data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The recurring decimal \\(0 . \\dot{x} y \\dot{z}\\), where \\(x, y, z\\) denote digits between 0 and 9 inclusive, is converted to a fraction in lowest term. How many different possible values may the numerator take?"
}
] | math | {
"style": "rule",
"ground_truth": "660"
} | {
"index": 34000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $a+b+c$.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nre... | math | {
"style": "rule",
"ground_truth": "-\\frac{15}{2}"
} | {
"index": 34001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (The grid lines are spaced one unit apart.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vector... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 34002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The red parabola shown is the graph of the equation $x = ay^2 + by + c$. Find $c$. (Assume that the graph has the standard unit scale.)\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nrea... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excel... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"index": 34004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region $G$ is bounded by the ellipsoid $\\frac{x^{2}}{16}+\\frac{y^{2}}{9}+\\frac{z^{2}}{4}=1$, and the region $g$ is bounded by this ellipsoid and the sphere $x^{2}+y^{2}+z^{2}=4$. A point is randomly chosen within the region $G$. What is the probability that it belongs to... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 34005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region between the graph of $y = f (x)$ and the $x$-axis, shaded in this figure, has an area of 10 square units. What will be the area between the graph of $y = 3f (x -2)$ and the $x$-axis?\n\n[asy]\ndefaultpen(linewidth(0.75));\nfill((10,0)..(30,20)..(40,15)--(50,40)..(58,... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 34006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region consisting of all points in three-dimensional space within $3$ units of line segment $\\overline{AB}$ has volume $216 \\pi$. What is the length $AB$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34007,
"solution": "1. **Understanding the Geometry**: The region described in the problem is the set of all points in three-dimensional space that are within $3$ units of the line segment $\\overline{AB}$. This region can be visualized as a cylinder with radius $3$ and axis along $\\overline{AB}$, cappe... |
DeepScaleR | [
{
"role": "user",
"content": "The region consisting of all points in three-dimensional space within $4$ units of line segment $\\overline{CD}$ has volume $544\\pi$. Calculate the length of $CD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{86}{3}"
} | {
"index": 34008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region consisting of all points in three-dimensional space within 3 units of line segment $\\overline{AB}$ has volume $216\\pi$. What is the length $\\textit{AB}$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 34009,
"solution": "1. **Understanding the Geometry**: The region described is a cylinder with hemispheres capping both ends. The radius of the cylinder and hemispheres is $3$ units.\n\n2. **Volume of Hemispheres**: Each hemisphere has a radius of $3$ units. The volume $V$ of a sphere is given by $V = \\... |
DeepScaleR | [
{
"role": "user",
"content": "The region consisting of all points in three-dimensional space within 4 units of line segment $\\overline{CD}$, plus a cone with the same height as $\\overline{CD}$ and a base radius of 4 units, has a total volume of $448\\pi$. Find the length of $\\textit{CD}$."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 34010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region shown is bounded by the arcs of circles having a radius 8 units, with each arc having a central angle measure of 45 degrees and intersecting at points of tangency. Determine the area of this region, which can be expressed in the form $a\\sqrt{b} + c\\pi$ square units... | math | {
"style": "rule",
"ground_truth": "-22"
} | {
"index": 34011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region shown is bounded by the arcs of circles having radius 4 units, having a central angle measure of 60 degrees and intersecting at points of tangency. The area of the region can be expressed in the form $a\\sqrt{b}+c\\pi$ square units, where $\\sqrt{b}$ is a radical in ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34012,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The region shown is bounded by the arcs of circles having radius 5 units, each with a central angle measure of 45 degrees, intersecting at points of tangency. The area of the region can be expressed in the form $a\\sqrt{b} + c\\pi$ square units, where $\\sqrt{b}$ is a radical i... | math | {
"style": "rule",
"ground_truth": "-12.625"
} | {
"index": 34013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The regular hexagon \\(ABCDEF\\) has diagonals \\(AC\\) and \\(CE\\). The internal points \\(M\\) and \\(N\\) divide these diagonals such that \\(AM: AC = CN: CE = r\\). Determine \\(r\\) if it is known that points \\(B\\), \\(M\\), and \\(N\\) are collinear."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"index": 34014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In... | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"index": 34015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The regular octagon \\( A B C D E F G H \\) is inscribed in a circle. Points \\( P \\) and \\( Q \\) are on the circle, with \\( P \\) between \\( C \\) and \\( D \\), such that \\( A P Q \\) is an equilateral triangle. It is possible to inscribe a regular \\( n \\)-sided polyg... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 34016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The regular price for a bicycle is $\\$320$. The bicycle is on sale for $20\\%$ off. The regular price for a helmet is $\\$80$. The helmet is on sale for $10\\%$ off. If Sandra bought both items on sale, what is her percentage savings on the total purchase?"
}
] | math | {
"style": "rule",
"ground_truth": "18\\%"
} | {
"index": 34017,
"solution": "Since the regular price for the bicycle is $\\$320$ and the savings are $20\\%$, then the amount of money that Sandra saves on the bicycle is $\\$320 \\times 20\\%=\\$320 \\times 0.2=\\$64$. Since the regular price for the helmet is $\\$80$ and the savings are $10\\%$, then the amount... |
DeepScaleR | [
{
"role": "user",
"content": "The regular tetrahedron, octahedron, and icosahedron have equal surface areas. How are their edges related?"
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{10} : \\sqrt{10} : 2"
} | {
"index": 34018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The relevant departments want to understand the popularization of knowledge about the prevention of H1N1 influenza in schools, so they designed a questionnaire with 10 questions and conducted a survey in various schools. Two classes, A and B, from a certain middle school were r... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{5}"
} | {
"index": 34019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The remainder can be defined for all real numbers $x$ and $y$ with $y \\neq 0$ by $\\text{rem} (x ,y)=x-y\\left \\lfloor \\frac{x}{y} \\right \\rfloor$ where $\\left \\lfloor \\tfrac{x}{y} \\right \\rfloor$ denotes the greatest integer less than or equal to $\\tfrac{x}{y}$. Wha... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{40}"
} | {
"index": 34020,
"solution": "1. **Definition of Remainder**: The remainder function $\\text{rem}(x, y)$ for real numbers $x$ and $y$ (with $y \\neq 0$) is defined as:\n \\[\n \\text{rem}(x, y) = x - y \\left\\lfloor \\frac{x}{y} \\right\\rfloor\n \\]\n where $\\left\\lfloor \\cdot \\right\\rfloor$ denotes... |
DeepScaleR | [
{
"role": "user",
"content": "The remainder can be defined for all real numbers $x$ and $y$ with $y \\neq 0$ by $\\text{rem} (x ,y)=x-y\\left \\lfloor \\frac{x}{y} \\right \\rfloor$ where $\\left \\lfloor \\tfrac{x}{y} \\right \\rfloor$ denotes the greatest integer less than or equal to $\\tfrac{x}{y}$. Wha... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{40}"
} | {
"index": 34021,
"solution": "1. **Definition of Remainder**: The remainder function $\\text{rem}(x, y)$ for real numbers $x$ and $y$ (with $y \\neq 0$) is defined as:\n \\[\n \\text{rem}(x, y) = x - y \\left\\lfloor \\frac{x}{y} \\right\\rfloor\n \\]\n where $\\left\\lfloor \\cdot \\right\\rfloor$ denotes... |
DeepScaleR | [
{
"role": "user",
"content": "The remainder when 111 is divided by 10 is 1. The remainder when 111 is divided by the positive integer $n$ is 6. How many possible values of $n$ are there?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 34022,
"solution": "Since the remainder when 111 is divided by $n$ is 6, then $111-6=105$ is a multiple of $n$ and $n>6$ (since, by definition, the remainder must be less than the divisor). Since $105=3 \\cdot 5 \\cdot 7$, the positive divisors of 105 are $1,3,5,7,15,21,35,105$. Therefore, the possible v... |
DeepScaleR | [
{
"role": "user",
"content": "The remainder when \\( 104^{2006} \\) is divided by 29 is ( )"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 34023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The remainder when the product $1492 \\cdot 1776 \\cdot 1812 \\cdot 1996$ is divided by 5 is"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 34024,
"solution": "1. **Identify the last digits of each number**: \n - The last digit of $1492$ is $2$.\n - The last digit of $1776$ is $6$.\n - The last digit of $1812$ is $2$.\n - The last digit of $1996$ is $6$.\n\n2. **Calculate the product of the last digits**:\n - Compute the product of... |
DeepScaleR | [
{
"role": "user",
"content": "The remainders when three positive integers are divided by 5 are 1, 2, and 3. Find the remainder when their product is divided by 5."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The repeating decimal \\( 0.\\dot{x}y\\dot{3} = \\frac{a}{27} \\), where \\( x \\) and \\( y \\) are distinct digits. Find the integer \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 34026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The repeating decimal for $\\frac{3}{11}$ is $0.ababab\\dots$ What is the value of the sum $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The repeating decimal for $\\frac{5}{13}$ is $0.cdc\\ldots$ What is the value of the sum $c+d$?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy\n\\[0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},\\]\n where $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$."
}
] | math | {
"style": "rule",
"ground_truth": "447"
} | {
"index": 34029,
"solution": "Note that $\\frac{33}{37}=\\frac{891}{999} = 0.\\overline{891}$. Also note that the period of $0.abab\\overline{ab}+0.abcabc\\overline{abc}$ is at most $6$. Therefore, we only need to worry about the sum $0.ababab+ 0.abcabc$. Adding the two, we get \\[\\begin{array}{ccccccc}&a&b&a&b&a... |
DeepScaleR | [
{
"role": "user",
"content": "The repeating decimals $0.abab\\overline{ab}$ and $0.abcabc\\overline{abc}$ satisfy\n\\[0.abab\\overline{ab}+0.abcabc\\overline{abc}=\\frac{33}{37},\\]\nwhere $a$, $b$, and $c$ are (not necessarily distinct) digits. Find the three digit number $abc$."
}
] | math | {
"style": "rule",
"ground_truth": "447"
} | {
"index": 34030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The residents of an accommodation need to pay the rent for the accommodation. If each of them contributes $10 \\mathrm{Ft}$, the amount collected falls $88 \\mathrm{Ft}$ short of the rent. However, if each of them contributes $10.80 \\mathrm{Ft}$, then the total amount collecte... | math | {
"style": "rule",
"ground_truth": "10.54"
} | {
"index": 34031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The restaurant has two types of tables: square tables that can seat 4 people, and round tables that can seat 9 people. If the number of diners exactly fills several tables, the restaurant manager calls this number a \"wealth number.\" Among the numbers from 1 to 100, how many \... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 34032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The result of the addition shown is\n\n```\n 300\n 2020\n+10001\n```"
}
] | math | {
"style": "rule",
"ground_truth": "12321"
} | {
"index": 34033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The results of asking 50 students if they participate in music or sports are shown in the Venn diagram. Calculate the percentage of the 50 students who do not participate in music and do not participate in sports."
}
] | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 34034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The right focus of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ (with $a>0$, $b>0$) is $F$, and $B$ is a point on the left branch of the hyperbola. The line segment $BF$ intersects with one asymptote of the hyperbola at point $A$, and it is given that $(\\vec{OF} - \... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 34035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The right triangles $\\triangle M D C$ and $\\triangle A D K$ share a common right angle $\\angle D$. Point $K$ is on $C D$ and divides it in the ratio $2: 3$ counting from point $C$. Point $M$ is the midpoint of side $A D$. Find the sum of $\\angle A K D$ and $\\angle M C D$, ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 34036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The right triangles \\(MDC\\) and \\(ADK\\) have a common right angle at \\(D\\). Point \\(K\\) lies on \\(CD\\) and divides it in the ratio \\(2:3\\) from point \\(C\\). Point \\(M\\) is the midpoint of side \\(AD\\). Find the sum of the degree measures of angles \\(AKD\\) and... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 34037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The right vertex of the ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ is $A$. A line $l$ passing through the origin intersects the ellipse $C$ at points $P$ and $Q$. If $|PQ| = a$ and $AP \\perpendicular PQ$, then the eccentricity of the ellipse $C$ is ______... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"index": 34038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rightmost nonzero digit in the decimal expansion of 101 ! is the same as the rightmost nonzero digit of $n$ !, where $n$ is an integer greater than 101. Find the smallest possible value of $n$."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 34039,
"solution": "101! has more factors of 2 than 5, so its rightmost nonzero digit is one of $2,4,6,8$. Notice that if the rightmost nonzero digit of 101 ! is $2 k(1 \\leq k \\leq 4)$, then 102 ! has rightmost nonzero digit $102(2 k) \\equiv 4 k(\\bmod 10)$, and 103 ! has rightmost nonzero digit $103(... |
DeepScaleR | [
{
"role": "user",
"content": "The road from Petya's house to the school takes 20 minutes. One day, on his way to school, he remembered that he forgot his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home to get t... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{20}"
} | {
"index": 34040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The robot vacuum cleaner is programmed to move on the floor according to the law:\n\n$$\\left\\{\\begin{array}{l} x = t(t-6)^2 \\\\ y = 0, \\quad 0 \\leq t \\leq 7 \\\\ y = (t-7)^2, \\quad t \\geq 7 \\end{array}\\right.$$\n\nwhere the axes are chosen parallel to the walls and t... | math | {
"style": "rule",
"ground_truth": "\\sqrt{445}"
} | {
"index": 34041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The robotics club has 30 members: 12 boys and 18 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{574,287}{593,775}"
} | {
"index": 34042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer r... | math | {
"style": "rule",
"ground_truth": "5491"
} | {
"index": 34043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The room numbers of a hotel are all three-digit numbers. The first digit represents the floor and the last two digits represent the room number. The hotel has rooms on five floors, numbered 1 to 5. It has 35 rooms on each floor, numbered $\\mathrm{n}01$ to $\\mathrm{n}35$ where... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 34044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The room numbers of a hotel are all three-digit numbers. The first digit represents the floor and the last two digits represent the room number. The hotel has rooms on five floors, numbered 1 to 5. It has 35 rooms on each floor, numbered $\\mathrm{n}01$ to $\\mathrm{n}35$ where... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 34045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The root(s) of $\\frac {15}{x^2 - 4} - \\frac {2}{x - 2} = 1$ is (are):"
}
] | math | {
"style": "rule",
"ground_truth": "-3 \\text{ and } 5"
} | {
"index": 34046,
"solution": "1. **Identify the common denominator and simplify the equation:**\n Given the equation:\n \\[\n \\frac{15}{x^2 - 4} - \\frac{2}{x - 2} = 1\n \\]\n We recognize that $x^2 - 4 = (x-2)(x+2)$. Thus, the common denominator for the fractions is $x^2 - 4$. Rewriting the second term... |
DeepScaleR | [
{
"role": "user",
"content": "The roots of\n\\[z^7 = -\\frac{1}{\\sqrt{2}} - \\frac{i}{\\sqrt{2}}\\]are $\\text{cis } \\theta_1$, $\\text{cis } \\theta_2$, $\\dots$, $\\text{cis } \\theta_7$, where $0^\\circ \\le \\theta_k < 360^\\circ$ for all $1 \\le k \\le 7$. Find $\\theta_1 + \\theta_2 + \\dots + \\th... | math | {
"style": "rule",
"ground_truth": "1305^\\circ"
} | {
"index": 34047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of $(x^{2}-3x+2)(x)(x-4)=0$ are:"
}
] | math | {
"style": "rule",
"ground_truth": "0, 1, 2 and 4"
} | {
"index": 34048,
"solution": "To find the roots of the equation $(x^{2}-3x+2)(x)(x-4)=0$, we need to analyze each factor separately.\n\n1. **Factorize $x^2 - 3x + 2$:**\n \\[\n x^2 - 3x + 2 = (x - 1)(x - 2)\n \\]\n This factorization is obtained by finding two numbers that multiply to $2$ (the constant ter... |
DeepScaleR | [
{
"role": "user",
"content": "The roots of $64x^3-144x^2+92x-15=0$ are in arithmetic progression. The difference between the largest and smallest roots is:"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 34049,
"solution": "1. **Identify the sum of the roots using Vieta's formulas**: \n The sum of the roots of the polynomial $64x^3 - 144x^2 + 92x - 15 = 0$ is given by $-\\frac{b}{a}$ where $a = 64$ and $b = -144$. Thus,\n \\[\n \\text{Sum of roots} = -\\frac{-144}{64} = \\frac{144}{64} = \\frac{9}{... |
DeepScaleR | [
{
"role": "user",
"content": "The roots of $z^{6}+z^{4}+z^{2}+1=0$ are the vertices of a convex polygon in the complex plane. Find the sum of the squares of the side lengths of the polygon."
}
] | math | {
"style": "rule",
"ground_truth": "12-4 \\sqrt{2}"
} | {
"index": 34050,
"solution": "Factoring the polynomial as $(z^{4}+1)(z^{2}+1)=0$, we find that the 6 roots are $e^{ \\pm i \\pi / 4}$, $e^{ \\pm i \\pi / 2}, e^{ \\pm i 3 \\pi / 4}$. The calculation then follows from the Law of Cosines or the distance formula."
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$ . Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\\frac{m}{n}$ , where $m$ , $n$ are... | math | {
"style": "rule",
"ground_truth": "2224"
} | {
"index": 34051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $2\\sqrt{x} + 2x^{-\\frac{1}{2}} = 5$ can be found by solving:"
}
] | math | {
"style": "rule",
"ground_truth": "4x^2-17x+4 = 0"
} | {
"index": 34052,
"solution": "1. **Substitute and Simplify**: Let $t = \\sqrt{x}$. Then, $t^2 = x$ and $x^{-\\frac{1}{2}} = \\frac{1}{t}$. Substitute these into the given equation:\n \\[\n 2\\sqrt{x} + 2x^{-\\frac{1}{2}} = 5 \\implies 2t + \\frac{2}{t} = 5.\n \\]\n\n2. **Form a Quadratic Equation**: Multiply... |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $2x^2 - 5x - 4 = 0$ can be written in the form $x = \\frac{m \\pm \\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. What is the value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 34053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $2x^2-mx+n=0$ sum to 6 and multiply to 10. What is the value of $m+n$?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 34054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if:"
}
] | math | {
"style": "rule",
"ground_truth": "c = a"
} | {
"index": 34055,
"solution": "1. **Start by normalizing the quadratic equation**: Given the quadratic equation $ax^2 + bx + c = 0$, we can divide each term by $a$ (assuming $a \\neq 0$) to simplify the equation:\n \\[\n x^2 + \\frac{b}{a}x + \\frac{c}{a} = 0\n \\]\n\n2. **Identify the roots**: Let the roots ... |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $x^2 + kx + 8 = 0$ differ by $\\sqrt{72}$. Find the greatest possible value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{26}"
} | {
"index": 34056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $x^2 + kx + 8 = 0$ differ by 10. Find the greatest possible value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{33}"
} | {
"index": 34057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $x^2+kx+5 = 0$ differ by $\\sqrt{61}$. Find the greatest possible value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 34058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The roots of the equation $x^{2}-2x = 0$ can be obtained graphically by finding the abscissas of the points of intersection of each of the following pairs of equations except the pair:\n[Note: Abscissas means x-coordinate.]"
}
] | math | {
"style": "rule",
"ground_truth": "$y = x$, $y = x-2$"
} | {
"index": 34059,
"solution": "To solve this problem, we need to find the roots of the equation $x^2 - 2x = 0$ and check which pair of equations does not yield these roots when their graphs intersect.\n\n1. **Finding the roots of the equation $x^2 - 2x = 0$:**\n \\[\n x^2 - 2x = 0 \\implies x(x - 2) = 0\n \\]... |
DeepScaleR | [
{
"role": "user",
"content": "The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \\mathrm{km} ... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 34060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The route not passing through the Zoo is 11 times shorter."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 34061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The ruler of a certain country, for purely military reasons, wanted there to be more boys than girls among his subjects. Under the threat of severe punishment, he decreed that each family should have no more than one girl. As a result, in this country, each woman's last - and o... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 34062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. [asy]\nimport olympiad; import geometry; si... | math | {
"style": "rule",
"ground_truth": "1442"
} | {
"index": 34063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. Assume the distances from A to the nearest ... | math | {
"style": "rule",
"ground_truth": "1803"
} | {
"index": 34064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rules for a race require that all runners start at $A$, touch any part of the 1500-meter wall, touch any part of the opposite 1500-meter wall, and stop at $B$. What is the minimum distance a participant must run? Assume that $A$ is 400 meters directly south of the first wal... | math | {
"style": "rule",
"ground_truth": "2915"
} | {
"index": 34065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The rules of table tennis competition stipulate: In a game, before the opponent's score reaches 10-all, one side serves twice consecutively, then the other side serves twice consecutively, and so on. Each serve, the winning side scores 1 point, and the losing side scores 0 poin... | math | {
"style": "rule",
"ground_truth": "0.3072"
} | {
"index": 34066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sale ad read: \"Buy three tires at the regular price and get the fourth tire for 3 dollars.\" Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire?"
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 34067,
"solution": "1. **Define the variable**: Let $x$ be the regular price of one tire.\n\n2. **Set up the equation**: According to the sale ad, Sam buys three tires at the regular price and gets the fourth tire for $3. Therefore, the total cost for four tires is:\n \\[\n 3x + 3 = 240\n \\]\n w... |
DeepScaleR | [
{
"role": "user",
"content": "The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from ... | math | {
"style": "rule",
"ground_truth": "$0"
} | {
"index": 34068,
"solution": "1. **Calculate Jack's total:**\n - Jack first calculates the total price including tax on the original price:\n \\[\n 90.00 \\times 1.06 = 95.40 \\text{ dollars}\n \\]\n - Then, he applies the 20% discount to this total:\n \\[\n 95.40 \\times 0.80 = 76.32 \\tex... |
DeepScaleR | [
{
"role": "user",
"content": "The same amount of steel used to create eight solid steel balls, each with a radius of 1 inch, is used to create one larger steel ball. What is the radius of the larger ball?\n\n[asy]\nsize(150);\nfilldraw(circle((0,0),1),gray);\nfilldraw(circle((.9,-.8),1),gray);\nfilldraw(cir... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 34069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The same amount of steel used to create six solid steel balls, each with a radius of 2 inches, is used to create one larger steel ball. What is the radius of the larger ball?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt[3]{2}"
} | {
"index": 34070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The same current passes through a copper voltameter and a $10 \\mathrm{ohm}$ resistance wire. In the voltameter, 1.1081 grams of copper are deposited on the negative electrode in 30 minutes. Calculate how many gram calories of heat are generated in the wire. The electrochemical... | math | {
"style": "rule",
"ground_truth": "8.48"
} | {
"index": 34071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The sampling group size is 10."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 34072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 34073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school assigns three people, A, B, and C, to participate in social practice activities in 7 different communities, with at most 2 people assigned to each community. There are $\\boxed{\\text{336}}$ different distribution schemes (answer in numbers)."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 34074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school choir originally had 36 members, and 9 more people joined. Calculate the total number of people in the choir now."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 34075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets."
}
] | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"index": 34076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school committee has organized a \"Chinese Dream, My Dream\" knowledge speech competition. There are 4 finalists, and each contestant can choose any one topic from the 4 backup topics to perform their speech. The number of scenarios where exactly one of the topics is not se... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 34077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school now introduces a new color, silver, for the flag design. Crestview's school colors are now purple, gold, and silver. The students are designing a flag using three solid-colored horizontal stripes. Using one, two, or all three of the school colors, how many different ... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school organized a picnic with several attendees. The school prepared many empty plates. Each attendee who arrives will count the empty plates and then take one plate for food (no one can take more than one plate). The first attendee will count all the empty plates, the sec... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"index": 34079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school organized a picnic with several participants. The school prepared many empty plates. Each attendee counts the empty plates and takes one empty plate to get food (each person can only take one plate, no more). The first attendee counts all the empty plates, the second... | math | {
"style": "rule",
"ground_truth": "1006"
} | {
"index": 34080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school plans to arrange 6 leaders to be on duty from May 1st to May 3rd. Each leader must be on duty for 1 day, with 2 leaders assigned each day. If leader A cannot be on duty on the 2nd, and leader B cannot be on duty on the 3rd, how many different methods are there to arr... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 34081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school plans to organize a movie viewing for the students either on January 4th or January 10th. After finalizing the date, the teacher informs the class leader. However, due to the similarity in pronunciation between \"four\" and \"ten,\" there is a 10% chance that the cla... | math | {
"style": "rule",
"ground_truth": "0.82"
} | {
"index": 34082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school plans to purchase 8 projectors and 32 computers, with each projector costing 7500 yuan and each computer costing 3600 yuan. How much money is needed in total?"
}
] | math | {
"style": "rule",
"ground_truth": "175200"
} | {
"index": 34083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school plans to schedule six leaders to be on duty from May 1st to May 3rd, with each leader on duty for one day and two leaders scheduled each day. Given that Leader A cannot be on duty on May 2nd and Leader B cannot be on duty on May 3rd, determine the number of different... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 34084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school plans to set up two computer labs, each equipped with one teacher's computer and several student computers. In a standard lab, the teacher's computer costs 8000 yuan, and each student computer costs 3500 yuan; in an advanced lab, the teacher's computer costs 11500 yu... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 34085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost?"
}
] | math | {
"style": "rule",
"ground_truth": "$5.84"
} | {
"index": 34086,
"solution": "1. **Define Variables:**\n Let $p$ be the cost of one pencil in dollars, and $n$ be the cost of one notebook in dollars.\n\n2. **Set Up Equations:**\n From the problem, we have two equations based on the given transactions:\n \\[\n 7p + 8n = 4.15 \\quad \\text{(Equation 1)}\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The school table tennis championship was held in an Olympic system format. The winner won six matches. How many participants in the tournament won more games than they lost? (In an Olympic system tournament, participants are paired up. Those who lose a game in the first round a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 34087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The school's boys basketball team has 16 players, including a set of twins, Bob and Bill, and a set of triplets, Chris, Craig, and Carl. In how many ways can we choose 7 starters if all three triplets must be in the starting lineup and both twins must either both be in the line... | math | {
"style": "rule",
"ground_truth": "385"
} | {
"index": 34088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The science club has 25 members: 10 boys and 15 girls. A 5-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{475}{506}"
} | {
"index": 34089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The scientific notation for 0.000048 is $4.8\\times 10^{-5}$."
}
] | math | {
"style": "rule",
"ground_truth": "4.8 \\times 10^{-5}"
} | {
"index": 34090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The scores (in points) of the 15 participants in the final round of a math competition are as follows: $56$, $70$, $91$, $98$, $79$, $80$, $81$, $83$, $84$, $86$, $88$, $90$, $72$, $94$, $78$. What is the $80$th percentile of these 15 scores?"
}
] | math | {
"style": "rule",
"ground_truth": "90.5"
} | {
"index": 34091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The scores on Trisha's first three tests were 88, 73 and 70. After two more tests, the mean score for all five tests was 81. Each test score was less than 90, and all of Trisha's test scores were different integer values. List Trisha's five test scores from the greatest to the ... | math | {
"style": "rule",
"ground_truth": "89, 88, 85, 73, 70"
} | {
"index": 34092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The scores on a $110$-point test were organized in the stem-and-leaf plot shown. $9 | 6$ represents $96$ points. What is the mode of the scores? \\begin{tabular}{c|lllllll}\n\\multicolumn{8}{c}{\\underline{Points on the Test}}\\\\\n5 &0 & 0 & & & & &\\\\\n6 &3 & & & & & &\\\\\n... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 34093,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The secant \\( ABC \\) intercepts an arc \\( BC \\), which contains \\( 112^\\circ \\); the tangent \\( AD \\) at point \\( D \\) divides this arc in the ratio \\( 7:9 \\). Find \\(\\angle BAD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "31.5"
} | {
"index": 34094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The second and fifth terms of an arithmetic sequence are 17 and 19, respectively. What is the eighth term?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 34095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term? Type the letter of the correct option.\n\nA. $-\\sqrt{3}$\n\nB. $-\\frac{2\\sqrt{3}}{3}$\n\nC. $-\\frac{\\sqrt{3}}{3}$\n\nD. $\\sqrt{3}$\n\nE. $3$"
}
] | math | {
"style": "rule",
"ground_truth": "B"
} | {
"index": 34096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The second and fourth terms of a geometric sequence are 2 and 6. Which of the following is a possible first term?"
}
] | math | {
"style": "rule",
"ground_truth": "$-\\frac{2\\sqrt{3}}{3}$"
} | {
"index": 34097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The second hand on a clock is 10 cm long. How far in centimeters does the tip of this second hand travel during a period of 15 minutes? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "300\\pi"
} | {
"index": 34098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The second hand on a clock is 8 cm long. How far in centimeters does the tip of the second hand travel during a period of 45 minutes? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "720\\pi"
} | {
"index": 34099,
"solution": ""
} |
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