| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "109", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "71", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "157", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "158", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "121", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "83", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "91", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "146", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "127", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "122", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "144", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "217", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "192", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "49", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40", "dataset": "MATH"} | |
| {"messages": [{"content": "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))).\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"} | |