data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ are two distinct points on the parabola $y=3-x^{2}$ that are symmetric with respect to the line $x+y=0$. Find $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "$3 \\sqrt{2}$"
} | {
"index": 100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ both live such that one lives $3 \\text{ km}$ further from the city than the other. They both arrived in the city at the same time, with $A$ traveling by car and $B$ by truck. They were picked up by the vehicles halfway to their destinations. $A$ walks $1.5$ times f... | math | {
"style": "rule",
"ground_truth": "16.5"
} | {
"index": 101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s... | math | {
"style": "rule",
"ground_truth": "5/6"
} | {
"index": 102,
"solution": "Let the speeds of $A$ and $B$ be $v_A$ and $v_B$ respectively, and let the distance from $O$ be measured in yards. Since $A$ and $B$ move along paths that intersect at right angles, we can consider their movements along the x-axis and y-axis respectively.\n\n1. **Initial Conditions:**\n... |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ together can do a job in $2$ days; $B$ and $C$ can do it in four days; and $A$ and $C$ in $2\\frac{2}{5}$ days.\nThe number of days required for A to do the job alone is:"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 103,
"solution": "1. **Define the rates of work**: Let $r_A$, $r_B$, and $r_C$ be the rates at which $A$, $B$, and $C$ can complete the job per day, respectively. The units are $\\frac{\\text{job}}{\\text{day}}$.\n\n2. **Set up the equations based on the given information**:\n - $A$ and $B$ together ca... |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 104,
"solution": "1. **Define Variables:**\n Let $x$ be half the circumference of the track. Therefore, the full circumference is $2x$.\n\n2. **Analyze First Meeting:**\n $A$ and $B$ start from diametrically opposite points on the track. When they first meet, $B$ has traveled $100$ yards. Since they ... |
DeepScaleR | [
{
"role": "user",
"content": "$A$ and $B$ travel around an elliptical track at uniform speeds in opposite directions, starting from the vertices of the major axis. They start simultaneously and meet first after $B$ has traveled $150$ yards. They meet a second time $90$ yards before $A$ completes one lap. Fi... | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$ can do a piece of work in $9$ days. $B$ is $50\\%$ more efficient than $A$. The number of days it takes $B$ to do the same piece of work is:"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 106,
"solution": "1. **Calculate $A$'s work rate**: If $A$ can complete the work in $9$ days, then $A$'s work rate is $\\frac{1}{9}$ of the work per day.\n\n2. **Determine $B$'s work rate**: Since $B$ is $50\\%$ more efficient than $A$, $B$'s work rate is $1.5$ times that of $A$. Therefore, $B$'s work ra... |
DeepScaleR | [
{
"role": "user",
"content": "$A$ is located 60 kilometers west of $B$. Individuals A and B depart from location $A$ while individuals C and D depart from location $B$ at the same time. A, B, and D all travel east, while C travels west. It is known that the speeds of A, B, C, and D form an arithmetic sequen... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 108,
"solution": "1. **Define Variables:**\n Let $A_n$ be the total number of rocks in pile $A$, and $B_n$ be the total number of rocks in pile $B$. Let the total weight of rocks in pile $C$ be $m$, and the total number of rocks in $C$ be $n$.\n\n2. **Use Given Averages:**\n - For piles $A$ and $B$, ... |
DeepScaleR | [
{
"role": "user",
"content": "$A$, $B$, $C$, and $D$ are points on a circle, and segments $\\overline{AC}$ and $\\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP < DP.$\n\n[asy]\n\nunitsize(0.6 inch);\n\ndraw(circle((0,0),1));\ndraw((-0.3,0.94)--(0.3,-0.94));\... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A$,$B$,$C$,$D$,$E$,$F$ are 6 students standing in a row to participate in a literary performance. If $A$ does not stand at either end, and $B$ and $C$ must be adjacent, then the total number of different arrangements is ____."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A, B, C$ , and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$ , $BC = 13$ , $CD = 37$ , and $AD = 47$ , what is the area of $ABCD$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A, B, C, D$ attended a meeting, and each of them received the same positive integer. Each person made three statements about this integer, with at least one statement being true and at least one being false. Their statements are as follows:\n\n$A:\\left(A_{1}\\right)$ The numb... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A, B, C, D, E, F, G$ are seven people sitting around a circular table. If $d$ is the total number of ways that $B$ and $G$ must sit next to $C$, find the value of $d$."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A, B, C, D,$ and $E$ are collinear in that order such that $AB = BC = 1, CD = 2,$ and $DE = 9$. If $P$ can be any point in space, what is the smallest possible value of $AP^2 + BP^2 + CP^2 + DP^2 + EP^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A,B,C$ are points in the plane such that $\\angle ABC=90^\\circ$ . Circles with diameters $BA$ and $BC$ meet at $D$ . If $BA=20$ and $BC=21$ , then the length of segment $BD$ can be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime p... | math | {
"style": "rule",
"ground_truth": "449"
} | {
"index": 116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$AB$ is a chord of length $6$ in a circle of radius $5$ and centre $O$ . A square is inscribed in the sector $OAB$ with two vertices on the circumference and two sides parallel to $ AB$ . Find the area of the square."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$AB$ and $AC$ are tangents to a circle, $\\angle BAC = 60^{\\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABC$ is a triangle with $AB = 33$ , $AC = 21$ and $BC = m$ , an integer. There are points $D$ , $E$ on the sides $AB$ , $AC$ respectively such that $AD = DE = EC = n$ , an integer. Find $m$ ."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$ , respectively. Find the largest possible value of $AB$ . \n\n*Team #6*"
}
] | math | {
"style": "rule",
"ground_truth": "58\\sqrt{3}"
} | {
"index": 120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABC$ is triangle. $l_1$ - line passes through $A$ and parallel to $BC$ , $l_2$ - line passes through $C$ and parallel to $AB$ . Bisector of $\\angle B$ intersect $l_1$ and $l_2$ at $X,Y$ . $XY=AC$ . What value can take $\\angle A- \\angle C$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABC$ is a triangle: $A=(0,0), B=(36,15)$ and both the coordinates of $C$ are integers. What is the minimum area $\\triangle ABC$ can have?\n$\\textbf{(A)}\\ \\frac{1}{2} \\qquad \\textbf{(B)}\\ 1 \\qquad \\textbf{(C)}\\ \\frac{3}{2} \\qquad \\textbf{(D)}\\ \\frac{13}{2}\\qquad... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABC$ is an equilateral triangle with sides equal to 2cm. $\\overline{BC}$ is extended its own length to $D$, and $E$ is the midpoint of $\\overline{AB}$. Suppose $\\overline{ED}$ meets $\\overline{AC}$ at $F$. Find the area of the quadrilateral $BEFC$ in square centimeters.\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABC$ is an isosceles triangle with base $AC$. $CD$ is the bisector of angle $C$, and $\\angle ADC = 150^\\circ$. Find $\\angle B$."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ forms a rhombus. $E$ is the intersection of $AC$ and $BD$ . $F$ lie on $AD$ such that $EF$ is perpendicular to $FD$ . Given $EF=2$ and $FD=1$ . Find the area of the rhombus $ABCD$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{51}"
} | {
"index": 126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangle with $AB = CD = 2$ . A circle centered at $O$ is tangent to $BC$ , $CD$ , and $AD$ (and hence has radius $1$ ). Another circle, centered at $P$ , is tangent to circle $O$ at point $T$ and is also tangent to $AB$ and $BC$ . If line $AT$ i... | math | {
"style": "rule",
"ground_truth": "3 - 2\\sqrt{2}"
} | {
"index": 127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$ , and $F$ is a point on $AD$ between $A$ and $D$ . The area of the triangle $EBC$ is $16$ , the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area o... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\\angle ABC$ is right, determine the radius of this incircle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{13}}{3}"
} | {
"index": 129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a parallelogram with $\\angle D$ obtuse. $M$ and $N$ are the feet of the perpendiculars from $D$ to $AB$ and $BC$ respectively. If $DB = DC = 50$ and $DA = 60$, find $DM + DN$."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\\overline{AB}$. $\\overline{PS} \\perp \\overline{BD}$ and $\\overline{PR} \\perp \\overline{AC}$. $\\overline{AF} \\perp \\overline{BD}$ and $\\overline{PQ} \\perp \\overline{AF}$. Then $PR + PS$ is e... | math | {
"style": "rule",
"ground_truth": "$AF$"
} | {
"index": 131,
"solution": "1. **Identify Parallel Lines**: Since $\\overline{PQ}$ and $\\overline{BD}$ are both perpendicular to $\\overline{AF}$, it follows that $\\overline{PQ} \\parallel \\overline{BD}$.\n\n2. **Angle Relationships**: Because $\\overline{PQ} \\parallel \\overline{BD}$ and $\\overline{AB}$ is a... |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangle whose area is 12 square units. How many square units are contained in the area of trapezoid $EFBA$?\n\n[asy]\n\nsize(4cm,4cm);\n\nfor(int i=0; i < 4; ++i){\nfor(int k=0; k < 5; ++k){\ndraw((0,i)--(4,i));\ndraw((k,0)--(k,3));\n} }\n\ndraw((0,0)--(1,3));\ndr... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 132,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangle; $P$ and $Q$ are the mid-points of $AB$ and $BC$ respectively. $AQ$ and $CP$ meet at $R$. If $AC = 6$ and $\\angle ARC = 150^{\\circ}$, find the area of $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"index": 133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ w... | math | {
"style": "rule",
"ground_truth": "293"
} | {
"index": 134,
"solution": "Use the prepared diagram for this solution.\n Call the intersection of DF and B'C' G. AB'E is an 8-15-17 right triangle, and so are B'DG and C'FG. Since C'F is 3, then using the properties of similar triangles GF is 51/8. DF is 22, so DG is 125/8. Finally, DB can to calculated to be 25/... |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangular sheet of paper. $E$ and $F$ are points on $AB$ and $CD$ respectively such that $BE < CF$. If $BCFE$ is folded over $EF$, $C$ maps to $C'$ on $AD$ and $B$ maps to $B'$ such that $\\angle{AB'C'} \\cong \\angle{B'EA}$. If $AB' = 5$ and $BE = 23$, then the a... | math | {
"style": "rule",
"ground_truth": "338"
} | {
"index": 135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a rectangular sheet of paper. Points $E$ and $F$ are located on edges $AB$ and $CD$, respectively, such that $BE < CF$. The rectangle is folded over line $EF$ so that point $C$ maps to $C'$ on side $AD$ and point $B$ maps to $B'$ on side $AD$ such that $\\angle{AB'C'}... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 136,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a regular tetrahedron (right triangular pyramid). If $M$ is the midpoint of $\\overline{CD}$, then what is $\\cos \\angle AMB$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a regular tetrahedron. Let $N$ be the midpoint of $\\overline{AB}$, and let $G$ be the centroid of triangle $ACD$. What is $\\cos \\angle BNG$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a square and $M$ and $N$ are the midpoints of $\\overline{BC}$ and $\\overline{CD},$ respectively. Find $\\sin \\theta.$\n\n[asy]\nunitsize(1.5 cm);\n\ndraw((0,0)--(2,0)--(2,2)--(0,2)--cycle);\ndraw((0,0)--(2,1));\ndraw((0,0)--(1,2));\nlabel(\"$A$\", (0,0), SW);\nlab... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a square of side length 1. $X$ and $Y$ are points on $BC$ and $CD$ respectively such that $CX = CY = m$. When extended, $AB$ meets $DX$ at $P$, $AD$ meets $BY$ at $Q$, $AX$ meets $DC$ at $R$, and $AY$ meets $BC$ at $S$. If $P, Q, R, S$ are collinear, find $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 - \\sqrt{5}}{2}"
} | {
"index": 140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a square that is made up of two identical rectangles and two squares of area $4 \\mathrm{~cm}^{2}$ and $16 \\mathrm{cm}^{2}$. What is the area, in $\\mathrm{cm}^{2}$, of the square $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 141,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a square where each side measures 4 units. $P$ and $Q$ are the midpoints of $\\overline{BC}$ and $\\overline{CD},$ respectively. Find $\\sin \\phi$ where $\\phi$ is the angle $\\angle APQ$.\n\n"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a trapezium such that $\\angle ADC=\\angle BCD=60^{\\circ}$ and $AB=BC=AD=\\frac{1}{2}CD$. If this trapezium is divided into $P$ equal portions $(P>1)$ and each portion is similar to trapezium $ABCD$ itself, find the minimum value of $P$.\n\nThe sum of tens and unit d... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a trapezoid with $AB \\parallel CD$, $AB=6$, and $CD=15$. If the area of $\\triangle AED=30$, what is the area of $\\triangle AEB?$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$ is a trapezoid with the measure of base $\\overline{AB}$ twice the measure of the base $\\overline{CD}$. Point $E$ is the point of intersection of the diagonals. The measure of diagonal $\\overline{AC}$ is 11. Find the length of segment $\\overline{EC}$. Express your ans... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{3}"
} | {
"index": 145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCD$, a rectangle with $AB = 12$ and $BC = 16$, is the base of pyramid $P$, which has a height of $24$. A plane parallel to $ABCD$ is passed through $P$, dividing $P$ into a frustum $F$ and a smaller pyramid $P'$. Let $X$ denote the center of the circumsphere of $F$, and let ... | math | {
"style": "rule",
"ground_truth": "177"
} | {
"index": 146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDE$ is a regular pentagon inscribed in a circle of radius 1. What is the area of the set of points inside the circle that are farther from $A$ than they are from any other vertex?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{5}"
} | {
"index": 147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDE$ is a regular pentagon. $AP$, $AQ$ and $AR$ are the perpendiculars dropped from $A$ onto $CD$, $CB$ extended and $DE$ extended, respectively. Let $O$ be the center of the pentagon. If $OP = 1$, then find $AO + AQ + AR$.\n\n[asy]\n\nunitsize(2 cm);\n\npair A, B, C, D, ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$ Compute $(1 - \\cos \\angle B)(1 - \\cos \\angle ACE).$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"index": 149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$ , different from $C$ . What is the length of the segment $IF$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\\angle{ACE}$ is $90^{\\circ}$ . What is the average of the measures, in degrees, of $\\angle{ABC}$ and $\\angle{CDE}$ ?\n\n*2018 CCA Math Bonanza Lightning Round #1.3*"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 151,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\\vartriangle ACE$ and $\\vartriangle BDF$ . What is the area of $R$ divided by the area of $ABCDEF$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ is a cube having a side length of 2. $P$ is the midpoint of $EF$, as shown. The area of $\\triangle APB$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{8}$"
} | {
"index": 153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ is a cube where each side has length $a$. Find $\\sin \\angle GAC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 154,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ is a cube. Find $\\cos \\angle GAC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ is a cube. Find $\\sin \\angle BAE$, where $E$ is the top vertex directly above $A$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{2}}"
} | {
"index": 156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ is a cube. Find $\\sin \\angle HAD$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"index": 157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ is a rectangular prism with $AB=CD=EF=GH=1$, $AD=BC=EH=FG=2$, and $AE=BF=CG=DH=3$. Find $\\sin \\angle GAC$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{\\sqrt{14}}"
} | {
"index": 158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ shown below is a cube with volume 1. Find the volume of pyramid $ABCH$.\n\n[asy]\nimport three;\ntriple A,B,C,D,EE,F,G,H;\nA = (0,0,0);\nB = (1,0,0);\nC = (1,1,0);\nD= (0,1,0);\nEE = (0,0,1);\nF = B+EE;\nG = C + EE;\nH = D + EE;\ndraw(B--C--D);\ndraw(B--A--D,dashed)... | math | {
"style": "rule",
"ground_truth": "\\frac16"
} | {
"index": 159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ shown below is a cube. Find $\\sin \\angle GAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH = D + EE;\n\ndraw(B--C--D);\n\ndraw(B--A--D,dashed)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$ABCDEFGH$ shown below is a cube. Find $\\sin \\angle HAC$.\n\n[asy]\n\nimport three;\n\ntriple A,B,C,D,EE,F,G,H;\n\nA = (0,0,0);\n\nB = (1,0,0);\n\nC = (1,1,0);\n\nD= (0,1,0);\n\nEE = (0,0,1);\n\nF = B+EE;\n\nG = C + EE;\n\nH = D + EE;\n\ndraw(B--C--D);\n\ndraw(B--A--D,dashed)... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$AM$ is the angle bisector of triangle $ABC$. $BM = 2$, $CM = 3$, and $D$ is the point where the extension of $AM$ intersects the circumcircle of the triangle. If $MD = 2$, find $AB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$ . Find the maximum value of $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$A_{2n}^{n+3} + A_{4}^{n+1} = \\boxed{\\text{\\_\\_\\_\\_\\_\\_\\_\\_}}$."
}
] | math | {
"style": "rule",
"ground_truth": "744"
} | {
"index": 165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$B$ and $C$ trisect $\\overline{AD}$ and $M$ is the midpoint of $\\overline{AD}$. $MC = 8$. How many units are in the length of $\\overline{AD}$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$BL$ is the angle bisector of triangle $ABC$. Find its area, given that $|AL| = 2$, $|BL| = 3\\sqrt{10}$, and $|CL| = 3$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15 \\sqrt{15}}{4}"
} | {
"index": 167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$C$ is a point on the extension of diameter $A B$, $C D$ is a tangent, and the angle $A D C$ is $110^{\\circ}$. Find the angular measure of arc $B D$."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 168,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$D$ is a point on side $AB$ of triangle $ABC$, satisfying $AD=2$ and $DB=8$. Let $\\angle ABC=\\alpha$ and $\\angle CAB=\\beta$. <br/>$(1)$ When $CD\\perp AB$ and $\\beta =2\\alpha$, find the value of $CD$; <br/>$(2)$ If $α+\\beta=\\frac{π}{4}$, find the maximum area of triangl... | math | {
"style": "rule",
"ground_truth": "5(\\sqrt{2} - 1)"
} | {
"index": 169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$ . Let $O$ be the centre of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$ . (See diagram). Given $EC=1$ , find the radius of the circle.\n\n[asy]\nsize(6cm);\n\npair O = (0,0), B = dir(110), D ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$E$ is the midpoint of side $BC$ of parallelogram $ABCD$. Line $AE$ intersects the diagonal $BD$ at point $G$. If the area of triangle $\\triangle BEG$ is 1, find the area of parallelogram $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$F$ is the right focus of the hyperbola $C: \\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1 \\left(a > 0, b > 0\\right)$. A perpendicular line is drawn from point $F$ to asymptote $C$, with the foot of the perpendicular denoted as $A$, intersecting another asymptote at point ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\sqrt{3}}{3}"
} | {
"index": 172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$F, G, H, I,$ and $J$ are collinear in that order such that $FG = 2, GH = 1, HI = 3,$ and $IJ = 7$. If $P$ can be any point in space, what is the smallest possible value of $FP^2 + GP^2 + HP^2 + IP^2 + JP^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "102.8"
} | {
"index": 173,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$H O W, B O W$, and $D A H$ are equilateral triangles in a plane such that $W O=7$ and $A H=2$. Given that $D, A, B$ are collinear in that order, find the length of $B A$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 174,
"solution": "Note that $H \\neq B$ since otherwise $D A B$ is an equilateral triangle. Let $M$ be the midpoint of $D A$, so $H B=7 \\sqrt{3}$ and $H M=\\sqrt{3}$, and $\\angle H M B=90^{\\circ}$. By the Pythagorean theorem, $$ B M=\\sqrt{(7 \\sqrt{3})^{2}-(\\sqrt{3})^{2}}=12 $$ Then $B A=B M-A M=11$... |
DeepScaleR | [
{
"role": "user",
"content": "$J K L M$ is a square. Points $P$ and $Q$ are outside the square such that triangles $J M P$ and $M L Q$ are both equilateral. The size, in degrees, of angle $P Q M$ is"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$JKLM$ is a square and $PQRS$ is a rectangle. If $JK$ is parallel to $PQ$, $JK = 8$ and $PS = 2$, then the total area of the shaded regions is:"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$K$ takes $30$ minutes less time than $M$ to travel a distance of $30$ miles. $K$ travels $\\frac {1}{3}$ mile per hour faster than $M$. If $x$ is $K$'s rate of speed in miles per hours, then $K$'s time for the distance is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{30}{x}"
} | {
"index": 177,
"solution": "Let's analyze the problem step by step:\n\n1. **Define Variables:**\n - Let $x$ be the speed of $K$ in miles per hour.\n - Then, $M$'s speed is $x - \\frac{1}{3}$ miles per hour, since $K$ travels $\\frac{1}{3}$ mile per hour faster than $M$.\n\n2. **Write the Time Equations:**\n ... |
DeepScaleR | [
{
"role": "user",
"content": "$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$ , respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$ . The perimeter of triangle $KLB$ is equal to $1$ . What is the area of triangle $MND$ ?"
}... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$M$ is an $8 \\times 8$ matrix. For $1 \\leq i \\leq 8$, all entries in row $i$ are at least $i$, and all entries on column $i$ are at least $i$. What is the minimum possible sum of the entries of $M$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "372"
} | {
"index": 179,
"solution": "Let $s_{n}$ be the minimum possible sum for an $n$ by $n$ matrix. Then, we note that increasing it by adding row $n+1$ and column $n+1$ gives $2 n+1$ additional entries, each of which has minimal size at least $n+1$. Consequently, we obtain $s_{n+1}=s_{n}+(2 n+1)(n+1)=s_{n}+2 n^{2}+3 n+... |
DeepScaleR | [
{
"role": "user",
"content": "$M$ is the midpoint of the side $AB$ in an equilateral triangle $\\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\\triangle ABC$ such that $\\angle CTA... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"index": 180,
"solution": "\nGiven an equilateral triangle \\(\\triangle ABC\\), where \\(M\\) is the midpoint of side \\(AB\\). We have a point \\(D\\) on side \\(BC\\) such that the segment \\(BD : DC = 3 : 1\\). We need to find \\(\\angle MTD\\) given that there is a point \\(T\\) on a line passing through \\(... |
DeepScaleR | [
{
"role": "user",
"content": "$N$ students are seated at desks in an $m \\times n$ array, where $m, n \\ge 3$ . Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $ handshakes, what is $N$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "280"
} | {
"index": 181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$O K R A$ is a trapezoid with $O K$ parallel to $R A$. If $O K=12$ and $R A$ is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to $O K$, through the intersection of the diagonals?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 182,
"solution": "Let $R A=x$. If the diagonals intersect at $X$, and the segment is $P Q$ with $P$ on $K R$, then $\\triangle P K X \\sim \\triangle R K A$ and $\\triangle O K X \\sim \\triangle R A X$ (by equal angles), giving $R A / P X=$ $A K / X K=1+A X / X K=1+A R / O K=(x+12) / 12$, so $P X=12 x /... |
DeepScaleR | [
{
"role": "user",
"content": "$O$ and $I$ are the circumcentre and incentre of $\\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$ , and $C$ . What is the magnitude of $\\angle B AC$ in degrees?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$O$ is the center of square $A B C D$, and $M$ and $N$ are the midpoints of $\\overline{B C}$ and $\\overline{A D}$, respectively. Points $A^{\\prime}, B^{\\prime}, C^{\\prime}, D^{\\prime}$ are chosen on $\\overline{A O}, \\overline{B O}, \\overline{C O}, \\overline{D O}$, res... | math | {
"style": "rule",
"ground_truth": "8634"
} | {
"index": 184,
"solution": "Assume without loss of generality that the side length of $A B C D$ is 1 so that the area of the square is also 1 . This also means that $O M=O N=\\frac{1}{2}$. As $A^{\\prime} B^{\\prime} M C^{\\prime} D^{\\prime} N$ is equiangular, it can be seen that $\\angle A^{\\prime} N O=60^{\\ci... |
DeepScaleR | [
{
"role": "user",
"content": "$O$ is the origin, and $F$ is the focus of the parabola $C:y^{2}=4x$. A line passing through $F$ intersects $C$ at points $A$ and $B$, and $\\overrightarrow{FA}=2\\overrightarrow{BF}$. Find the area of $\\triangle OAB$."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{2}}{2}"
} | {
"index": 185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$P$ is a point inside triangle $A B C$, and lines $A P, B P, C P$ intersect the opposite sides $B C, C A, A B$ in points $D, E, F$, respectively. It is given that $\\angle A P B=90^{\\circ}$, and that $A C=B C$ and $A B=B D$. We also know that $B F=1$, and that $B C=999$. Find ... | math | {
"style": "rule",
"ground_truth": "499 / 500"
} | {
"index": 186,
"solution": "Let $A C=B C=s, A B=B D=t$. Since $B P$ is the altitude in isosceles triangle $A B D$, it bisects angle $B$. So, the Angle Bisector Theorem in triangle $A B C$ given $A E / E C=A B / B C=t / s$. Meanwhile, $C D / D B=(s-t) / t$. Now Ceva's theorem gives us $$ \\begin{gathered} \\frac{A ... |
DeepScaleR | [
{
"role": "user",
"content": "$P$ is a point interior to rectangle $ABCD$ and such that $PA=3$ inches, $PD=4$ inches, and $PC=5$ inches. Then $PB$, in inches, equals:"
}
] | math | {
"style": "rule",
"ground_truth": "$3\\sqrt{2}$"
} | {
"index": 187,
"solution": "1. **Identify the relationships given by the problem:**\n - $PA = 3$ inches, $PD = 4$ inches, $PC = 5$ inches, and we need to find $PB = x$ inches.\n\n2. **Use the perpendicular distances from $P$ to the sides of the rectangle:**\n - Let $a, b, c, d$ be the perpendicular distances f... |
DeepScaleR | [
{
"role": "user",
"content": "$P(x)$ is a polynomial of degree $3n$ such that\n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1) = 0, \\quad\\text{ and }\\\\ && P(3n+1) = 730.\\end{eqnarray*} \nDetermine $n$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ n = 4 \\]"
} | {
"index": 188,
"solution": "By Lagrange Interpolation Formula $f(x) = 2\\sum_{p=0}^{n}\\left ( \\prod_{0\\leq r\\neq3p\\leq 3n}^{{}}\\frac{x-r}{3p-r} \\right )+ \\sum_{p=1}^{n}\\left ( \\prod_{0\\leq r\\neq3p-2\\leq 3n}^{{}} \\frac{x-r}{3p-2-r}\\right )$ \nand hence $f(3n+1) = 2\\sum_{p=0}^{n}\\left ( \\prod_{0\\l... |
DeepScaleR | [
{
"role": "user",
"content": "$P(x)=ax^2+bx+c$ has exactly $1$ different real root where $a,b,c$ are real numbers. If $P(P(P(x)))$ has exactly $3$ different real roots, what is the minimum possible value of $abc$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$P, A, B, C,$ and $D$ are five distinct points in space such that $\\angle APB = \\angle BPC = \\angle CPD = \\angle DPA = \\theta$, where $\\theta$ is a given acute angle. Determine the greatest and least values of $\\angle APC + \\angle BPD$."
}
] | math | {
"style": "rule",
"ground_truth": "0^\\circ \\text{ and } 360^\\circ"
} | {
"index": 190,
"solution": "\nConsider five distinct points \\( P, A, B, C, \\) and \\( D \\) in space where the angles formed at \\( P \\) satisfy \\( \\angle APB = \\angle BPC = \\angle CPD = \\angle DPA = \\theta \\). We are tasked with finding the greatest and least possible values of the sum of angles \\( \\a... |
DeepScaleR | [
{
"role": "user",
"content": "$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"index": 191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. The length of $Q R$ is"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{6}$"
} | {
"index": 192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$R$ varies directly as $S$ and inversely as $T$. When $R = \\frac{4}{3}$ and $T = \\frac{9}{14}$, $S = \\frac{3}{7}$. Find $S$ when $R = \\sqrt{48}$ and $T = \\sqrt{75}$."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 193,
"solution": "1. **Establish the relationship**: Given that $R$ varies directly as $S$ and inversely as $T$, we can express this relationship mathematically as:\n \\[\n R = c \\cdot \\frac{S}{T}\n \\]\n where $c$ is a constant.\n\n2. **Determine the constant $c$**: Using the values $R = \\fra... |
DeepScaleR | [
{
"role": "user",
"content": "$S$ is a set of complex numbers such that if $u, v \\in S$, then $u v \\in S$ and $u^{2}+v^{2} \\in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most 1 is finite. What is the largest possible value of $N$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 194,
"solution": "First, if $S$ contained some $u \\neq 0$ with absolute value $<1$, then (by the first condition) every power of $u$ would be in $S$, and $S$ would contain infinitely many different numbers of absolute value $<1$. This is a contradiction. Now suppose $S$ contains some number $u$ of absol... |
DeepScaleR | [
{
"role": "user",
"content": "$S$ is a subset of the set $\\{1, 2, \\cdots, 2023\\}$, such that the sum of the squares of any two elements is not a multiple of 9. What is the maximum value of $|S|$? (Here, $|S|$ represents the number of elements in $S$.)"
}
] | math | {
"style": "rule",
"ground_truth": "1350"
} | {
"index": 195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$S=\\{1,2, \\ldots, 6\\} .$ Then find out the number of unordered pairs of $(A, B)$ such that $A, B \\subseteq S$ and $A \\cap B=\\phi$"
}
] | math | {
"style": "rule",
"ground_truth": "365"
} | {
"index": 196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "$X, Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets \n$X, Y, Z, X \\cup Y, X \\cup Z$ and $Y \\cup Z$ are $37, 23, 41, 29, 39.5$ and $33$ respectively. \nFind the average age of the people in set $X \\cup Y \\cup Z$."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 197,
"solution": "1. **Define Variables:**\n Let $X$, $Y$, and $Z$ represent the sums of the ages of the people in sets $X$, $Y$, and $Z$ respectively. Let $x$, $y$, and $z$ represent the numbers of people in sets $X$, $Y$, and $Z$ respectively.\n\n2. **Use Given Averages:**\n - The average age of pe... |
DeepScaleR | [
{
"role": "user",
"content": "$[x-(y-z)] - [(x-y) - z] = $"
}
] | math | {
"style": "rule",
"ground_truth": "2z"
} | {
"index": 198,
"solution": "We start by simplifying the given expression step by step:\n\n1. **Expand and simplify the inner parentheses:**\n \\[\n [x-(y-z)] - [(x-y) - z] = [x-y+z] - [x-y+z]\n \\]\n Here, we used the property that subtracting a negative is equivalent to adding, i.e., $-(y-z) = -y+z$ and $... |
DeepScaleR | [
{
"role": "user",
"content": "$\\Delta ABC$ is isosceles with $AC = BC$. If $m\\angle C = 40^{\\circ}$, what is the number of degrees in $m\\angle CBD$? [asy] pair A,B,C,D,E;\nC = dir(65); B = C + dir(-65); D = (1.5,0); E = (2,0);\ndraw(B--C--A--E); dot(D);\nlabel(\"$A$\",A,S); label(\"$B$\",B,S); label(\"$... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 199,
"solution": ""
} |
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