data_source stringclasses 1 value | prompt listlengths 1 1 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|
[
{
"content": "In the figure, $\\overline{DC}$ is parallel to $\\overline{AB}$. We have $\\angle DCA = 40^\\circ$ and $\\angle ABC = 73^\\circ$. Find $\\angle ACB$.\n\n[asy]\npair A,B,C,D;\nA = (0,0);\nB= (1,0);\nD = (-0.1,0.6);\nC = (0.8,0.6);\ndraw((-0.2,0)--(1.2,0));\ndraw((-0.2,0.6)--(1.2,0.6));\ndraw(A--C--B);\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,N);\nlabel(\"$D$\",D,N);\ndot(A);dot(B);dot(C);dot(D);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "67^\\circ",
"style": "rule"
} | {
"index": 900,
"split": "train"
} | |
[
{
"content": "In Idaho, 472 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the ``Pop'' sector of the graph is $251^\\circ$, to the nearest whole degree. How many of the people surveyed chose ``Pop''? Express your answer as a whole number.\n\n[asy]import graph;\nsize(150);\nreal w = 10pt,linew = 1;\nfilldraw(Arc((0,0),1,0,251)--(0,0)--cycle,gray(.4),black+linewidth(linew));\nfilldraw(Arc((0,0),1,251,280)--(0,0)--cycle,white,black+linewidth(linew));\nfilldraw(Arc((0,0),1,280,350)--(0,0)--cycle,gray(.7),black+linewidth(linew));\nfilldraw(Arc((0,0),1,350,360)--(0,0)--cycle,white,black+linewidth(linew));\nlabel(\"\\textbf{POP}\",expi(2.19),expi(2.19),fontsize(w));\nlabel(\"\\textbf{COKE}\",expi(4.63),expi(4.63),fontsize(w));\nlabel(\"\\textbf{SODA}\",expi(5.50),expi(5.50),fontsize(w));\nlabel(\"\\textbf{OTHER}\",expi(6.20),expi(6.20),fontsize(w));[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "329",
"style": "rule"
} | {
"index": 901,
"split": "train"
} | |
[
{
"content": "If Jeff picks one letter randomly from the alphabet, what is the probability that the letter is in the word `PROBABILITY'? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{9}{26}",
"style": "rule"
} | {
"index": 902,
"split": "train"
} | |
[
{
"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 & & & & & &\\\\\n7 &7 & 8 & & & & &\\\\\n8 &2 & 6 & 7 & 9 & 9 & 9 & 9\\\\\n9 &1 & 4 & 4 & 4 & 6 & &\\\\\n10 &0 & 0 & 0 & & & &\\\\\n\\end{tabular} Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "89",
"style": "rule"
} | {
"index": 903,
"split": "train"
} | |
[
{
"content": "What is the smallest four-digit palindrome that is divisible by 4? (A palindrome is a number that reads the same forwards and backwards, like 61216.) Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "2112",
"style": "rule"
} | {
"index": 904,
"split": "train"
} | |
[
{
"content": "A sports conference has 14 teams in two divisions of 7. How many games are in a complete season for the conference if each team must play every other team in its own division twice and every team in the other division once? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "133",
"style": "rule"
} | {
"index": 905,
"split": "train"
} | |
[
{
"content": "In the diagram, $ABCD$ is a square with side length $6,$ and $WXYZ$ is a rectangle with $ZY=10$ and $XY=6.$ Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to half of the area of $WXYZ,$ what is the length of $AP?$\n\n[asy]\ndraw((0,0)--(10,0)--(10,6)--(0,6)--cycle,black+linewidth(1));\ndraw((1.5,1)--(7.5,1)--(7.5,7)--(1.5,7)--cycle,black+linewidth(1));\nfilldraw((1.5,1)--(7.5,1)--(7.5,6)--(1.5,6)--cycle,gray,black+linewidth(1));\nlabel(\"$W$\",(0,6),NW);\nlabel(\"$X$\",(10,6),NE);\nlabel(\"$Y$\",(10,0),SE);\nlabel(\"$Z$\",(0,0),SW);\nlabel(\"$A$\",(1.5,7),NW);\nlabel(\"$B$\",(7.5,7),NE);\nlabel(\"$C$\",(7.5,1),E);\nlabel(\"$D$\",(1.5,1),W);\nlabel(\"$P$\",(1.5,6),SW);\nlabel(\"6\",(1.5,7)--(7.5,7),N);\nlabel(\"6\",(10,0)--(10,6),E);\nlabel(\"10\",(0,0)--(10,0),S);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 906,
"split": "train"
} | |
[
{
"content": "Find all $y \\in [0,\\pi]$ for which \\[\\sin(x+y)\\leq \\sin(x)+\\sin(y)\\]for every $x$ between $0$ and $\\pi$, inclusive. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "[0,\\pi]",
"style": "rule"
} | {
"index": 907,
"split": "train"
} | |
[
{
"content": "There exist two distinct unit vectors $\\mathbf{v}$ such that the angle between $\\mathbf{v}$ and $\\begin{pmatrix} 2 \\\\ 2 \\\\ -1 \\end{pmatrix}$ is $45^\\circ,$ and the angle between $\\mathbf{v}$ and $\\begin{pmatrix} 0 \\\\ 1 \\\\ -1 \\end{pmatrix}$ is $60^\\circ.$ Let $\\mathbf{v}_1$ and $\\mathbf{v}_2$ be these vectors. Find $\\|\\mathbf{v}_1 - \\mathbf{v}_2\\|.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": 908,
"split": "train"
} | |
[
{
"content": "For real numbers $t,$ the point of intersection of the lines $tx - 2y - 3t = 0$ and $x - 2ty + 3 = 0$ is plotted. All the plotted points lie on what kind of curve?\n\n(A) Line\n(B) Circle\n(C) Parabola\n(D) Ellipse\n(E) Hyperbola\n\nEnter the letter of the correct option. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\text{(E)}",
"style": "rule"
} | {
"index": 909,
"split": "train"
} | |
[
{
"content": "In coordinate space, $A = (1,2,3),$ $B = (5,3,1),$ and $C = (3,4,5).$ Find the orthocenter of triangle $ABC.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\left( \\frac{5}{2}, 3, \\frac{7}{2} \\right)",
"style": "rule"
} | {
"index": 910,
"split": "train"
} | |
[
{
"content": "The curve $y = \\sin x$ cuts the line whose equation is $y = \\sin 70^\\circ$ into segments having the successive ratios\n\\[\\dots p : q : p : q \\dots\\]with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p,q).$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "(1,8)",
"style": "rule"
} | {
"index": 911,
"split": "train"
} | |
[
{
"content": "Find the smallest positive solution to\n\\[\\tan 2x + \\tan 3x = \\sec 3x\\]in radians. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\pi}{14}",
"style": "rule"
} | {
"index": 912,
"split": "train"
} | |
[
{
"content": "Find the range of the function\n\\[f(x) = \\left( \\arccos \\frac{x}{2} \\right)^2 + \\pi \\arcsin \\frac{x}{2} - \\left( \\arcsin \\frac{x}{2} \\right)^2 + \\frac{\\pi^2}{12} (x^2 + 6x + 8).\\] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\left[ \\frac{\\pi^2}{4}, \\frac{9 \\pi^2}{4} \\right]",
"style": "rule"
} | {
"index": 913,
"split": "train"
} | |
[
{
"content": "Find the sum of all positive real solutions $x$ to the equation \\[2\\cos2x \\left(\\cos2x - \\cos{\\left( \\frac{2014\\pi^2}{x} \\right) } \\right) = \\cos4x - 1,\\]where $x$ is measured in radians. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "1080 \\pi",
"style": "rule"
} | {
"index": 914,
"split": "train"
} | |
[
{
"content": "Compute $\\tan\\left(\\frac{\\pi}{7}\\right)\\tan\\left(\\frac{2\\pi}{7}\\right)\\tan\\left(\\frac{3\\pi}{7}\\right)$. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": 915,
"split": "train"
} | |
[
{
"content": "The projection of $\\begin{pmatrix} 0 \\\\ 1 \\\\ 4 \\end{pmatrix}$ onto a certain vector $\\mathbf{w}$ is $\\begin{pmatrix} 1 \\\\ -1/2 \\\\ 1/2 \\end{pmatrix}.$ Find the projection of $\\begin{pmatrix} 3 \\\\ 3 \\\\ -2 \\end{pmatrix}$ onto $\\mathbf{w}.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix} 1/3 \\\\ -1/6 \\\\ 1/6 \\end{pmatrix}",
"style": "rule"
} | {
"index": 916,
"split": "train"
} | |
[
{
"content": "Compute\n\\[\\begin{vmatrix} 1 & -3 & 3 \\\\ 0 & 5 & -1 \\\\ 4 & -2 & 1 \\end{vmatrix}.\\] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "-45",
"style": "rule"
} | {
"index": 917,
"split": "train"
} | |
[
{
"content": "Let $x = \\cos \\frac{2 \\pi}{7} + i \\sin \\frac{2 \\pi}{7}.$ Compute the value of\n\\[(2x + x^2)(2x^2 + x^4)(2x^3 + x^6)(2x^4 + x^8)(2x^5 + x^{10})(2x^6 + x^{12}).\\] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "43",
"style": "rule"
} | {
"index": 918,
"split": "train"
} | |
[
{
"content": "In triangle $ABC$, $3 \\sin A + 4 \\cos B = 6$ and $4 \\sin B + 3 \\cos A = 1$. Find all possible values of $\\angle C,$ in degrees. Enter all the possible values, separated by commas. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "30^\\circ",
"style": "rule"
} | {
"index": 919,
"split": "train"
} | |
[
{
"content": "Find the minimum value of\n\\[(\\sin x + \\csc x)^2 + (\\cos x + \\sec x)^2\\]for $0 < x < \\frac{\\pi}{2}.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 920,
"split": "train"
} | |
[
{
"content": "Find the unit vector $\\mathbf{v},$ lying in the $xz$-plane, which makes an angle of $45^\\circ$ with $\\begin{pmatrix} 2 \\\\ 2 \\\\ -1 \\end{pmatrix},$ and an angle of $60^\\circ$ with $\\begin{pmatrix} 0 \\\\ 1 \\\\ - 1 \\end{pmatrix}.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix} \\sqrt{2}/2 \\\\ 0 \\\\ -\\sqrt{2}/2 \\end{pmatrix}",
"style": "rule"
} | {
"index": 921,
"split": "train"
} | |
[
{
"content": "Let $a,$ $b,$ $c$ be integers such that\n\\[\\mathbf{A} = \\frac{1}{5} \\begin{pmatrix} -3 & a \\\\ b & c \\end{pmatrix}\\]and $\\mathbf{A}^2 = \\mathbf{I}.$ Find the largest possible value of $a + b + c.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 922,
"split": "train"
} | |
[
{
"content": "Let $D$ be the determinant of the matrix whose column vectors are $\\mathbf{a},$ $\\mathbf{b},$ and $\\mathbf{c}.$ Find the determinant of the matrix whose column vectors are $\\mathbf{a} + \\mathbf{b},$ $\\mathbf{b} + \\mathbf{c},$ and $\\mathbf{c} + \\mathbf{a},$ in terms of $D.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "2D",
"style": "rule"
} | {
"index": 923,
"split": "train"
} | |
[
{
"content": "A line is parameterized by a parameter $t,$ so that the vector on the line at $t = -1$ is $\\begin{pmatrix} 1 \\\\ 3 \\\\ 8 \\end{pmatrix},$ and the vector on the line at $t = 2$ is $\\begin{pmatrix} 0 \\\\ -2 \\\\ -4 \\end{pmatrix}.$ Find the vector on the line at $t = 3.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix} -1/3 \\\\ -11/3 \\\\ -8 \\end{pmatrix}",
"style": "rule"
} | {
"index": 924,
"split": "train"
} | |
[
{
"content": "Compute\n\\[\\cos^2 0^\\circ + \\cos^2 1^\\circ + \\cos^2 2^\\circ + \\dots + \\cos^2 90^\\circ.\\] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{91}{2}",
"style": "rule"
} | {
"index": 925,
"split": "train"
} | |
[
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1,$ $\\|\\mathbf{c}\\| = 2,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{c}) + \\mathbf{b} = \\mathbf{0}.\\]Find the smallest possible angle between $\\mathbf{a}$ and $\\mathbf{c},$ in degrees. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "30^\\circ",
"style": "rule"
} | {
"index": 926,
"split": "train"
} | |
[
{
"content": "Line segment $\\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$\n\n[asy]\nunitsize(1 cm);\n\npair A, B, P;\n\nA = (0,0);\nB = (5,1);\nP = interp(A,B,10/7);\n\ndraw(A--P);\n\ndot(\"$A$\", A, S);\ndot(\"$B$\", B, S);\ndot(\"$P$\", P, S);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\left( -\\frac{3}{7}, \\frac{10}{7} \\right)",
"style": "rule"
} | {
"index": 927,
"split": "train"
} | |
[
{
"content": "Let\n\\[\\mathbf{A} = \\begin{pmatrix} 4 & 1 \\\\ -9 & -2 \\end{pmatrix}.\\]Compute $\\mathbf{A}^{100}.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix} 301 & 100 \\\\ -900 & -299 \\end{pmatrix}",
"style": "rule"
} | {
"index": 928,
"split": "train"
} | |
[
{
"content": "In triangle $ABC,$ $AB = 20$ and $BC = 15.$ Find the largest possible value of $\\tan A.$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3 \\sqrt{7}}{7}",
"style": "rule"
} | {
"index": 929,
"split": "train"
} | |
[
{
"content": "Given that $(1+\\sin t)(1+\\cos t)=5/4$ and\n$(1-\\sin t)(1-\\cos t)=\\frac mn-\\sqrt{k},$\nwhere $k, m,$ and $n$ are positive integers with $m$ and $n$ relatively prime, find $k+m+n.$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "27",
"style": "rule"
} | {
"index": 930,
"split": "train"
} | |
[
{
"content": "Consider two lines: line $l$ parametrized as\n\\begin{align*} \nx &= 1 + 4t,\\\\\ny &= 4 + 3t\n\\end{align*}and the line $m$ parametrized as\n\\begin{align*} \nx &=-5 + 4s\\\\\ny &= 6 + 3s.\n\\end{align*}Let $A$ be a point on line $l$, $B$ be a point on line $m$, and let $P$ be the foot of the perpendicular from $A$ to line $m$.\n\nThen $\\overrightarrow{PA}$ is the projection of $\\overrightarrow{BA}$ onto some vector $\\begin{pmatrix} v_1\\\\v_2\\end{pmatrix}$ such that $v_1+v_2 = 2$. Find $\\begin{pmatrix}v_1 \\\\ v_2 \\end{pmatrix}$. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix}-6 \\\\ 8 \\end{pmatrix}",
"style": "rule"
} | {
"index": 931,
"split": "train"
} | |
[
{
"content": "Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate\n\\[\\begin{vmatrix} \\sin^2 A & \\cot A & 1 \\\\ \\sin^2 B & \\cot B & 1 \\\\ \\sin^2 C & \\cot C & 1 \\end{vmatrix}.\\] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 932,
"split": "train"
} | |
[
{
"content": "There are three real numbers $x$ that are not in the domain of $$f(x) = \\frac{1}{1+\\frac{1}{1+\\frac 1x}}.$$ What is the sum of those three numbers? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac 32",
"style": "rule"
} | {
"index": 933,
"split": "train"
} | |
[
{
"content": "I had been planning to work for 20 hours a week for 12 weeks this summer to earn $\\$3000$ to buy a used car. Unfortunately, I got sick for the first two weeks of the summer and didn't work any hours. How many hours a week will I have to work for the rest of the summer if I still want to buy the car? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "24",
"style": "rule"
} | {
"index": 934,
"split": "train"
} | |
[
{
"content": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl}\n\\frac{x}{21} & \\text{ if }x\\text{ is a multiple of 3 and 7}, \\\\\n3x & \\text{ if }x\\text{ is only a multiple of 7}, \\\\\n7x & \\text{ if }x\\text{ is only a multiple of 3}, \\\\\nx+3 & \\text{ if }x\\text{ is not a multiple of 3 or 7}.\n\\end{array}\n\\right.\\]If $f^a(x)$ means the function is nested $a$ times (for example, $f^2(x)=f(f(x))$), what is the smallest value of $a$ greater than 1 that satisfies $f(2)=f^a(2)$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 935,
"split": "train"
} | |
[
{
"content": "A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "19",
"style": "rule"
} | {
"index": 936,
"split": "train"
} | |
[
{
"content": "Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + \\left(b + \\frac 1b\\right)x + c = 0$. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 937,
"split": "train"
} | |
[
{
"content": "Consider the lines \\begin{align*}\ny&=3x+5 \\\\ 2y&=4x+5 \\\\ 3y&=9x-2 \\\\ 2y&=x-3 \\\\ 4y&=x-5.\n\\end{align*}Let's say that a pair of lines is $\\emph{good}$ if the two lines are either parallel or perpendicular to each other. Among all the pairs of lines shown, how many pairs are good? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 938,
"split": "train"
} | |
[
{
"content": "Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 9$? Express your answer in terms of $\\pi$. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac94\\pi",
"style": "rule"
} | {
"index": 939,
"split": "train"
} | |
[
{
"content": "Write the following expression as a polynomial: $$(2x^2+3x+7)(x+1)-(x+1)(x^2+4x-63)+(3x-14)(x+1)(x+5).$$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "4x^3+4x^2",
"style": "rule"
} | {
"index": 940,
"split": "train"
} | |
[
{
"content": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt{2} + \\sqrt{3} + \\sqrt{7}}$, and write your answer in the form \\[\n\\frac{A\\sqrt{2} + B\\sqrt{3} + C\\sqrt{7} + D\\sqrt{E}}{F},\n\\]where everything is in simplest radical form and the fraction is in lowest terms, and $F$ is positive. What is $A + B + C + D + E + F$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "57",
"style": "rule"
} | {
"index": 941,
"split": "train"
} | |
[
{
"content": "A certain organization consists of five leaders and some number of regular members. Every year, the current leaders are kicked out of the organization. Next, each regular member must find two new people to join as regular members. Finally, five new people are elected from outside the organization to become leaders. In the beginning, there are fifteen people in the organization total. How many people total will be in the organization five years from now? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "2435",
"style": "rule"
} | {
"index": 942,
"split": "train"
} | |
[
{
"content": "All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "137",
"style": "rule"
} | {
"index": 943,
"split": "train"
} | |
[
{
"content": "The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Three identical point charges start at the vertices of an equilateral triangle, and this configuration stores 15 Joules of energy. How much more energy, in Joules, would be stored if one of these charges was moved to the midpoint of the opposite side? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 944,
"split": "train"
} | |
[
{
"content": "Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "217",
"style": "rule"
} | {
"index": 945,
"split": "train"
} | |
[
{
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": 946,
"split": "train"
} | |
[
{
"content": "For how many integer values of $a$ does the equation $$x^2 + ax + 8a = 0$$ have integer solutions for $x$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 947,
"split": "train"
} | |
[
{
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{18}$, what is the smallest possible value for $x + y$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "75",
"style": "rule"
} | {
"index": 948,
"split": "train"
} | |
[
{
"content": "Compute the domain of the function $$f(x)=\\frac{1}{\\lfloor x^2-7x+13\\rfloor}.$$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "(-\\infty,3] \\cup [4,\\infty)",
"style": "rule"
} | {
"index": 949,
"split": "train"
} | |
[
{
"content": "The first $20$ numbers of an arrangement are shown below. What would be the value of the $40^{\\mathrm{th}}$ number if the arrangement were continued?\n\n$\\bullet$ Row 1: $2,$ $2$\n\n$\\bullet$ Row 2: $4,$ $4,$ $4,$ $4$\n\n$\\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$\n\n$\\bullet$ Row 4: $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8,$ $8$ Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 950,
"split": "train"
} | |
[
{
"content": "There are positive integers that have these properties:\n\n$\\bullet$ I. The sum of the squares of their digits is $50,$ and\n\n$\\bullet$ II. Each digit is larger than the one on its left.\n\nWhat is the product of the digits of the largest integer with both properties? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 951,
"split": "train"
} | |
[
{
"content": "In how many ways can 8 people be seated in a row of chairs if three of the people, John, Wilma, and Paul, refuse to sit in three consecutive seats? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "36000",
"style": "rule"
} | {
"index": 952,
"split": "train"
} | |
[
{
"content": "In how many different ways can 3 men and 4 women be placed into two groups of two people and one group of three people if there must be at least one man and one woman in each group? Note that identically sized groups are indistinguishable. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 953,
"split": "train"
} | |
[
{
"content": "Ryan has 3 red lava lamps and 3 blue lava lamps. He arranges them in a row on a shelf randomly, and then randomly turns 3 of them on. What is the probability that the leftmost lamp is blue and off, and the rightmost lamp is red and on? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{9}{100}",
"style": "rule"
} | {
"index": 954,
"split": "train"
} | |
[
{
"content": "The probability of getting rain on any given day in June in Capital City is $\\frac{1}{10}$. What is the probability that it rains on at most 2 days in June? Express your answer as a decimal to the nearest thousandth. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "0.411",
"style": "rule"
} | {
"index": 955,
"split": "train"
} | |
[
{
"content": "The lottery in our state consists of two drawings. First, a MegaBall is picked from among 27 numbered balls. Second, five WinnerBalls are picked from among 44 numbered balls. To win the lottery, you must pick the MegaBall number correctly and also pick the numbers on the five WinnerBalls (but you don't need to get the order right for the WinnerBalls). What is the probability that the ticket I hold has the winning numbers? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{1}{29,\\!322,\\!216}",
"style": "rule"
} | {
"index": 956,
"split": "train"
} | |
[
{
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that both cards are numbers (2 through 10) totaling to 12? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{35}{663}",
"style": "rule"
} | {
"index": 957,
"split": "train"
} | |
[
{
"content": "In how many ways is it possible to arrange the digits of 1150 to get a four-digit multiple of 5? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 958,
"split": "train"
} | |
[
{
"content": "The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 959,
"split": "train"
} | |
[
{
"content": "Two distinct positive integers from 1 to 50 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 5? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{89}{245}",
"style": "rule"
} | {
"index": 960,
"split": "train"
} | |
[
{
"content": "We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "240",
"style": "rule"
} | {
"index": 961,
"split": "train"
} | |
[
{
"content": "There are two colors Jessica can use to color this 2 by 2 grid. If non-overlapping regions that share a side cannot be the same color, how many possible ways are there for Jessica to color the grid?\n\n[asy]\nsize(101);\ndraw(unitsquare);\ndraw((0,.5)--(1,.5));\ndraw((.5,0)--(.5,1));\nlabel(\"$A$\",(0,1),NW); label(\"$B$\",(1,1),NE); label(\"$C$\",(1,0),SE); label(\"$D$\",(0,0),SW);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 962,
"split": "train"
} | |
[
{
"content": "A gardener plants three maple trees, four oaks, and five birch trees in a row. He plants them in random order, each arrangement being equally likely. Let $\\frac m n$ in lowest terms be the probability that no two birch trees are next to one another. Find $m+n$.\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "106",
"style": "rule"
} | {
"index": 963,
"split": "train"
} | |
[
{
"content": "Nine tiles are numbered $1, 2, 3, \\cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 964,
"split": "train"
} | |
[
{
"content": "Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\le3,$ and $0\\le z\\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "200",
"style": "rule"
} | {
"index": 965,
"split": "train"
} | |
[
{
"content": "A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "441",
"style": "rule"
} | {
"index": 966,
"split": "train"
} | |
[
{
"content": "Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "831",
"style": "rule"
} | {
"index": 967,
"split": "train"
} | |
[
{
"content": "Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \\le x \\le 15$. The probability that $\\lfloor\\sqrt{P(x)}\\rfloor = \\sqrt{P(\\lfloor x \\rfloor)}$ is equal to $\\frac{\\sqrt{a} + \\sqrt{b} + \\sqrt{c} - d}{e}$ , where $a$, $b$, $c$, $d$, and $e$ are positive integers. Find $a + b + c + d + e$.\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "850",
"style": "rule"
} | {
"index": 968,
"split": "train"
} | |
[
{
"content": "A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adjacent?\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "588",
"style": "rule"
} | {
"index": 969,
"split": "train"
} | |
[
{
"content": "A palindrome is an integer that reads the same forward and backward, such as 3663. What percent of the palindromes between 100 and 500 contain at least one 5? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "10\\%",
"style": "rule"
} | {
"index": 970,
"split": "train"
} | |
[
{
"content": "Two cards are dealt at random from a standard deck of 52 cards (13 hearts, 13 clubs, 13 spades, and 13 diamonds). What is the probability that the first card is a 6 and the second card is a Queen? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\dfrac{4}{663}",
"style": "rule"
} | {
"index": 971,
"split": "train"
} | |
[
{
"content": "Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\\{C,H,L,P,R\\},$ the second from $\\{A,I,O\\},$ and the third from $\\{D,M,N,T\\}.$\n\nWhen Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of ADDITIONAL license plates that can be made by adding two letters? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 972,
"split": "train"
} | |
[
{
"content": "Determine the number of ways to arrange the letters of the word ALABAMA. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "210",
"style": "rule"
} | {
"index": 973,
"split": "train"
} | |
[
{
"content": "Given that $\\binom{17}{7}=19448$, $\\binom{17}{8}=24310$ and $\\binom{17}{9}=24310$, calculate $\\binom{19}{9}$. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "92378",
"style": "rule"
} | {
"index": 974,
"split": "train"
} | |
[
{
"content": "How many three-digit numbers satisfy the property that the middle digit is the average of the first and the last digits? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 975,
"split": "train"
} | |
[
{
"content": "A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2, 2, 3, 7, and 9, and the rock's age begins with an odd digit.\n\nHow many possibilities are there for the rock's age? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 976,
"split": "train"
} | |
[
{
"content": "The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \\div k$ is an integer value? Express your answer as a common fraction. Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": 977,
"split": "train"
} | |
[
{
"content": "In a trapezoid $ABCD$ with $AB$ parallel to $CD$, the diagonals $AC$ and $BD$ intersect at $E$. If the area of triangle $ABE$ is 50 square units, and the area of triangle $ADE$ is 20 square units, what is the area of trapezoid $ABCD$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "98",
"style": "rule"
} | {
"index": 978,
"split": "train"
} | |
[
{
"content": "A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles as shown, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC$?\n\n[asy]\nunitsize(0.7cm);\npair A,B,C;\nA=(0,8);\nB=(-2.8,0);\nC=(2.8,0);\ndraw(A--B--C--cycle,linewidth(0.7));\ndraw(Circle((0,2),2),linewidth(0.7));\ndraw(Circle((0,5),1),linewidth(0.7));\ndraw((0,2)--(2,2));\ndraw((0,5)--(1,5));\nlabel(\"2\",(1,2),N);\nlabel(\"1\",(0.5,5),N);\nlabel(\"$A$\",A,N);\nlabel(\"$B$\",B,SW);\nlabel(\"$C$\",C,SE);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "16\\sqrt{2}",
"style": "rule"
} | {
"index": 979,
"split": "train"
} | |
[
{
"content": "Let $P$ be a point inside triangle $ABC$. Let $G_1$, $G_2$, and $G_3$ be the centroids of triangles $PBC$, $PCA$, and $PAB$, respectively. If the area of triangle $ABC$ is 18, then find the area of triangle $G_1 G_2 G_3$.\n\n[asy]\nimport geometry;\n\nunitsize(2 cm);\n\npair A, B, C, P;\npair[] G;\n\nA = (1,3);\nB = (0,0);\nC = (4,0);\nP = (2,1);\nG[1] = (P + B + C)/3;\nG[2] = (P + C + A)/3;\nG[3] = (P + A + B)/3;\n\ndraw(A--B--C--cycle);\ndraw(A--P);\ndraw(B--P);\ndraw(C--P);\ndraw(G[1]--G[2]--G[3]--cycle);\n\nlabel(\"$A$\", A, dir(90));\nlabel(\"$B$\", B, SW);\nlabel(\"$C$\", C, SE);\ndot(\"$G_1$\", G[1], S);\ndot(\"$G_2$\", G[2], SE);\ndot(\"$G_3$\", G[3], NW);\nlabel(\"$P$\", P, S);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "2",
"style": "rule"
} | {
"index": 980,
"split": "train"
} | |
[
{
"content": "Let $\\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 1$ and $BD = 4,$ then what is $CD$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 981,
"split": "train"
} | |
[
{
"content": "What is the degree measure of angle $LOQ$ when polygon $\\allowbreak LMNOPQ$ is a regular hexagon? [asy]\ndraw((-2,0)--(-1,1.73205081)--(1,1.73205081)--(2,0)--(1,-1.73205081)--(-1,-1.73205081)--cycle);\ndraw((-1,-1.73205081)--(1,1.73205081)--(1,-1.73205081)--cycle);\nlabel(\"L\",(-1,-1.73205081),SW);\nlabel(\"M\",(-2,0),W);\nlabel(\"N\",(-1,1.73205081),NW);\nlabel(\"O\",(1,1.73205081),N);\nlabel(\"P\",(2,0),E);\nlabel(\"Q\",(1,-1.73205081),S);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "30^\\circ",
"style": "rule"
} | {
"index": 982,
"split": "train"
} | |
[
{
"content": "In right triangle $ABC$, shown below, $\\cos{C}=\\frac{9\\sqrt{130}}{130}$. Find $AC$.\n\n[asy]\ndraw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));\ndraw(rightanglemark((7,0),(0,0),(0,9),20),black+linewidth(1));\nlabel(\"$A$\",(0,0),W);\nlabel(\"$B$\",(7,0),E);\nlabel(\"$C$\",(0,9),W);\nlabel(\"$\\sqrt{130}$\",(7,0)--(0,9),NE);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 983,
"split": "train"
} | |
[
{
"content": "Medians $\\overline{DP}$ and $\\overline{EQ}$ of $\\triangle DEF$ are perpendicular. If $DP= 18$ and $EQ = 24$, then what is ${DE}$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 984,
"split": "train"
} | |
[
{
"content": "A cube has side length $6$. Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)\n\n[asy]\nimport three;\nreal t=-0.05;\ntriple A,B,C,D,EE,F,G,H;\nA = (0,0,0);\nB = (cos(t),sin(t),0);\nD= (-sin(t),cos(t),0);\nC = B+D;\nEE = (0,0,1);\nF = B+EE;\nG = C + EE;\nH = D + EE;\ndraw(surface(B--EE--G--cycle),rgb(.6,.3,.6),nolight);\ndraw(surface(B--D--G--cycle),rgb(.7,.4,.7),nolight);\ndraw(surface(D--EE--G--cycle),rgb(.8,.5,.8),nolight);\ndraw(B--C--D);\ndraw(EE--F--G--H--EE);\ndraw(B--F);\ndraw(C--G);\ndraw(D--H);\npen pu=rgb(.5,.2,.5)+8; pen bk=black+8;\ndot(B,pu); dot(C,bk); dot(D,pu); dot(EE,pu); dot(F,bk); dot(G,pu); dot(H,bk);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "72",
"style": "rule"
} | {
"index": 985,
"split": "train"
} | |
[
{
"content": "In the diagram below, points $A$, $B$, $C$, and $P$ are situated so that $PA=2$, $PB=3$, $PC=4$, and $BC=5$. What is the maximum possible area of $\\triangle ABC$? [asy]\ndefaultpen(linewidth(0.8)); size(150);\npair B = (0,0), C = (5,0), A = (2,3), P = (2.2,2);\ndraw(A--B--C--cycle^^B--P^^C--P^^A--P);\nlabel(\"$A$\",A,N); label(\"$B$\",B,S); label(\"$C$\",C,S); label(\"$P$\",P,S);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 986,
"split": "train"
} | |
[
{
"content": "Polygon $ABCDEF$ is a regular hexagon. What is the measure in degrees of angle $ABF$? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "30",
"style": "rule"
} | {
"index": 987,
"split": "train"
} | |
[
{
"content": "Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 2$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the nearest tenth.\n\n[asy]\npath a=(7,13)..(0,0)--(20,20)..cycle;\npath b=(13,7)..(0,0)--(20,20)..cycle;\ndraw(a);\ndraw(b);\ndot((8.6,3.5));\nlabel(\"F\",(8.6,3.5),SE);\nlabel(\"E\",(11.4,16.5),NW);\ndot((11.4,16.5));\ndraw((0,0)--(20,0)--(20,20)--(0,20)--cycle);\nlabel(\"$A$\",(0,0),SW);\nlabel(\"$B$\",(0,20),NW);\nlabel(\"$C$\",(20,20),NE);\nlabel(\"$D$\",(20,0),SE);\nlabel(\"I\",(2,19),S);\nlabel(\"II\",(9,13),S);\nlabel(\"III\",(11,7),N);\nlabel(\"IV\",(18,1),N);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "2.3",
"style": "rule"
} | {
"index": 988,
"split": "train"
} | |
[
{
"content": "Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures 20 degrees. What is number of degrees in the measure of the largest interior angle of triangle $ABC$?\n\n[asy]\ndraw((-20,0)--(0,8)--(20,0)--cycle);\nlabel(\"$20^{\\circ}$\",(-13,-0.7),NE);\nlabel(\"$A$\",(-20,0),W);\nlabel(\"$B$\",(0,8),N);\nlabel(\"$C$\",(20,0),E);\n\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "140",
"style": "rule"
} | {
"index": 989,
"split": "train"
} | |
[
{
"content": "A right square pyramid with base edges of length $8\\sqrt{2}$ units each and slant edges of length 10 units each is cut by a plane that is parallel to its base and 3 units above its base. What is the volume, in cubic units, of the new pyramid that is cut off by this plane? [asy]\nimport three;\nsize(2.5inch);\ncurrentprojection = orthographic(1/2,-1,1/4);\ntriple A = (0,0,6);\ntriple[] base = new triple[4];\nbase[0] = (-4, -4, 0);\n\nbase[1] = (4, -4, 0);\nbase[2] = (4, 4, 0);\nbase[3] = (-4, 4, 0);\ntriple[] mid = new triple[4];\nfor(int i=0; i < 4; ++i)\nmid[i] = (.6*xpart(base[i]) + .4*xpart(A), .6*ypart(base[i]) + .4*ypart(A), .6*zpart(base[i]) + .4*zpart(A));\nfor(int i=0; i < 4; ++i)\n{\ndraw(A--base[i]);\ndraw(base[i]--base[(i+1)%4]);\ndraw(mid[i]--mid[(i+1)%4], dashed);\n}\nlabel(\"$8\\sqrt{2}$ units\", base[0]--base[1]);\nlabel(\"10 units\", base[0]--A, 2*W);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 990,
"split": "train"
} | |
[
{
"content": "Find $AX$ in the diagram if $CX$ bisects $\\angle ACB$.\n\n[asy]\nimport markers;\n\nreal t=.56;\npair A=(0,0);\npair B=(3,2);\npair C=(.5,1.5);\npair X=t*A+(1-t)*B;\n\ndraw(C--A--B--C--X);\n\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,E);\nlabel(\"$C$\",C,N);\nlabel(\"$X$\",X,SE);\n\n//markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));\n//markangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));\n\nlabel(\"$28$\",.5*(B+X),SE);\nlabel(\"$30$\",.5*(B+C),N);\nlabel(\"$21$\",.5*(A+C),NW);\n\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{98}5",
"style": "rule"
} | {
"index": 991,
"split": "train"
} | |
[
{
"content": "In triangle $ABC,\\,$ angle $C$ is a right angle and the altitude from $C\\,$ meets $\\overline{AB}\\,$ at $D.\\,$ The lengths of the sides of $\\triangle ABC\\,$ are integers, $BD=29^3,\\,$ and $\\cos B=m/n\\,$, where $m\\,$ and $n\\,$ are relatively prime positive integers. Find $m+n.\\,$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "450",
"style": "rule"
} | {
"index": 992,
"split": "train"
} | |
[
{
"content": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100$.\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "181",
"style": "rule"
} | {
"index": 993,
"split": "train"
} | |
[
{
"content": "In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "28",
"style": "rule"
} | {
"index": 994,
"split": "train"
} | |
[
{
"content": "Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$\n Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 995,
"split": "train"
} | |
[
{
"content": "In $\\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC =1:2$. Point $F$ is on $AD$ such that $AF:FD=3:1$. If the area of $\\triangle DEF$ is 17, determine the area of $\\triangle ABC$. [asy]\nsize(6cm);defaultpen(fontsize(11));\npair b =(0,0);pair c = (10, 0);pair a=(4, 6);\npair d= midpoint(b--c);\npair e = a + (1/3) * (c - a);\npair f = d + 0.25 * (a - d);\ndraw(a--b--c--cycle--d--e--f);\n\nlabel(\"$A$\", a, N);\nlabel(\"$B$\", b, SW);\nlabel(\"$C$\", c, SE);\nlabel(\"$D$\", d, S);\nlabel(\"$E$\", e, NE);\nlabel(\"$F$\", f, SW);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "408",
"style": "rule"
} | {
"index": 996,
"split": "train"
} | |
[
{
"content": "Let $\\triangle ABC$ be an isosceles triangle such that $BC = 30$ and $AB = AC.$ We have that $I$ is the incenter of $\\triangle ABC,$ and $IC = 18.$ What is the length of the inradius of the triangle? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "3\\sqrt{11}",
"style": "rule"
} | {
"index": 997,
"split": "train"
} | |
[
{
"content": "In the figure, triangles $ABC$ and $BCD$ are equilateral triangles. What is the value of $AD \\div BC$ when expressed in simplest radical form?\n\n[asy]\ndraw((0,0)--(5,8.7)--(10,0)--cycle);\ndraw((10,0)--(15,8.7)--(5,8.7));\nlabel(\"$A$\",(0,0),SW);\nlabel(\"$B$\",(5,8.7),N);\nlabel(\"$C$\",(10,0),SE);\nlabel(\"$D$\",(15,8.7),NE);\n[/asy] Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": 998,
"split": "train"
} | |
[
{
"content": "Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area? Please reason step by step, and put your final answer within \\boxed{}.\n",
"role": "user"
}
] | math | {
"ground_truth": "25\\sqrt{3}",
"style": "rule"
} | {
"index": 999,
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.