/a/96ab646de7704969b91c76a214126b45f2b07b25.png" width="9" /> have greatest common divisor equal to <img alt="$1$" height="14" src="https://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" width="11" />, and <img alt="$c$" height="8" src="https://latex.artofproblemsolving.com/3/3/7/3372c1cb6d68cf97c2d231acc0b47b95a9ed04cc.png" width="8" /> is not divisible by the square of any prime. Find <img alt="$a + b + c + d$" height="14" src="https://latex.artofproblemsolving.com/f/4/e/f4e7cee8fe39683f5495d0b4daf7056f4a964643.png" width="101" />.

Question No: 408

A piecewise linear function is defined by $f(x) = \begin{cases} x & \operatorname{if} ~ -1 \leq x < 1 \\ 2 - x & \operatorname{if} ~ 1 \leq x < 3\end{cases}$ and $f(x + 4) = f(x)$ for all real numbers $x$ . The graph of $f(x)$ has the sawtooth pattern depicted below.

$[asy] import graph; size(300); Label f; f.p=fontsize(6); yaxis(-2,2,Ticks(f, 2.0)); xaxis(-6.5,6.5,Ticks(f, 2.0)); draw((0, 0)..(1/4,sqrt(1/136))..(1/2,sqrt(1/68))..(0.75,sqrt(0.75/34))..(1, sqrt(1/34))..(2, sqrt(2/34))..(3, sqrt(3/34))..(4, sqrt(4/34))..(5, sqrt(5/34))..(6, sqrt(6/34))..(7, sqrt(7/34))..(8, sqrt(8/34)), red); draw((0, 0)..(1/4,-sqrt(1/136))..(0.5,-sqrt(1/68))..(0.75,-sqrt(0.75/34))..(1, -sqrt(1/34))..(2, -sqrt(2/34))..(3, -sqrt(3/34))..(4, -sqrt(4/34))..(5, -sqrt(5/34))..(6, -sqrt(6/34))..(7, -sqrt(7/34))..(8, -sqrt(8/34)), red); draw((-7,0)--(7,0), black+0.8bp); draw((0,-2.2)--(0,2.2), black+0.8bp); draw((-6,-0.1)--(-6,0.1), black); draw((-4,-0.1)--(-4,0.1), black); draw((-2,-0.1)--(-2,0.1), black); draw((0,-0.1)--(0,0.1), black); draw((2,-0.1)--(2,0.1), black); draw((4,-0.1)--(4,0.1), black); draw((6,-0.1)--(6,0.1), black); draw((-7,1)..(-5,-1), blue); draw((-5,-1)--(-3,1), blue); draw((-3,1)--(-1,-1), blue); draw((-1,-1)--(1,1), blue); draw((1,1)--(3,-1), blue); draw((3,-1)--(5,1), blue); draw((5,1)--(7,-1), blue); [/asy]$

The parabola $x = 34y^{2}$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$ -coordinates of all these intersection points can be expressed in the form $\tfrac{a + b\sqrt{c}}{d}$ , where $a$ , $b$ , $c$ , and $d$ are positive integers such that $a$ , $b$ , $d$ have greatest common divisor equal to $1$ , and $c$ is not divisible by the square of any prime. Find $a + b + c + d$ .

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Tags: Algebra Geometry

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