[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(0.5 PI x))) and x = 5g(f(cos(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 3g(f(sin(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 4g(f(sin(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(1.5 PI x))) and x = 4g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(0.5 PI x))) and x = 4g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(cos(1.5 PI x))) and x = 4g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 3g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 4g(f(cos(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 3g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(cos(0.5 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(cos(2.0 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(sin(0.5 PI x))) and x = 5g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(cos(1.5 PI x))) and x = 5g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(sin(0.5 PI x))) and x = 5g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 4g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(0.5 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 5g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(0.5 PI x))) and x = 4g(f(cos(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(0.5 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(1.5 PI x))) and x = 5g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(2.0 PI x))) and x = 4g(f(cos(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(1.0 PI x))) and x = 4g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(cos(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(sin(1.5 PI x))) and x = 5g(f(cos(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.5 PI x))) and x = 5g(f(cos(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(cos(1.5 PI x))) and x = 5g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.5 PI x))) and x = 5g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(sin(2.0 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 4g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(1.0 PI x))) and x = 3g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 4g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(0.5 PI x))) and x = 5g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(cos(1.5 PI x))) and x = 3g(f(cos(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 5g(f(cos(1.0 PI x))) and x = 5g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(cos(2.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 3g(f(cos(2.0 PI x))) and x = 3g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 5g(f(sin(2.0 PI x))) and x = 5g(f(cos(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.0 PI x))) and x = 5g(f(sin(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/4| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 3g(f(sin(1.5 PI x))) and x = 3g(f(sin(1.0 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 4g(f(sin(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(sin(2.0 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/4|. Find the number of intersections of the graphs of y = 4g(f(sin(1.5 PI x))) and x = 5g(f(cos(0.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}


[User]
Define f(x) = ||x| - 1/3| and g(x) = ||x| - 1/3|. Find the number of intersections of the graphs of y = 4g(f(cos(1.0 PI x))) and x = 5g(f(sin(1.5 PI y))).

Present the answer in LaTex format: \boxed{Your answer}