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+{"messages": [{"content": "Considering the functions f(x) = ax and g(x) = px^2 + qx, where a, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73", "dataset": "MATH"}
+{"messages": [{"content": "If a, b, p are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = asin(b\u03c0x) + 2 and g(x) = px intersect at least 6 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px^2 + qx have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "388", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = asin(b\u03c0x) and g(x) = pcos(q\u03c0x) - 3 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "545", "dataset": "MATH"}
+{"messages": [{"content": "If a, b, p are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = px + 3 intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "62", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px have at least 2 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = asin(b\u03c0x) and g(x) = psin(q\u03c0x), where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 7 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "508", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px^2 + qx - 2 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "390", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = ax and g(x) = px + 3, where a, p can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax and g(x) = px^2 + qx have at least 2 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63", "dataset": "MATH"}
+{"messages": [{"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = acos(b\u03c0x) and g(x) = psin(q\u03c0x) intersect at least 2 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "562", "dataset": "MATH"}
+{"messages": [{"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = pcos(q\u03c0x) intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "515", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = asin(b\u03c0x) and g(x) = pcos(q\u03c0x) have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "546", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax and g(x) = px^2 + qx have at least 3 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = ax^2 + bx and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "404", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "78", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = acos(b\u03c0x) and g(x) = pcos(q\u03c0x) - 2, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 5 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "501", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = ax^2 + bx and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "414", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax + 1 and g(x) = px^2 + qx have at least 3 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = acos(b\u03c0x) and g(x) = pcos(q\u03c0x) + 2 have at least 7 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "510", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = ax^2 + bx + 3 and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "397", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px^2 + qx - 5 have at least 2 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "277", "dataset": "MATH"}
+{"messages": [{"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = px^2 + qx intersect at least 3 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "If a, p are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax and g(x) = px intersect at least 2 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = acos(b\u03c0x) - 1 and g(x) = psin(q\u03c0x) - 1 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "549", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax - 5 and g(x) = px^2 + qx - 1 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "72", "dataset": "MATH"}
+{"messages": [{"content": "Pick values for a, p from 1 to 5, how many combinations can we have so that the function f(x) = ax and g(x) = px have at least 3 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
+{"messages": [{"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = px^2 + qx intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "392", "dataset": "MATH"}
+{"messages": [{"content": "Considering the functions f(x) = ax^2 + bx - 3 and g(x) = px^2 + qx, where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "395", "dataset": "MATH"}