messages listlengths 1 1 | ground_truth stringlengths 1 3 | dataset stringclasses 1 value |
|---|---|---|
[
{
"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}",
"... | 73 | MATH |
[
{
"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(bπx) + 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"
}
] | 53 | MATH |
[
{
"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"
}
] | 388 | MATH |
[
{
"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(bπx) and g(x) = pcos(qπx) - 3 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 545 | MATH |
[
{
"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"
}
] | 62 | MATH |
[
{
"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"
}
] | 60 | MATH |
[
{
"content": "Considering the functions f(x) = asin(bπx) and g(x) = psin(qπx), 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 answe... | 508 | MATH |
[
{
"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"
}
] | 390 | MATH |
[
{
"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"
}
] | 65 | MATH |
[
{
"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":... | 20 | MATH |
[
{
"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"
}
] | 63 | MATH |
[
{
"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(bπx) and g(x) = psin(qπx) intersect at least 2 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 562 | MATH |
[
{
"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(qπx) intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 515 | MATH |
[
{
"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(bπx) and g(x) = pcos(qπx) have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 546 | MATH |
[
{
"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"
}
] | 0 | MATH |
[
{
"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 answe... | 404 | MATH |
[
{
"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"
}
] | 78 | MATH |
[
{
"content": "Considering the functions f(x) = acos(bπx) and g(x) = pcos(qπx) - 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 a... | 501 | MATH |
[
{
"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 answe... | 414 | MATH |
[
{
"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"
}
] | 0 | MATH |
[
{
"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(bπx) and g(x) = pcos(qπx) + 2 have at least 7 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 510 | MATH |
[
{
"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 a... | 397 | MATH |
[
{
"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"
}
] | 277 | MATH |
[
{
"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"
}
... | 0 | MATH |
[
{
"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"
}
] | 0 | MATH |
[
{
"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(bπx) - 1 and g(x) = psin(qπx) - 1 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 549 | MATH |
[
{
"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"
}
] | 72 | MATH |
[
{
"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"
}
] | 0 | MATH |
[
{
"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"
}
... | 392 | MATH |
[
{
"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 a... | 395 | MATH |
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