data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The grade received on a certain teacher's 100-point test varies in direct proportion to the amount of time a student spends preparing for the test. If a student receives 72 points on a test for which she spent 3 hours preparing, what score would she receive on the next test if ... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 32600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were: \\(\\begin{tabular}[t]{lllllllll} 89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75, \\\\ 63, & 84, & 78, & 71, & 80, & 90. & & & \\\\ \\end{tabular}\\)\nIn Mr. Freeman's cla... | math | {
"style": "rule",
"ground_truth": "33\\frac{1}{3}\\%"
} | {
"index": 32601,
"solution": "1. **Identify the scores that fall into the C range (75 - 84):**\n - From the given scores: 89, 72, 54, 97, 77, 92, 85, 74, 75, 63, 84, 78, 71, 80, 90.\n - The scores that fall within the C range are: 77, 75, 84, 78, 80.\n\n2. **Count the number of students who received a C:**\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The graph below shows a portion of the curve defined by the quartic polynomial $P(x)=x^4+ax^3+bx^2+cx+d$.\n\n[asy]\nunitsize(0.8 cm);\n\nint i;\n\nreal func (real x) {\n return(0.5*(x^4/4 - 2*x^3/3 - 3/2*x^2) + 2.7);\n}\n\ndraw(graph(func,-4.5,4.5));\ndraw((-4.5,0)--(4.5,0));\... | math | {
"style": "rule",
"ground_truth": "\\text{C}"
} | {
"index": 32602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?\n\n[asy]\ndraw((0,0)--(0,7)--(24,7)--(24,0)--cycle);\nlabel(\"KEY:\",(3,5));\nfill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 32603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 32604,
"solution": "1. **Identify the Relevant Data Points**: From the problem, we need to determine the total amount spent during June, July, and August. This requires us to find the total amount spent by the end of August and subtract the total amount spent by the end of May (since June 1st spending wo... |
DeepScaleR | [
{
"role": "user",
"content": "The graph of\n\\[r = -2 \\cos \\theta + 6 \\sin \\theta\\]is a circle. Find the area of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "10 \\pi"
} | {
"index": 32605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$ is an ellipse in the first quadrant of the $xy$-plane. Let $a$ and $b$ be the maximum and minimum values of $\\frac{y}{x}$ over all points $(x,y)$ on the ellipse. What is the value of $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 32606,
"solution": "1. **Identify the problem**: We need to find the maximum and minimum values of $\\frac{y}{x}$ for points $(x, y)$ on the ellipse given by the equation $2x^2 + xy + 3y^2 - 11x - 20y + 40 = 0$. The ratio $\\frac{y}{x}$ represents the slope of a line passing through the origin.\n\n2. **S... |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $r = \\cos \\theta$ is a circle. Find the smallest value of $t$ so that when $r = \\cos \\theta$ is plotted for $0 \\le \\theta \\le t,$ the resulting graph is the entire circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 32607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $r = \\sin \\theta$ is also seen in a circular form but pertains to a vertical alignment. Find the smallest value of $t$ so that when $r = \\sin \\theta$ is plotted for $0 \\le \\theta \\le t,$ the resulting graph forms a complete circle."
}
] | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"index": 32608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $x^{4}=x^{2} y^{2}$ is a union of $n$ different lines. What is the value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32609,
"solution": "The equation $x^{4}-x^{2} y^{2}=0$ factors as $x^{2}(x+y)(x-y)=0$, so its graph is the union of the three lines $x=0, x+y=0$, and $x-y=0$."
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $xy = 1$ is a hyperbola. Find the distance between the foci of this hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 32611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = \\frac{p(x)}{q(x)}$ is shown below, where $p(x)$ and $q(x)$ are quadratic. (Assume that the grid lines are at integers.)\n\n[asy]\nunitsize(0.6 cm);\n\nreal func (real x) {\n return (2*(x - 1)/(x + 2));\n}\n\nint i;\n\nfor (i = -8; i <= 8; ++i) {\n draw((i,... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 32612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = \\frac{p(x)}{q(x)}$ is shown below, where $p(x)$ is linear and $q(x)$ is quadratic. (Assume that the grid lines are at integers.)\n\n[asy]\nunitsize(0.6 cm);\n\nreal func (real x) {\n return (2*x/((x - 2)*(x + 3)));\n}\n\nint i;\n\nfor (i = -5; i <= 5; ++i) ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = \\frac{p(x)}{q(x)}$ is shown, where $p(x)$ and $q(x)$ are quadratic polynomials. The horizontal asymptote is $y = 2$, and the vertical asymptote is $x = -3$. There is a hole in the graph at $x=4$. \n\nFind $\\frac{p(5)}{q(5)}$ if the graph passes through $(2,0... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 32614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = \\frac{p(x)}{q(x)}$ where $p(x)$ is quadratic and $q(x)$ is quadratic is given conceptually (imagine a graph with necessary features). The function has vertical asymptotes at $x = -4$ and $x = 1$. The graph passes through the point $(0,0)$ and $(2,-1)$. Determ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 32615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ has a maximum value of 54, and passes through the points $(-2,0)$ and $(4,0).$ Find $a + b + c.$"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 32616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ has a maximum value of 72, and passes through the points $(0, -1)$ and $(6, -1)$. Find $a + b + c$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{356}{9}"
} | {
"index": 32617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ has a maximum value of 75, and passes through the points $(-3,0)$ and $(3,0)$. Find the value of $a + b + c$ at $x = 2$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{3}"
} | {
"index": 32618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = ax^2 + bx + c$ is shown, where $a$, $b$, and $c$ are integers. The vertex of the parabola is at $(-2, 3)$, and the point $(1, 6)$ lies on the graph. Determine the value of $a$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\left( 1, \\frac{1}{2}, -4 \\right)"
} | {
"index": 32621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "f(4 - x)"
} | {
"index": 32622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.3 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{B}"
} | {
"index": 32623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{C}"
} | {
"index": 32624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{E}"
} | {
"index": 32625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{E}"
} | {
"index": 32626,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{D}"
} | {
"index": 32627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{C}"
} | {
"index": 32628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{A}"
} | {
"index": 32629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (... | math | {
"style": "rule",
"ground_truth": "\\text{D}"
} | {
"index": 32630,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=\\frac{5x^2-9}{3x^2+5x+2}$ has a horizontal asymptote at $y=a$. What is $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac53"
} | {
"index": 32631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=\\frac{5x^2-9}{3x^2+5x+2}$ has vertical asymptotes at $x = a$ and $x = b$. Find $a + b$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{3}."
} | {
"index": 32632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a$.\n\n[asy]\nsize(140);\nLabel f;\n\nf.p=fontsize(4);\n\nxaxis(-3,3,Ticks(f, 1.0));\n\nyaxis(-4,4,Ticks(f, 1.0));\n\nreal f(real x)\n\n{\n\nreturn -2x^2+4x+1;\n\n}\n\ndraw(graph(f,-.7,2.7),... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 32633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a$.\n\n[asy]\nsize(140);\nLabel f;\n\nf.p=fontsize(4);\n\nxaxis(-3,3,Ticks(f, 1.0));\n\nyaxis(-4,4,Ticks(f, 1.0));\n\nreal f(real x)\n\n{\n\nreturn x^2+2x-1;\n\n}\n\ndraw(graph(f,-2.7,.7),li... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a-b+c$.\n\n[asy]\nsize(150);\n\nLabel f;\n\nf.p=fontsize(4);\n\nxaxis(-3,3,Ticks(f, 1.0));\n\nyaxis(-4,4,Ticks(f, 1.0));\n\nreal f(real x)\n\n{\n\nreturn x^2+2x-1;\n\n}\n\ndraw(graph(f,-2.7,... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 32635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=f(x)$ for $-3\\le x\\le 3$ is shown below. On this interval, what is the range of $f(x)-x$? Express your answer in interval notation.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=blac... | math | {
"style": "rule",
"ground_truth": "(-1,0]"
} | {
"index": 32636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=f(x)$ is shown below, with $1$ unit between grid lines. Assume $f(x)$ is defined only on the domain shown.\n\nWhat is the sum of all integers $c$ for which the equation $f(x)=c$ has exactly $6$ solutions?\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\... | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 32637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=g(x)$, defined on a limited domain shown, is conceptualized through the function $g(x) = \\frac{(x-6)(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{945} - 2.5$. If each horizontal grid line represents a unit interval, determine the sum of all integers $d$ for which the equation ... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 32638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32639,
"solution": "\n1. **Identify the Polynomial and Intersection Points**:\n The problem states that the graph of the polynomial $y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$ intersects the line $y = bx + c$ at three values of $x$. These intersections are the roots of the polynomial $x^6 - 10x^5 + 29x^4 -... |
DeepScaleR | [
{
"role": "user",
"content": "The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region?"
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 32640,
"solution": "The equation given can be rewritten as:\n$40|x| = - y^2 - 2xy + 400$\n We can split the equation into a piecewise equation by breaking up the absolute value:\n$40x = -y^2 - 2xy + 400\\quad\\quad x\\ge 0$\n$40x = y^2 + 2xy - 400 \\quad \\quad x < 0$\n Factoring the first one: (alternat... |
DeepScaleR | [
{
"role": "user",
"content": "The graph of \\(y^2 + 2xy + 60|x| = 900\\) partitions the plane into several regions. What is the area of the bounded region?"
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"index": 32641,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of \\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]has its foci at $(0,\\pm 4),$ while the graph of \\[\\frac{x^2}{a^2}-\\frac{y^2}{b^2} = 1\\]has its foci at $(\\pm 6,0).$ Compute the value of $|ab|.$"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{65}"
} | {
"index": 32642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of \\[\\sqrt{(x-1)^2+(y+2)^2} - \\sqrt{(x-5)^2+(y+2)^2} = 3\\]consists of one branch of a hyperbola. Compute the positive value for the slope of an asymptote of the hyperbola."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt7}{3}"
} | {
"index": 32643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of \\[y^4 - 4x^4 = 2y^2 - 1\\]is the union of the graphs of two different conic sections. Which two types of conic sections are they?\n\n(Write your answer as a list, with \"C\" for circle, \"E\" for ellipse, \"H\" for hyperbola, and \"P\" for parabola. For example, \... | math | {
"style": "rule",
"ground_truth": "\\text{H, E}"
} | {
"index": 32644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of a certain quadratic $y = ax^2 + bx + c$ is a parabola with vertex $(-4,0)$ which passes through the point $(1,-75)$. What is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 32645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of a parabola has the following properties:\n\n$\\bullet$ It passes through the point $(1,5).$\n\n$\\bullet$ The $y$-coordinate of the focus is 3.\n\n$\\bullet$ Its axis of symmetry is parallel to the $x$-axis.\n\n$\\bullet$ Its vertex lies on the $y$-axis.\n\nExpress... | math | {
"style": "rule",
"ground_truth": "y^2 - 4x - 6y + 9 = 0"
} | {
"index": 32646,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of an equation \\[\\sqrt{(x-3)^2 + (y+4)^2} + \\sqrt{(x+5)^2 + (y-8)^2} = 20.\\]is an ellipse. What is the distance between its foci?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{13}"
} | {
"index": 32647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation $10x + 270y = 2700$ is drawn on graph paper where each square represents one unit in each direction. A second line defined by $x + y = 10$ also passes through the graph. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 32648,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation $9x+223y=2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?"
}
] | math | {
"style": "rule",
"ground_truth": "888"
} | {
"index": 32649,
"solution": "We know that the number of squares intersected in an $m\\times{n}$ rectangle is $m + n -\\mbox{gcd}(m,n)$. So if we apply that here, we get that the number of intersected squares is:\n$9 + 223 - 1 = 231$.\n Now just subtract that from the total number of squares and divide by 2, since... |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation $x + 2y + 3 = 0$ is perpendicular to the graph of the equation $ax + 2y + 3 = 0$. What is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 32650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation $x+y=\\left\\lfloor x^{2}+y^{2}\\right\\rfloor$ consists of several line segments. Compute the sum of their lengths."
}
] | math | {
"style": "rule",
"ground_truth": "4+\\sqrt{6}-\\sqrt{2}"
} | {
"index": 32651,
"solution": "We split into cases on the integer $k=\\left\\lfloor x^{2}+y^{2}\\right\\rfloor$. Note that $x+y=k$ but $x^{2}+y^{2} \\geq$ $\\frac{1}{2}(x+y)^{2}=\\frac{1}{2} k^{2}$ and $x^{2}+y^{2}<k+1$, which forces $k \\leq 2$. If $k=0$, the region defined by $0 \\leq x^{2}+y^{2}<1$ and $x+y=0$ i... |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation $y = \\frac{x}{x^3 + Ax^2 + Bx + C}$, where $A,B,C$ are integers, is shown below. Find $A + B + C$. [asy]\nimport graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.52,xmax=4.62,ymin=-3.6... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation $y = |x| - 3$ is translated two units to the left and three units down. What are the coordinates of the minimum point of the new graph?"
}
] | math | {
"style": "rule",
"ground_truth": "(-2,-6)"
} | {
"index": 32653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation \\[ x^2 + 4y^2 - 10x + 56y = k\\]is a non-degenerate ellipse if and only if $k > a.$ What is $a?$"
}
] | math | {
"style": "rule",
"ground_truth": "-221"
} | {
"index": 32654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the equation \\[\\sqrt{x^2+y^2} + |y-1| = 3\\]consists of portions of two different parabolas. Compute the distance between the vertices of the parabolas."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x) = \\log_2 x$ is shifted 1 unit to the left, and then the part below the $x$-axis is reflected across the $x$-axis to obtain the graph of function $g(x)$. Suppose real numbers $m$ and $n$ ($m < n$) satisfy $g(m) = g\\left(-\\frac{n+1}{n+2}\\right)... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{15}"
} | {
"index": 32656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)$ is shown below. How many values of $x$ satisfy $f(f(x)) = 3$? [asy]\nimport graph; size(7.4cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.4,xmax=5.66,ymin=-1.05,ymax=6.16;\n\nfor(int i = -4; i... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)$ is symmetric about the $y$-axis, and for any $x \\in \\mathbb{R}$, it holds that $f(x+3)=-f(x)$. If $f(x)=(\\frac{1}{2})^{x}$ when $x \\in \\left( \\frac{3}{2}, \\frac{5}{2} \\right)$, then find $f(2017)$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"index": 32658,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=3 \\sqrt {2}\\cos (x+\\varphi)+\\sin x$, where $x\\in \\mathbb{R}$ and $\\varphi\\in\\left(- \\frac {\\pi}{2}, \\frac {\\pi}{2}\\right)$, passes through the point $\\left( \\frac {\\pi}{2},4\\right)$. Find the minimum value of $f(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"index": 32659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\cos(\\omega x+\\frac{π}{4})$ $(\\omega>0)$ is transformed to an odd function by shifting it to the left by $\\frac{π}{3}$ units. Determine the minimum value of the real number $\\omega$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 32660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\frac{x}{x+a}$ is symmetric about the point $(1,1)$, and the function $g(x)=\\log_{10}(10^x+1)+bx$ is even. Find the value of $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 32661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin (2x+\\varphi )$ $(|\\varphi| < \\frac{\\pi}{2})$ is shifted to the left by $\\frac{\\pi}{6}$ units and becomes an even function. Let the sequence $\\{a_n\\}$ be defined by the formula $a_n=f(\\frac{n\\pi}{6})$. Compute the sum of the first ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 32662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin (2x+\\varphi )$ $(|\\varphi| < \\frac{\\pi}{2})$ is shifted to the left by $\\frac{\\pi}{6}$ units, and the resulting graph corresponds to an even function. Find the minimum value of $m$ such that there exists $x \\in \\left[ 0,\\frac{\\pi}... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 32663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin (\\omega x+\\frac{\\pi}{3})$ ($\\omega>0$) is shifted to the left by $\\frac{\\pi}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, determine the minimum value of $\\omega$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin (\\omega x+\\frac{π}{3})$ ($\\omega\\ \\ \\gt 0$) is shifted to the left by $\\frac{π}{2}$ units to obtain the curve $C$. If $C$ is symmetric about the $y$-axis, then find the minimum value of $\\omega$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{12}$ units and then becomes symmetric about the $y$-axis. Determine the maximum value of the function $f(x)$ in the interval $\\left[0, \\frac{\\pi}{4}\\right]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin(2x+\\varphi)$ is translated to the right by $\\frac{\\pi}{6}$ units, and the resulting graph is symmetric about the origin. Determine the value of $\\varphi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 32667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin(\\omega x+\\varphi)$, where $(\\omega>0, |\\varphi|<\\frac{\\pi}{2})$, passes through the point $(0,-\\frac{1}{2})$. Find the minimum value of $\\omega$ if the graph of this function is shifted to the right by $\\frac{\\pi}{3}$ units and be... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 32668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sin({ωx-\\frac{π}{6}})$, where $0<ω<6$, is shifted to the right by $\\frac{π}{6}$ units to obtain the graph of the function $g(x)$. If $\\left(0,\\frac{π}{ω}\\right)$ is a monotone interval of $g(x)$, and $F(x)=f(x)+g(x)$, determine the maximum... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 32669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $f(x)=\\sqrt{3}\\cos 2x-\\sin 2x$ can be obtained by translating the graph of the function $f(x)=2\\sin 2x$ by an unspecified distance."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{6}"
} | {
"index": 32670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y= \\sqrt {3}\\cos x+\\sin x, (x\\in \\mathbb{R})$ is translated to the right by $\\theta$ ($\\theta>0$) units. Determine the minimum value of $\\theta$ such that the resulting graph is symmetric about the y-axis."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"index": 32671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=\\log_a(x+3)-1$ ($a > 0$ and $a \\neq 1$) always passes through a fixed point $A$. If the point $A$ lies on the line $mx+ny+1=0$, where $mn > 0$, then the minimum value of $\\frac{1}{m}+\\frac{1}{n}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3+2\\sqrt{2}"
} | {
"index": 32672,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=\\sin (2x+\\varphi) (0 < \\varphi < \\pi)$ is shifted to the right by $\\frac{\\pi}{8}$ and then is symmetric about the $y$-axis. Determine the possible value(s) of $\\varphi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{4}"
} | {
"index": 32673,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=\\sin (2x+\\varphi)$ is translated to the left by $\\dfrac {\\pi}{8}$ units along the $x$-axis and results in a graph of an even function, then determine one possible value of $\\varphi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {\\pi}{4}"
} | {
"index": 32674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=\\sin 2x-\\sqrt{3}\\cos 2x$ can be obtained by shifting the graph of the function $y=\\sin 2x+\\sqrt{3}\\cos 2x$ to the right by $\\frac{\\pi}{3}$ units."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 32675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=\\sin(\\omega x+ \\frac {5\\pi}{6})$ where $0<\\omega<\\pi$ intersects with the coordinate axes at points closest to the origin, which are $(0, \\frac {1}{2})$ and $( \\frac {1}{2}, 0)$. Determine the axis of symmetry of this graph closest to the y-... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 32676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=f(x)$ is shown below. For all $x > 4$, it is true that $f(x) > 0.4$. If $f(x) = \\frac{x^2}{Ax^2 + Bx + C}$, where $A,B,$ and $C$ are integers, then find $A+B+C$. [asy]\nimport graph; size(10.9cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); ... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"index": 32677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=f(x)$ is symmetric around the line $y=x$. After moving it left by one unit, the graph is still symmetric around the line $y=x$. If $f(1)=0$, then $f(2011)=$ __(A)__."
}
] | math | {
"style": "rule",
"ground_truth": "-2010"
} | {
"index": 32678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=g(x)$ is given. For all $x > 5$, it holds that $g(x) > 0.5$. The function $g(x)$ is defined as $g(x) = \\frac{x^2}{Ax^2 + Bx + C}$ where $A$, $B$, and $C$ are integers. The vertical asymptotes of $g$ are at $x = -3$ and $x = 4$, and the horizontal a... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"index": 32679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=g(x)$ is given. For all $x > 5$, it is observed that $g(x) > 0.1$. If $g(x) = \\frac{x^2}{Ax^2 + Bx + C}$, where $A, B, C$ are integers, determine $A+B+C$ knowing that the vertical asymptotes occur at $x = -3$ and $x = 4$."
}
] | math | {
"style": "rule",
"ground_truth": "-108"
} | {
"index": 32680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function $y=g(x)$ is shown below. For all $x > 5$, it is true that $g(x) > 0.5$. If $g(x) = \\frac{x^2}{Dx^2 + Ex + F}$, where $D, E,$ and $F$ are integers, then find $D+E+F$. Assume the function has vertical asymptotes at $x = -3$ and $x = 4$ and a horizontal ... | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"index": 32681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function \\( y = x^2 + ax + b \\) is drawn on a board. Julia drew two lines parallel to the \\( O x \\) axis on the same diagram. The first line intersects the graph at points \\( A \\) and \\( B \\), and the second line intersects the graph at points \\( C \\)... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function f(x) = sin(2x) is translated to the right by $\\frac{\\pi}{6}$ units to obtain the graph of the function g(x). Find the analytical expression for g(x). Also, find the minimum value of $|x_1 - x_2|$ for $x_1$ and $x_2$ that satisfy $|f(x_1) - g(x_2)| = ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"index": 32683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function in the form \\( y=\\frac{b}{|x|-a} \\) (where \\( a, b > 0 \\)) resembles the Chinese character \"唄\". It is referred to as the \"唄 function\", and the point symmetric to its intersection with the y-axis about the origin is called the \"目 point\". A ci... | math | {
"style": "rule",
"ground_truth": "3\\pi"
} | {
"index": 32684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the function y=sin(2x+φ) is shifted to the left by π/6 units along the x-axis, resulting in an even function graph. Determine the value of φ such that the equation 2(x + π/6) + φ = -x + 2πk is satisfied for some integer k."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 32685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the line $x+y=b$ intersects the line segment from $(2,5)$ to $(4,9)$ at its midpoint. What is the value of $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 32686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(0,3)$ to $(6,9)$. What is the value of b?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 32687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(1,3)$ to $(5,7)$. What is the value of b?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 32688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the parabola $x = 2y^2 - 6y + 3$ has an $x$-intercept $(a,0)$ and two $y$-intercepts $(0,b)$ and $(0,c)$. Find $a + b + c$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the parabola defined by the equation $y=(x-2)^2+3$ is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at $x=a$ and $x=b$. What is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 32690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the polynomial\n\n$P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e$\n\nhas five distinct $x$-intercepts, one of which is at $(0,0)$. Which of the following coefficients cannot be zero?\n\n$\\textbf{(A)}\\ a \\qquad \\textbf{(B)}\\ b \\qquad \\textbf{(C)}\\ c \\qquad \\tex... | math | {
"style": "rule",
"ground_truth": "\\text{(D)}"
} | {
"index": 32691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the power function $f(x)$ passes through the point $(3, \\frac{1}{9})$, find the maximum value of the function $g(x) = (x-1)f(x)$ on the interval $[1,3]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 32692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 32693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ has these properties: (1) The maximum value of $y = ax^2 + bx + c$ is 4, which occurs at $x = 2$. (2) The graph passes through the point $(0,-16)$. If the graph also passes through the point $(5,n)$, what is the value of $n$?"
}
... | math | {
"style": "rule",
"ground_truth": "-41"
} | {
"index": 32694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the quadratic $y = ax^2 + bx + c$ is a parabola that passes through the points $(-1,7)$, $(5,7)$, and $(6,10)$. What is the $x$-coordinate of the vertex of the parabola?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{1}{q(x)}$ is shown below. If $q(x)$ is a quadratic and $q(2) = 6$, find $q(x).$\n\n[asy]\nsize(8cm);\nimport graph;\n\nLabel f; \nf.p=fontsize(6); \n\nreal f(real x) {return 1/(2*(x+1)*(x-1));}\n\nint gridsize = 5;\ndraw((-gridsize,0)-... | math | {
"style": "rule",
"ground_truth": "2x^2 - 2"
} | {
"index": 32696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{2x^6+3x^5 - x^2 - 1}{q(x)}$ has a horizontal asymptote. What is the smallest possible degree of $q(x)$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote of $y = 0$ and a vertical asymptote of $ x=-1 $. If $q(x)$ is quadratic, $p(2)=1$, and $q(2) = 3$, find $p(x) + q(x).$\n[asy]\nsize(8cm);\nimport graph;\n\nLabel f; \nf.p=fontsiz... | math | {
"style": "rule",
"ground_truth": "x^2 + x - 2"
} | {
"index": 32698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The graph of the rational function $\\frac{p(x)}{q(x)}$ is shown below. If $q(x)$ is quadratic, $p(3)=3$, and $q(2) = 2$, find $p(x) + q(x)$.\n\n[asy]\nsize(8cm);\nimport graph;\n\nLabel f; \nf.p=fontsize(6); \n//xaxis(-5,5,Ticks(f, 1.0)); \n//yaxis(-5,5,Ticks(f, 1.0));\ndraw((... | math | {
"style": "rule",
"ground_truth": "x^2"
} | {
"index": 32699,
"solution": ""
} |
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