data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A rectangle can be divided into \\( n \\) equal squares. The same rectangle can also be divided into \\( n + 76 \\) equal squares. Find all possible values of \\( n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 2900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a length of $\\frac{3}{5}$ and an area of $\\frac{1}{3}$. What is the width of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\\\frac{5}{9}"
} | {
"index": 2901,
"solution": "In a rectangle, length times width equals area, so width equals area divided by length. Therefore, the width is $\\frac{1}{3} \\div \\frac{3}{5}=\\frac{1}{3} \\times \\frac{5}{3}=\\frac{5}{9}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a length to width ratio of 5:2. Within this rectangle, a right triangle is formed by drawing a line from one corner to the midpoint of the opposite side. If the length of this line (hypotenuse of the triangle) is measured as $d$, find the constant $k$ such that ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{13}"
} | {
"index": 2902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a perimeter of 120 inches and each side has an even integer length. How many non-congruent rectangles meet these criteria?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a perimeter of 30 units and its dimensions are whole numbers. What is the maximum possible area of the rectangle in square units?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 2904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a perimeter of 80 cm and each side has an integer length. How many non-congruent rectangles meet these criteria?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has a perimeter of 80 inches and each side has an integer length. Additionally, one dimension must be at least twice as long as the other. How many non-congruent rectangles meet these criteria?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has an area of $A$. The midpoints of each side of the rectangle are connected to form a new, smaller rectangle inside the original. What is the ratio of the area of the smaller rectangle to the area of the original rectangle? Express your answer as a common fraction... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 2909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has an area of 400 $\\text{cm}^2$. If the rectangle is a square, what is its perimeter?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 2910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has dimensions $4$ and $2\\sqrt{3}$. Two equilateral triangles are contained within this rectangle, each with one side coinciding with the longer side of the rectangle. The triangles intersect, forming another polygon. What is the area of this polygon?\nA) $2\\sqrt{... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"index": 2911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has dimensions $8 \\times 12$, and a circle centered at one of its corners has a radius of 10. Calculate the area of the union of the regions enclosed by the rectangle and the circle."
}
] | math | {
"style": "rule",
"ground_truth": "96 + 75\\pi"
} | {
"index": 2912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has dimensions 12 by 15, and a circle centered at one of its vertices has a radius of 15. What is the area of the union of the regions enclosed by the rectangle and the circle? Express your answer in terms of \\( \\pi \\)."
}
] | math | {
"style": "rule",
"ground_truth": "180 + 168.75\\pi"
} | {
"index": 2913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has integer length and width, and a perimeter of 120 units. Determine the number of square units in the greatest possible area, given that one of the dimensions must be a prime number."
}
] | math | {
"style": "rule",
"ground_truth": "899"
} | {
"index": 2914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has its length increased by $30\\%$ and its width increased by $15\\%$. Determine the percentage increase in the area of the rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "49.5\\%"
} | {
"index": 2915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has its length increased by $30\\%$ and its width increased by $15\\%$. What is the percentage increase in the area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "49.5\\%"
} | {
"index": 2916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has length $AC = 40$ and width $AE = 24$. Point $B$ is positioned one-third of the way along $AC$ from $A$ to $C$, and point $F$ is halfway along $AE$. Find the total area enclosed by rectangle $ACDE$ and the semicircle with diameter $AC$ minus the area of quadrilat... | math | {
"style": "rule",
"ground_truth": "800 + 200\\pi"
} | {
"index": 2917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has length 13 and width 10. The length and the width of the rectangle are each increased by 2. By how much does the area of the rectangle increase?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 2918,
"solution": "The area of the original rectangle is $13 \times 10=130$. When the dimensions of the original rectangle are each increased by 2, we obtain a rectangle that is 15 by 12. The area of the new rectangle is $15 \times 12=180$, and so the area increased by $180-130=50$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has length 8 cm and width $\\pi$ cm. A semi-circle has the same area as the rectangle. What is its radius?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2919,
"solution": "A rectangle with length 8 cm and width $\\pi$ cm has area $8\\pi \\mathrm{cm}^{2}$. Suppose that the radius of the semi-circle is $r \\mathrm{~cm}$. The area of a circle with radius $r \\mathrm{~cm}$ is $\\pi r^{2} \\mathrm{~cm}^{2}$ and so the area of the semi-circle is $\\frac{1}{2} ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has one side of length 5 and the other side less than 4. When the rectangle is folded so that two opposite corners coincide, the length of the crease is \\(\\sqrt{6}\\). Calculate the length of the other side."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 2920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has positive integer side lengths and an area of 24. What perimeter of the rectangle cannot be?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2921,
"solution": "Since the rectangle has positive integer side lengths and an area of 24, its length and width must be a positive divisor pair of 24. Therefore, the length and width must be 24 and 1, or 12 and 2, or 8 and 3, or 6 and 4. Since the perimeter of a rectangle equals 2 times the sum of the l... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has side lengths $6$ and $8$ . There are relatively prime positive integers $m$ and $n$ so that $\\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal o... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 2922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon a... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 2923,
"solution": "Alternatively, use basic geometry. First, scale everything down by dividing everything by 6. Let $a/6=p$. Then, the dimensions of the central rectangle in the hexagon is p x 4, and the original rectangle is 6 x p. By Pythagorean theorem and splitting the end triangles of the hexagon in... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle has width $x$ and length $y$. The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of $x+y$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2924,
"solution": "The sum of the lengths of the horizontal line segments is $4x$, because the tops of the four small rectangles contribute a total of $2x$ to their combined perimeter and the bottoms contribute a total of $2x$. Similarly, the sum of the lengths of the vertical line segments is $4y$. In o... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle having integer length and width has a perimeter of 100 units. What is the number of square units in the least possible area?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 2925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\\tfrac{1}{4}$."
}
] | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 2926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle is called cool if the number of square units in its area is equal to twice the number of units in its perimeter. A cool rectangle also must have integer side lengths. What is the sum of all the different possible areas of cool rectangles?"
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"index": 2927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew coloured all the squares in the middle row. How many squares did he not colour?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2929,
"solution": "Label the lengths of the vertical and horizontal segments as $a, b, c, d$. Rectangle W is $b$ by $c$, so its perimeter is $2 b+2 c$, which equals 2. Rectangle X is $b$ by $d$, so its perimeter is $2 b+2 d$, which equals 3. Rectangle Y is $a$ by $c$, so its perimeter is $2 a+2 c$, which... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle is inscribed in a triangle if its vertices all lie on the boundary of the triangle. Given a triangle \\( T \\), let \\( d \\) be the shortest diagonal for any rectangle inscribed in \\( T \\). Find the maximum value of \\( \\frac{d^2}{\\text{area } T} \\) for all tr... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{7}"
} | {
"index": 2930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. What percent larger is the area of the large semicircles than the area of the small semicircles? Express your... | math | {
"style": "rule",
"ground_truth": "178\\%"
} | {
"index": 2931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle of size $1000 \\times 1979$ is divided into cells. Into how many parts will it be divided if one diagonal is drawn in it?"
}
] | math | {
"style": "rule",
"ground_truth": "2978"
} | {
"index": 2932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"index": 2933,
"solution": "Note that the diagonal of the rectangle with minimum perimeter must have the diagonal along the middle segment of length 8 of the rectangle (any other inscribed rectangle can be rotated a bit, then made smaller; this one can't because then the rectangle cannot be inscribed since its lo... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle was cut into three rectangles, two of which have dimensions 9 m x 12 m and 10 m x 15 m. What is the maximum possible area of the original rectangle? Express your answer in square meters."
}
] | math | {
"style": "rule",
"ground_truth": "330"
} | {
"index": 2934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with a diagonal of length $x$ is twice as long as it is wide. What is the area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}x^2"
} | {
"index": 2935,
"solution": "1. **Define the dimensions of the rectangle**: Let the width of the rectangle be $l$ and the length be $2l$ (since the length is twice the width).\n\n2. **Expression for the area of the rectangle**: The area $A$ of the rectangle is given by the product of its length and width. Thus,\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with a length of 6 cm and a width of 4 cm is rotated around one of its sides. Find the volume of the resulting geometric solid (express the answer in terms of $\\pi$)."
}
] | math | {
"style": "rule",
"ground_truth": "144\\pi"
} | {
"index": 2936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with a perimeter of 100 cm was divided into 70 identical smaller rectangles by six vertical cuts and nine horizontal cuts. What is the perimeter of each smaller rectangle if the total length of all cuts equals 405 cm?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with diagonal length $x$ is twice as long as it is wide. What is the area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5} x^2"
} | {
"index": 2938,
"solution": "1. **Define the dimensions of the rectangle**: Let the width of the rectangle be $w$. Since the rectangle's length is twice its width, the length is $2w$.\n\n2. **Apply the Pythagorean Theorem**: The diagonal of a rectangle forms a right triangle with the rectangle's length and width. ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with dimensions $8 \\times 2 \\sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\pi + 4"
} | {
"index": 2939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with dimensions $8 \\times 2 \\sqrt{2}$ and a circle with a radius of 2 have a common center. What is the area of their overlapping region?"
}
] | math | {
"style": "rule",
"ground_truth": "$2 \\pi + 4$"
} | {
"index": 2940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with dimensions 100 cm by 150 cm is tilted so that one corner is 20 cm above a horizontal line, as shown. To the nearest centimetre, the height of vertex $Z$ above the horizontal line is $(100+x) \\mathrm{cm}$. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 2941,
"solution": "We enclose the given rectangle in a larger rectangle with horizontal and vertical sides so that the vertices of the smaller rectangle lie on the sides of the larger rectangle. We will also remove the units from the problem and deal with dimensionless quantities. Since $V W Y Z$ is a re... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with dimensions \\(24 \\times 60\\) is divided into unit squares by lines parallel to its sides. Into how many parts will this rectangle be divided if its diagonal is also drawn?"
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"index": 2942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with integer length and width has a perimeter of 120 units. What is the number of square units in the least possible area, assuming at least one of the dimensions is a prime number?"
}
] | math | {
"style": "rule",
"ground_truth": "116"
} | {
"index": 2943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangle with sides of length $4$ and $2$ is rolled into the lateral surface of a cylinder. The volume of the cylinder is $\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\pi}"
} | {
"index": 2944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row and each column contains the same number of chairs as every other column. If there must be at least two chairs in every ro... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incr... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 2946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular board of 12 columns and 12 rows has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 12, row two is 13 through 24, and so on. Determine which number of the form $n^2$ is the first to ensure that at least... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 2947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular board of 8 columns has squares numbered beginning in the upper left corner and moving left to right so row one is numbered 1 through 8, row two is 9 through 16, and so on. A student shades square 1, then skips one square and shades square 3, skips two squares and ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 2948,
"solution": "1. **Identify the Pattern of Shaded Squares**: The shaded squares follow the sequence of triangular numbers, which are given by the formula $T_n = \\frac{n(n+1)}{2}$ for positive integers $n$. This sequence starts with $T_1 = 1$, $T_2 = 3$, $T_3 = 6$, $T_4 = 10$, and so on.\n\n2. **Und... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box $Q$ is inscribed in a sphere of radius $s$. The surface area of $Q$ is 576, and the sum of the lengths of its 12 edges is 168. Determine the radius $s$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{33}"
} | {
"index": 2950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals."
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt{10}"
} | {
"index": 2951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?"
}
] | math | {
"style": "rule",
"ground_truth": "20\\sqrt{2}"
} | {
"index": 2952,
"solution": "1. **Define the variables**: Let the side lengths of the rectangular box be $x$, $y$, and $z$.\n\n2. **Use the given total edge length**: The sum of the lengths of all the edges of the box is given by:\n \\[\n 4(x+y+z) = 48\n \\]\n Dividing both sides by 4, we find:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box has a volume of 108 cubic feet. How many cubic yards are in the volume of this box?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes?"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 2954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle wit... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 2955,
"solution": "It isn't hard to see that the triangle connecting the centers of the faces of the rectangular prism is congruent to the triangle connecting the midpoints of three edges that concur. So we can now apply de Gua's theorem (https://en.wikipedia.org/wiki/De_Gua%27s_theorem) to see that:\n$3... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2956,
"solution": "1. **Equating Volume and Surface Area**: Given a rectangular box with dimensions $a \\times b \\times c$, the volume is $abc$ and the surface area is $2(ab + bc + ca)$. Setting these equal gives:\n \\[\n 2(ab + bc + ca) = abc.\n \\]\n Dividing both sides by $2abc$:\n \\[\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular cake with dimensions $4$ inches, $3$ inches, and $2$ inches (length, width, height respectively) is iced on the sides, the top, and the bottom. The cake is cut from the top center vertex across to the midpoint of the bottom edge on the opposite side face, creating... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 2957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular chessboard of size \\( m \\times n \\) is composed of unit squares (where \\( m \\) and \\( n \\) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as ... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 2958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular chocolate bar is made of equal squares. Irena breaks off two complete strips of squares and eats the 12 squares she obtains. Later, Jack breaks off one complete strip of squares from the same bar and eats the 9 squares he obtains. How many squares of chocolate are... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 2959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is $400$ feet long. The fence costs $\\$5$ per foot, and $\\$1,\\!200$ altogether. To the nearest foot, find the length of the side parallel to the barn that w... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 2960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular cuboid \\(A B C D-A_{1} B_{1} C_{1} D_{1}\\) has \\(A A_{1} = 2\\), \\(A D = 3\\), and \\(A B = 251\\). The plane \\(A_{1} B D\\) intersects the lines \\(C C_{1}\\), \\(C_{1} B_{1}\\), and \\(C_{1} D_{1}\\) at points \\(L\\), \\(M\\), and \\(N\\) respectively. Wha... | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"index": 2961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular field has a length of 20 metres and a width of 5 metres. If its length is increased by 10 m, by how many square metres will its area be increased?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 2962,
"solution": "Since the field originally has length 20 m and width 5 m, then its area is $20 \\times 5=100 \\mathrm{~m}^{2}$. The new length of the field is $20+10=30 \\mathrm{~m}$, so the new area is $30 \\times 5=150 \\mathrm{~m}^{2}$. The increase in area is $150-100=50 \\mathrm{~m}^{2}$. (Altern... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular field is 300 feet wide and 400 feet long. Random sampling indicates that there are, on the average, three ants per square inch through out the field. [12 inches = 1 foot.] Of the following, the number that most closely approximates the number of ants in the field ... | math | {
"style": "rule",
"ground_truth": "50000000"
} | {
"index": 2963,
"solution": "1. **Convert dimensions from feet to inches**: \n - The width of the field in inches is:\n \\[\n 300 \\text{ feet} \\times \\frac{12 \\text{ inches}}{1 \\text{ foot}} = 3600 \\text{ inches}\n \\]\n - The length of the field in inches is:\n \\[\n 400 \\text{ feet... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular field is half as wide as it is long, and it is completely enclosed by 54 meters of fencing. What is the number of square meters in the area of the field?"
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 2964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. Determine the total number of different flags th... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"index": 2965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. How many different flags can be made?"
}
] | math | {
"style": "rule",
"ground_truth": "260"
} | {
"index": 2966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor."
}
] | math | {
"style": "rule",
"ground_truth": "841"
} | {
"index": 2967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the floor. The unpainted part of the floor forms a border of width $1$ foot around the painte... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 2968,
"solution": "1. **Identify the area of the painted and unpainted regions**: Given that the unpainted border is 1 foot wide and surrounds the entire painted rectangle, the dimensions of the painted rectangle are $(a-2)$ by $(b-2)$. The total area of the floor is $ab$, and the area of the painted rec... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $1$ foot aro... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 2969,
"solution": "1. **Define the areas**: Let $A_{\\text{outer}}$ be the area of the entire floor and $A_{\\text{inner}}$ be the area of the painted rectangle. Given that the unpainted border is 1 foot wide, the dimensions of the painted rectangle are $(a-2)$ and $(b-2)$.\n \n Therefore, we have:\n... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 2970,
"solution": "To solve this problem, we need to determine how many tiles the bug crosses as it walks in a straight line from one corner of the rectangle to the opposite corner. The rectangle is $10$ feet wide and $17$ feet long, and the tiles are $1$ foot square each.\n\n1. **Understanding the Path*... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular floor that is $12$ feet wide and $19$ feet long is tiled with rectangular tiles that are $1$ foot by $2$ feet. Find the number of tiles a bug visits when walking from one corner to the diagonal opposite corner."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?"
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 2972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular garden 60 feet long and 15 feet wide is enclosed by a fence. To utilize the same fence but change the shape, the garden is altered to an equilateral triangle. By how many square feet does this change the area of the garden?"
}
] | math | {
"style": "rule",
"ground_truth": "182.53"
} | {
"index": 2973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular garden measures $12$ meters in width and $20$ meters in length. It is paved with tiles that are $2$ meters by $2$ meters each. A cat runs from one corner of the rectangular garden to the opposite corner but must leap over a small pond that exactly covers one tile ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular garden measuring 60 feet by 20 feet is enclosed by a fence. In a redesign to maximize the area using the same amount of fencing, its shape is changed to a circle. How many square feet larger or smaller is the new garden compared to the old one?"
}
] | math | {
"style": "rule",
"ground_truth": "837.62"
} | {
"index": 2975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular garden needs to be enclosed on three sides using a 70-meter rock wall as one of the sides. Fence posts are placed every 10 meters along the fence, including at the ends where the fence meets the rock wall. If the area of the garden is 2100 square meters, calculate... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 2976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver."
... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of pos... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2978,
"solution": "1. **Identify the dimensions and orientation of the grazing area**: The grazing area is a rectangle with dimensions $36$ m and $60$ m. To minimize the number of fence posts, we use the longer side ($60$ m) along the existing rock wall. This means the two $36$ m sides and one $60$ m sid... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular grid consists of 5 rows and 6 columns with equal square blocks. How many different squares can be traced using the lines in the grid?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 2979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular grid is given. Let's call two cells neighboring if they share a common side. Count the number of cells that have exactly four neighboring cells. There are 23 such cells. How many cells have three neighboring cells?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 2980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular metal plate measuring \\(10\\) cm by \\(8\\) cm has a circular piece of maximum size cut out, followed by cutting a rectangular piece of maximum size from the circular piece. Calculate the total metal wasted in this process."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 2981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular parking lot has a diagonal of $25$ meters and an area of $168$ square meters. In meters, what is the perimeter of the parking lot?"
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 2983,
"solution": "1. **Identify the given information and set up equations:**\n Let the sides of the rectangular parking lot be $a$ and $b$. The problem states that the diagonal is $25$ meters and the area is $168$ square meters. Using the Pythagorean theorem for the diagonal, we have:\n \\[\n a^2... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular pasture is to be fenced off on three sides using part of a 100 meter rock wall as the fourth side. Fence posts are to be placed every 15 meters along the fence including at the points where the fence meets the rock wall. Given the dimensions of the pasture are 36 ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 2984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches?"
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 2985,
"solution": "1. **Calculate the area of the photograph**: \n The photograph measures $8$ inches in height and $10$ inches in width. Therefore, the area of the photograph is calculated as:\n \\[\n \\text{Area of photograph} = 8 \\times 10 = 80 \\text{ square inches}\n \\]\n\n2. **Determine t... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular picture frame is constructed from 1.5-inch-wide pieces of wood. The area of just the frame is \\(27\\) square inches, and the length of one of the interior edges of the frame is \\(4.5\\) inches. Determine the sum of the lengths of the four interior edges of the f... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular picture frame is made from one-inch-wide pieces of wood. The area of just the frame is $18$ square inches, and one of the outer edges of the frame is $5$ inches long. What is the sum of the lengths of the four interior edges of the frame?\n\n[asy]\n\nsize(5cm,5c... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 2987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \\mathrm{~cm}$ remained. The given drawing indicat... | math | {
"style": "rule",
"ground_truth": "258"
} | {
"index": 2988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\\angle 1 = 22^\\circ$, find the measure of $\\angle 2$."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 2990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper $A B C D$ is folded and flattened such that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ landing on side $A B$. Given that $\\angle 1 = 22^{\\circ}$, find $\\angle 2$."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 2991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper $P Q R S$ has $P Q=20$ and $Q R=15$. The piece of paper is glued flat on the surface of a large cube so that $Q$ and $S$ are at vertices of the cube. What is the shortest distance from $P$ to $R$, as measured through the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "18.4"
} | {
"index": 2992,
"solution": "Since $P Q R S$ is rectangular, then $\\angle S R Q=\\angle S P Q=90^{\\circ}$. Also, $S R=P Q=20$ and $S P=Q R=15$. By the Pythagorean Theorem in $\\triangle S P Q$, since $Q S>0$, we have $Q S=\\sqrt{S P^{2}+P Q^{2}}=\\sqrt{15^{2}+20^{2}}=\\sqrt{225+400}=\\sqrt{625}=25$. Draw perpend... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle $\\theta$ is"
}
] | math | {
"style": "rule",
"ground_truth": "$3\\sec ^2\\theta\\csc\\theta$"
} | {
"index": 2993,
"solution": "1. **Identify the Geometry and Variables**: \n - Let the rectangle be $ABCD$ with $A$ at the top left, $B$ at the top right, $C$ at the bottom right, and $D$ at the bottom left.\n - The crease $BE$ is formed such that $E$ lies on side $CD$.\n - Define $F$ on $AD$ such that $F$ is... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if\n (i) all four sides of the rectangle are segments of drawn line segments,... | math | {
"style": "rule",
"ground_truth": "896"
} | {
"index": 2994,
"solution": "Denote the number of horizontal lines drawn as $x$, and the number of vertical lines drawn as $y$. The number of basic rectangles is $(x - 1)(y - 1)$. $5x + 4y = 2007 \\Longrightarrow y = \\frac{2007 - 5x}{4}$. Substituting, we find that $(x - 1)\\left(-\\frac 54x + \\frac{2003}4\\righ... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper whose length is $\\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The pape... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 2995,
"solution": "1. **Assumptions and Setup**: Assume the width of the paper is $w$ and the length is $\\sqrt{3}w$. The area of the paper, $A$, is given by:\n \\[\n A = w \\times \\sqrt{3}w = \\sqrt{3}w^2.\n \\]\n Without loss of generality, let $w = 1$. Then the area $A = \\sqrt{3}$.\n\n2. **D... |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper with a length of 20 cm and a width of 12 cm is folded along its diagonal (refer to the diagram). What is the perimeter of the shaded region formed?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 2996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimete... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 2997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines so that the folded flaps just touch at the corners as indicated by the dotted lines. Find the area of the resulting trapezoid."
}
] | math | {
"style": "rule",
"ground_truth": "55/2"
} | {
"index": 2998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A rectangular piece of paper with vertices $A B C D$ is being cut by a pair of scissors. The pair of scissors starts at vertex $A$, and then cuts along the angle bisector of $D A B$ until it reaches another edge of the paper. One of the two resulting pieces of paper has 4 times... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 2999,
"solution": "Without loss of generality, let $A B>A D$, and let $x=A D, y=A B$. Let the cut along the angle bisector of $\\angle D A B$ meet $C D$ at $E$. Note that $A D E$ is a $45-45-90$ triangle, so $D E=A D=x$, and $E C=y-x$. Now, $[A D E]=\\frac{x^{2}}{2}$, and $[A E C B]=x\\left(y-\\frac{x}{2... |
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