data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Transform the following expression into a product: 447. \\(\\sin 75^{\\circ} + \\sin 15^{\\circ}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 35900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Translate the function $f(x) = \\sin 2x + \\sqrt{3}\\cos 2x$ to the left by $\\varphi$ ($\\varphi > 0$) units. If the resulting graph is symmetric about the y-axis, then the minimum value of $\\varphi$ is \\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"index": 35901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Translate the function $y=\\sqrt{3}\\cos x+\\sin x$ $(x\\in\\mathbb{R})$ to the left by $m$ $(m > 0)$ units, and the resulting graph is symmetric about the $y$-axis. Find the minimum value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 35902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Translate the graph of $y= \\sqrt {2}\\sin (2x+ \\frac {\\pi}{3})$ to the right by $\\phi(0 < \\phi < \\pi)$ units to obtain the graph of the function $y=2\\sin x(\\sin x-\\cos x)-1$. Then, $\\phi=$ ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {13\\pi}{24}"
} | {
"index": 35903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Translate the graph of $y= \\sqrt{2}\\sin(2x+ \\frac{\\pi}{3})$ to the right by $\\varphi$ ($0<\\varphi<\\pi$) units to obtain the graph of the function $y=2\\sin x(\\sin x-\\cos x)-1$. Find the value of $\\varphi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13\\pi}{24}"
} | {
"index": 35904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Translate the graph of the function $f(x)=\\sin(2x+\\varphi)$ ($|\\varphi| < \\frac{\\pi}{2}$) to the left by $\\frac{\\pi}{6}$ units. If the resulting graph is symmetric about the origin, determine the minimum value of the function $f(x)$ on the interval $\\left[0, \\frac{\\pi... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"index": 35905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Translate the graph of the function y = 2sin( $$\\frac {π}{3}$$ - x) - cos( $$\\frac {π}{6}$$ + x) by shifting it to the right by $$\\frac {π}{4}$$ units. Determine the minimum value of the corresponding function."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 35906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Transport Teams A and B need to deliver a batch of relief supplies to an earthquake-stricken area. Team A can transport 64.4 tons per day, which is 75% more than Team B can transport per day. If both teams transport the supplies simultaneously, when Team A has transported half ... | math | {
"style": "rule",
"ground_truth": "644"
} | {
"index": 35907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $A B C D$ is inscribed in the parabola $y=x^{2}$ such that $A=\\left(a, a^{2}\\right), B=\\left(b, b^{2}\\right)$, $C=\\left(-b, b^{2}\\right)$, and $D=\\left(-a, a^{2}\\right)$ for some positive reals $a, b$ with $a>b$. If $A D+B C=A B+C D$, and $A B=\\frac{3}{4}$, w... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{40}"
} | {
"index": 35908,
"solution": "Let $t=\\frac{3}{4}$, so $2 a+2 b=2 \\sqrt{(a-b)^{2}+\\left(a^{2}-b^{2}\\right)^{2}}=2 t$ gives $a=b=t$ and $t^{2}=(a-b)^{2}\\left[1+(a+b)^{2}\\right]=(a-b)^{2}\\left[1+t^{2}\\right]$. Thus $a=\\frac{t+\\frac{t}{\\sqrt{1+t^{2}}}}{2}=\\frac{\\frac{3}{4}+\\frac{3}{5}}{2}=\\frac{27}{40}$... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $A B C D$, with bases $A B$ and $C D$, has side lengths $A B=28, B C=13, C D=14$, and $D A=15$. Let diagonals $A C$ and $B D$ intersect at $P$, and let $E$ and $F$ be the midpoints of $A P$ and $B P$, respectively. Find the area of quadrilateral $C D E F$."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 35909,
"solution": "Note that $E F$ is a midline of triangle $A P B$, so $E F$ is parallel to $A B$ and $E F=\\frac{1}{2} A B=14=C D$. We also have that $E F$ is parallel to $C D$, and so $C D E F$ is a parallelogram. From this, we have $E P=P C$ as well, so $\\frac{C E}{C A}=\\frac{2}{3}$. It follows th... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\\angle DBA = 23^{\\circ}$, and $\\angle BDC = 46^{\\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 35910,
"solution": "\n#### Step 1: Extend Lines and Identify Angles\nExtend $\\overline{AB}$ and $\\overline{DC}$ to meet at point $E$. Since $\\angle DBA = 23^\\circ$ and $\\angle BDC = 46^\\circ$, we can calculate $\\angle EDB$ and $\\angle DBE$:\n- $\\angle EDB = 180^\\circ - \\angle BDC = 180^\\circ ... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}, BC=CD=43$, and $\\overline{AD} \\perp \\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the le... | math | {
"style": "rule",
"ground_truth": "194"
} | {
"index": 35911,
"solution": "1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\\overline{AB}\\parallel\\overline{CD}$ and $BC=CD=43$. Also, $\\overline{AD}\\perp\\overline{BD}$, which implies that $\\triangle ABD$ is a right triangle.\n\n2. **Examine the diagonals and inters... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD}, BC=CD=43$, and $\\overline{AD}\\perp\\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length... | math | {
"style": "rule",
"ground_truth": "194"
} | {
"index": 35912,
"solution": "1. **Identify the properties of the trapezoid**: Given that $ABCD$ is a trapezoid with $\\overline{AB}\\parallel\\overline{CD}$ and $BC=CD=43$. Also, $\\overline{AD}\\perp\\overline{BD}$, which implies that $\\triangle ABD$ is a right triangle.\n\n2. **Intersection and midpoint proper... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has base $AB = 20$ units and base $CD = 30$ units. Diagonals $AC$ and $BD$ intersect at $X$. If the area of trapezoid $ABCD$ is $300$ square units, what is the area of triangle $BXC$?"
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 35913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has parallel sides $\\overline{AB}$ of length $33$ and $\\overline {CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "$25$"
} | {
"index": 35914,
"solution": "1. **Identify the right triangles and set up equations using the Pythagorean theorem:**\n - Let $\\overline{DE} \\perp \\overline{AB}$ and $\\overline{FC} \\perp \\overline{AB}$.\n - Denote $\\overline{AE} = x$ and $\\overline{DE} = h$. In right triangle $AED$, we have:\n \\[\... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "164"
} | {
"index": 35915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$, $BC=50^{}_{}$, $CD=19^{}_{}$, and $AD=70^{}_{}$, with $AB^{}_{}$ parallel to $CD^{}_{}$. A circle with center $P^{}_{}$ on $AB^{}_{}$ is drawn tangent to $BC^{}_{}$ and $AD^{}_{}$. Given that $AP^{}_{}=\\frac mn$, where $m^{}_{}$ ... | math | {
"style": "rule",
"ground_truth": "164"
} | {
"index": 35916,
"solution": "From $(1)$ above, $x = \\frac{70r}{h}$ and $92-x = \\frac{50r}{h}$. Adding these equations yields $92 = \\frac{120r}{h}$. Thus, $x = \\frac{70r}{h} = \\frac{7}{12}\\cdot\\frac{120r}{h} = \\frac{7}{12}\\cdot92 = \\frac{161}{3}$, and $m+n = \\boxed{164}$.\n We can use $(1)$ from Solutio... |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has base $EF = 15$ units and base $GH = 25$ units. Diagonals $EG$ and $FH$ intersect at $Y$. If the area of trapezoid $EFGH$ is $200$ square units, what is the area of triangle $FYG$?"
}
] | math | {
"style": "rule",
"ground_truth": "46.875"
} | {
"index": 35917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has base $EF = 24$ units and base $GH = 36$ units. Diagonals $EG$ and $FH$ intersect at point $Y$. If the area of trapezoid $EFGH$ is $360$ square units, what is the area of triangle $FYH$?"
}
] | math | {
"style": "rule",
"ground_truth": "86.4"
} | {
"index": 35918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has sides $EF=100$, $FG=55$, $GH=23$, and $HE=75$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the length of $EQ$ if it is expressed as a fraction $\\frac{p}{q}$, where $p$ and $q$ are relatively prime in... | math | {
"style": "rule",
"ground_truth": "\\frac{750}{13}"
} | {
"index": 35919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has sides $EF=105$, $FG=45$, $GH=21$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find the exact length of $EQ$ using fractions."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{336}{5}"
} | {
"index": 35920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $EFGH$ has sides $EF=86$, $FG=60$, $GH=26$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find $EQ$, expressed as a fraction $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and co... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"index": 35921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid $PQRS$ has vertices $P(-3, 3)$, $Q(3, 3)$, $R(5, -1)$, and $S(-5, -1)$. If a point is selected at random from the region determined by the trapezoid, what is the probability that the point is above the $x$-axis?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 35922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trapezoid \\(ABCD\\) is inscribed in the parabola \\(y = x^2\\) such that \\(A = (a, a^2)\\), \\(B = (b, b^2)\\), \\(C = (-b, b^2)\\), and \\(D = (-a, a^2)\\) for some positive reals \\(a, b\\) with \\(a > b\\). If \\(AD + BC = AB + CD\\), and \\(AB = \\frac{3}{4}\\), what is \... | math | {
"style": "rule",
"ground_truth": "\\frac{27}{40}"
} | {
"index": 35923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Travel along the alley clockwise.\n\nIn 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \\pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.\n\nIn the first case, when moving along the alley, the... | math | {
"style": "rule",
"ground_truth": "0.21"
} | {
"index": 35924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, ... | math | {
"style": "rule",
"ground_truth": "5979"
} | {
"index": 35925,
"solution": "1. **Identify the pattern of numbers said by Tadd**: Tadd's turns involve saying an increasing number of consecutive numbers. The first few blocks of numbers Tadd says are:\n - 1st block: [1]\n - 2nd block: [7-10] (4 numbers)\n - 3rd block: [22-28] (7 numbers)\n - 4th block: [... |
DeepScaleR | [
{
"role": "user",
"content": "Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started?"
}
] | math | {
"style": "rule",
"ground_truth": "7/27"
} | {
"index": 35926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trevor and Edward play a game in which they take turns adding or removing beans from a pile. On each turn, a player must either add or remove the largest perfect square number of beans that is in the heap. The player who empties the pile wins. For example, if Trevor goes first ... | math | {
"style": "rule",
"ground_truth": "0, 5, 20, 29, 45, 80, 101, 116, 135, 145, 165, 173, 236, 257, 397, 404, 445, 477, 540, 565, 580, 629, 666, 836, 845, 885, 909, 944, 949, 954, 975"
} | {
"index": 35927,
"solution": "The correct answers are 0 (worth imaginary points), 5 (worth 0 points), 20 (4 points), 29, 45 (5 points), 80 (6 points), 101, 116, 135, 145, 165, 173 (7 points), 236, 257 (8 points), 397, 404, 445, 477, 540, 565, 580, 629, 666 (9 points), 836, 845, 885, 909, 944, 949, 954, 975 (10 poi... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB = 2$ , $BC = 3$ , $CA = 4$ , and circumcenter $O$ . If the sum of the areas of triangles $AOB$ , $BOC$ , and $COA$ is $\\tfrac{a\\sqrt{b}}{c}$ for positive integers $a$ , $b$ , $c$ , where $\\gcd(a, c) = 1$ and $b$ is not divisible by t... | math | {
"style": "rule",
"ground_truth": "152"
} | {
"index": 35928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=25$ , $AC=29$ , and $BC=36$ . Additionally, $\\Omega$ and $\\omega$ are the circumcircle and incircle of $\\triangle ABC$ . Point $D$ is situated on $\\Omega$ such that $AD$ is a diameter of $\\Omega$ , and line $AD$ intersects $\\om... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 35929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $\\angle BAC=90^\\circ$ . A semicircle with diameter $XY$ is inscribed inside $\\triangle ABC$ such that it is tangent to a point $D$ on side $BC$ , with $X$ on $AB$ and $Y$ on $AC$ . Let $O$ be the midpoint of $XY$ . Given that $AB=3$ , ... | math | {
"style": "rule",
"ground_truth": "39/32"
} | {
"index": 35930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $\\angle{A}=90^{\\circ}$ , $AB=2$ , and $AC=4$ . Circle $\\omega_1$ has center $C$ and radius $CA$ , while circle $\\omega_2$ has center $B$ and radius $BA$ . The two circles intersect at $E$ , different from point $A$ . Point $M$ is on $\... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has a right angle at $C$ , and $D$ is the foot of the altitude from $C$ to $AB$ . Points $L, M,$ and $N$ are the midpoints of segments $AD, DC,$ and $CA,$ respectively. If $CL = 7$ and $BM = 12,$ compute $BN^2$ ."
}
] | math | {
"style": "rule",
"ground_truth": "193"
} | {
"index": 35932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has sidelengths $AB=1$ , $BC=\\sqrt{3}$ , and $AC=2$ . Points $D,E$ , and $F$ are chosen on $AB, BC$ , and $AC$ respectively, such that $\\angle EDF = \\angle DFA = 90^{\\circ}$ . Given that the maximum possible value of $[DEF]^2$ can be expressed a... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 35933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is equilateral with side length $12$ . Point $D$ is the midpoint of side $\\overline{BC}$ . Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$ . Point $E$ lies on segment $\\overline{AD}$ and circle $E$ is tangent to circles ... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} - 6"
} | {
"index": 35934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AB=AC$ . The bisectors of angles $ABC$ and $ACB$ meet at $I$ . If the measure of angle $CIA$ is $130^\\circ$ , compute the measure of angle $CAB$ .\n\n*Proposed by Connor Gordon*"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 35935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is right angled at $A$ . The circle with center $A$ and radius $AB$ cuts $BC$ and $AC$ internally at $D$ and $E$ respectively. If $BD = 20$ and $DC = 16$ , determine $AC^2$ ."
}
] | math | {
"style": "rule",
"ground_truth": "936"
} | {
"index": 35936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\vartriangle ABC$ has circumcenter $O$ and orthocenter $H$ . Let $D$ be the foot of the altitude from $A$ to $BC$ , and suppose $AD = 12$ . If $BD = \\frac14 BC$ and $OH \\parallel BC$ , compute $AB^2$ .\n."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 35937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has $A B=1, B C=\\sqrt{7}$, and $C A=\\sqrt{3}$. Let $\\ell_{1}$ be the line through $A$ perpendicular to $A B, \\ell_{2}$ the line through $B$ perpendicular to $A C$, and $P$ the point of intersection of $\\ell_{1}$ and $\\ell_{2}$. Find $P C$."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 35938,
"solution": "By the Law of Cosines, $\\angle B A C=\\cos ^{-1} \\frac{3+1-7}{2 \\sqrt{3}}=\\cos ^{-1}\\left(-\\frac{\\sqrt{3}}{2}\\right)=150^{\\circ}$. If we let $Q$ be the intersection of $\\ell_{2}$ and $A C$, we notice that $\\angle Q B A=90^{\\circ}-\\angle Q A B=90^{\\circ}-30^{\\circ}=60^{\... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has $A B=10, B C=17$, and $C A=21$. Point $P$ lies on the circle with diameter $A B$. What is the greatest possible area of $A P C$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{189}{2}"
} | {
"index": 35939,
"solution": "To maximize $[A P C]$, point $P$ should be the farthest point on the circle from $A C$. Let $M$ be the midpoint of $A B$ and $Q$ be the projection of $M$ onto $A C$. Then $P Q=P M+M Q=\\frac{1}{2} A B+\\frac{1}{2} h_{B}$, where $h_{B}$ is the length of the altitude from $B$ to $A C$. ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has $A B=4, B C=3$, and a right angle at $B$. Circles $\\omega_{1}$ and $\\omega_{2}$ of equal radii are drawn such that $\\omega_{1}$ is tangent to $A B$ and $A C, \\omega_{2}$ is tangent to $B C$ and $A C$, and $\\omega_{1}$ is tangent to $\\omega_{2}$. Find ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 35940,
"solution": "Denote by $r$ the common radius of $\\omega_{1}, \\omega_{2}$, and let $O_{1}, O_{2}$ be the centers of $\\omega_{1}$ and $\\omega_{2}$ respectively. Suppose $\\omega_{i}$ hits $A C$ at $B_{i}$ for $i=1,2$, so that $O_{1} O_{2}=B_{1} B_{2}=2 r$. Extend angle bisector $A O_{1}$ to hit ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has $A B=4, B C=5$, and $C A=6$. Points $A^{\\prime}, B^{\\prime}, C^{\\prime}$ are such that $B^{\\prime} C^{\\prime}$ is tangent to the circumcircle of $\\triangle A B C$ at $A, C^{\\prime} A^{\\prime}$ is tangent to the circumcircle at $B$, and $A^{\\prime} ... | math | {
"style": "rule",
"ground_truth": "\\frac{80}{3}"
} | {
"index": 35941,
"solution": "Note that by equal tangents, $B^{\\prime} A=B^{\\prime} C, C^{\\prime} A=C^{\\prime} B$, and $A^{\\prime} B=A^{\\prime} C$. Moreover, since the line segments $A^{\\prime} B^{\\prime}, B^{\\prime} C^{\\prime}$, and $C^{\\prime} A^{\\prime}$ are tangent to the circumcircle of $A B C$ at... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has incircle $\\omega$ which touches $A B$ at $C_{1}, B C$ at $A_{1}$, and $C A$ at $B_{1}$. Let $A_{2}$ be the reflection of $A_{1}$ over the midpoint of $B C$, and define $B_{2}$ and $C_{2}$ similarly. Let $A_{3}$ be the intersection of $A A_{2}$ with $\\omeg... | math | {
"style": "rule",
"ground_truth": "14/65"
} | {
"index": 35942,
"solution": "Notice that $A_{2}$ is the point of tangency of the excircle opposite $A$ to $B C$. Therefore, by considering the homothety centered at $A$ taking the excircle to the incircle, we notice that $A_{3}$ is the intersection of $\\omega$ and the tangent line parallel to $B C$. It follows t... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has perimeter 1. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\\min (A B, B C, C A)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{3-\\sqrt{5}}{4}, \\frac{1}{3}\\right]"
} | {
"index": 35943,
"solution": "Let $a, b, c$ denote the side lengths $B C, C A$, and $A B$, respectively. Without loss of generality, assume $a \\leq b \\leq c$; we are looking for the possible range of $a$. First, note that the maximum possible value of $a$ is $\\frac{1}{3}$, which occurs when $A B C$ is equilater... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ has side lengths $A B=15, B C=18, C A=20$. Extend $C A$ and $C B$ to points $D$ and $E$ respectively such that $D A=A B=B E$. Line $A B$ intersects the circumcircle of $C D E$ at $P$ and $Q$. Find the length of $P Q$."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 35944,
"solution": "WLOG suppose that $P$ is closer to $A$ than to $B$. Let $D A=A B=B E=c=15, B C=a=18, C A=$ $b=20, P A=x$, and $Q B=y$. By Power of a Point on $B$ and $A$, we get $a c=(x+c) y$ and $b c=(y+c) x$, respectively. Subtracting the two equations gives $c y-c x=a c-b c \\Rightarrow y-x=a-b$. ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ is given in the plane. Let $A D$ be the angle bisector of $\\angle B A C$; let $B E$ be the altitude from $B$ to $A D$, and let $F$ be the midpoint of $A B$. Given that $A B=28, B C=33, C A=37$, what is the length of $E F$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 35945,
"solution": "$14 \\triangle A B E$ is a right triangle, and $F$ is the midpoint of the hypotenuse (and therefore the circumcenter), so $E F=B F=A F=14$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ is given with $A B=13, B C=14, C A=15$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively. Let $G$ be the foot of the altitude from $A$ in triangle $A F E$. Find $A G$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{396}{65}"
} | {
"index": 35946,
"solution": "By Heron's formula we have $[A B C]=\\sqrt{21(8)(7)(6)}=84$. Let $D$ be the foot of the altitude from $A$ to $B C$; then $A D=2 \\cdot \\frac{84}{14}=12$. Notice that because $\\angle B F C=\\angle B E C, B F E C$ is cyclic, so $\\angle A F E=90-\\angle E F C=90-\\angle E B C=\\angle ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ obeys $A B=2 A C$ and $\\angle B A C=120^{\\circ}$. Points $P$ and $Q$ lie on segment $B C$ such that $$\\begin{aligned} A B^{2}+B C \\cdot C P & =B C^{2} \\\\ 3 A C^{2}+2 B C \\cdot C Q & =B C^{2} \\end{aligned}$$ Find $\\angle P A Q$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "40^{\\circ}"
} | {
"index": 35947,
"solution": "We have $A B^{2}=B C(B C-C P)=B C \\cdot B P$, so triangle $A B C$ is similar to triangle $P B A$. Also, $A B^{2}=B C(B C-2 C Q)+A C^{2}=(B C-C Q)^{2}-C Q^{2}+A C^{2}$, which rewrites as $A B^{2}+C Q^{2}=$ $B Q^{2}+A C^{2}$. We deduce that $Q$ is the foot of the altitude from $A$. Thu... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $A B C$ satisfies $\\angle B>\\angle C$. Let $M$ be the midpoint of $B C$, and let the perpendicular bisector of $B C$ meet the circumcircle of $\\triangle A B C$ at a point $D$ such that points $A, D, C$, and $B$ appear on the circle in that order. Given that $\\angle... | math | {
"style": "rule",
"ground_truth": "86^{\\circ}"
} | {
"index": 35948,
"solution": "Extend $D M$ to hit the circumcircle at $E$. Then, note that since $A D E B$ is a cyclic quadrilateral, $\\angle A B E=180^{\\circ}-\\angle A D E=180^{\\circ}-\\angle A D M=180^{\\circ}-68^{\\circ}=112^{\\circ}$. We also have that $\\angle M E C=\\angle D E C=\\angle D A C=64^{\\circ}... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ and point $P$ in the same plane are given. Point $P$ is equidistant from $A$ and $B$, angle $APB$ is twice angle $ACB$, and $\\overline{AC}$ intersects $\\overline{BP}$ at point $D$. If $PB = 3$ and $PD= 2$, then $AD\\cdot CD =$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 35949,
"solution": "1. **Identify the Circle and Key Points**: Since $P$ is equidistant from $A$ and $B$, $P$ lies on the perpendicular bisector of $\\overline{AB}$. Given that $\\angle APB = 2\\angle ACB$, and $P$ is equidistant from $A$ and $B$, we can infer that $A$, $B$, and $C$ lie on a circle cente... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB = 13, BC = 14$, and $AC = 15$. The points $D, E$, and $F$ are the midpoints of $\\overline{AB}, \\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\neq E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. What is ... | math | {
"style": "rule",
"ground_truth": "\\frac{195}{8}"
} | {
"index": 35950,
"solution": "1. **Identify the Midpoints and Congruent Triangles**:\n - Given $D, E, F$ are midpoints of $\\overline{AB}, \\overline{BC}, \\overline{AC}$ respectively.\n - The segments $AD = DB = \\frac{13}{2}$, $BE = EC = 7$, and $CF = FA = \\frac{15}{2}$.\n\n2. **Congruence of Triangles**:\n... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB = 15, BC = 16$, and $AC = 17$. The points $D, E$, and $F$ are the midpoints of $\\overline{AB}, \\overline{BC}$, and $\\overline{AC}$ respectively. Let $X \\neq E$ be the intersection of the circumcircles of $\\triangle BDE$ and $\\triangle CEF$. Determin... | math | {
"style": "rule",
"ground_truth": "\\frac{960 \\sqrt{39}}{14}"
} | {
"index": 35951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$"
}
] | math | {
"style": "rule",
"ground_truth": "12\\sqrt2"
} | {
"index": 35952,
"solution": "1. **Assign Coordinates to Points**:\n - Place $A$, $B$, and $C$ on the Cartesian plane such that $B$ is at the origin $(0,0)$.\n - Using the distance formula and the given side lengths, we find coordinates for $A$ and $C$:\n - $AB = 13$, $BC = 14$, and $AC = 15$.\n - By p... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$?... | math | {
"style": "rule",
"ground_truth": "90^{\\circ}"
} | {
"index": 35953,
"solution": "1. **Given Information and Setup**: \n - Triangle $ABC$ with $AB = 2 \\cdot AC$.\n - Points $D$ and $E$ are on $\\overline{AB}$ and $\\overline{BC}$ respectively, such that $\\angle BAE = \\angle ACD = x$.\n - $F$ is the intersection of segments $AE$ and $CD$.\n - $\\triangle ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=21$, $AC=22$ and $BC=20$. Points $D$ and $E$ are located on $\\overline{AB}$ and $\\overline{AC}$, respectively, such that $\\overline{DE}$ is parallel to $\\overline{BC}$ and contains the center of the inscribed circle of triangle $ABC$. Then $DE=m/n$, w... | math | {
"style": "rule",
"ground_truth": "923"
} | {
"index": 35954,
"solution": "More directly than Solution 2, we have \\[DE=BC\\left(\\frac{h_a-r}{h_a}\\right)=20\\left(1-\\frac{r}{\\frac{[ABC]}{\\frac{BC}{2}}}\\right)=20\\left(1-\\frac{10r}{sr}\\right)=20\\left(1-\\frac{10}{\\frac{63}{2}}\\right)=\\frac{860}{63}\\implies \\boxed{923}.\\]"
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\\triangle ABC$. What is $BI$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35955,
"solution": "1. **Identify the triangle and its sides**: We are given a triangle $ABC$ with sides $AB = 27$, $AC = 26$, and $BC = 25$.\n\n2. **Incircle and angle bisectors**: Let $I$ be the incenter of $\\triangle ABC$, which is the point of intersection of the internal angle bisectors. The incirc... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=40,AC=31,$ and $\\sin{A}=\\frac{1}{5}$. This triangle is inscribed in rectangle $AQRS$ with $B$ on $\\overline{QR}$ and $C$ on $\\overline{RS}$. Find the maximum possible area of $AQRS$."
}
] | math | {
"style": "rule",
"ground_truth": "744"
} | {
"index": 35956,
"solution": "Note that if angle $BAC$ is obtuse, it would be impossible for the triangle to inscribed in a rectangle. This can easily be shown by drawing triangle ABC, where $A$ is obtuse. Therefore, angle A is acute. Let angle $CAS=n$ and angle $BAQ=m$. Then, $\\overline{AS}=31\\cos(n)$ and $\\ov... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?"
}
] | math | {
"style": "rule",
"ground_truth": "820"
} | {
"index": 35957,
"solution": "We can start how we did above in solution 4 to get $\\frac{9}{4} * \\sqrt{(81x^2-1)(81-x^2)}$. Then, we can notice the inside is a quadratic in terms of $x^2$, which is $-81(x^2)^2+6562x^2-81$. This is maximized when $x^2 = \\frac{3281}{81}$.If we plug it into the equation, we get $\\... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC = 450$ and $BC = 300$ . Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$ , and $\\overline{CL}$ is the angle bisector of angle $C$ . Let $P$ be the point of intersection of $\\overline{BK}$ and $\\over... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 35958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$, and $\\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overlin... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 35959,
"solution": "Using the diagram above, we can solve this problem by using mass points. By angle bisector theorem: \\[\\frac{BL}{CB}=\\frac{AL}{CA}\\implies\\frac{BL}{300}=\\frac{AL}{450}\\implies 3BL=2AL\\] So, we can weight $A$ as $2$ and $B$ as $3$ and $L$ as $5$. Since $K$ is the midpoint of $A$... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC = 450$ and $BC = 300$. Points $K$ and $L$ are located on $\\overline{AC}$ and $\\overline{AB}$ respectively so that $AK = CK$, and $\\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\\overline{BK}$ and $\\overlin... | math | {
"style": "rule",
"ground_truth": "072"
} | {
"index": 35960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $AC=3$, $BC=4$, and $AB=5$. Point $D$ is on $\\overline{AB}$, and $\\overline{CD}$ bisects the right angle. The inscribed circles of $\\triangle ADC$ and $\\triangle BCD$ have radii $r_a$ and $r_b$, respectively. What is $r_a/r_b$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{28} \\left(10 - \\sqrt{2}\\right)"
} | {
"index": 35961,
"solution": "1. **Identify the type of triangle and use the Pythagorean theorem**: Triangle $ABC$ with sides $AC=3$, $BC=4$, and $AB=5$ is a right triangle because $3^2 + 4^2 = 5^2$. Thus, $\\angle ACB = 90^\\circ$.\n\n2. **Use the Angle Bisector Theorem**: Since $\\overline{CD}$ bisects the right... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 35962,
"solution": "WLOG let E be be between C & D (as in solution 1). Assume $AD = 3m$. We use power of a point to get that $AG = DE = \\sqrt{2}m$ and $AB = AG + GB = AG + BE = 10+2\\sqrt{2} m$\n Since now we have $AC = 10$, $BC = 20, AB = 10+2\\sqrt{2} m$ in triangle $\\triangle ABC$ and cevian $AD = 3... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $\\angle BAC = 60^{\\circ}$, $\\angle CBA \\leq 90^{\\circ}$, $BC=1$, and $AC \\geq AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\\triangle ABC$, respectively. Assume that the area of pentagon $BCOIH$ is the maximum possible. ... | math | {
"style": "rule",
"ground_truth": "80^{\\circ}"
} | {
"index": 35963,
"solution": "1. **Define the angles and known values**: Let $\\angle CAB = A = 60^\\circ$, $\\angle ABC = B$, and $\\angle BCA = C$. We know $BC = 1$ and $AC \\geq AB$.\n\n2. **Properties of triangle centers**: \n - $\\angle BOC = 2A = 120^\\circ$ (Central angle is twice the inscribed angle).\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has $\\angle C = 60^{\\circ}$ and $BC = 4$. Point $D$ is the midpoint of $BC$. What is the largest possible value of $\\tan{\\angle BAD}$?\n$\\mathrm{(A)}\\ \\frac{\\sqrt{3}}{6}\\qquad\\mathrm{(B)}\\ \\frac{\\sqrt{3}}{3}\\qquad\\mathrm{(C)}\\ \\frac{\\sqrt{3}}{2\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4\\sqrt{2}-3}"
} | {
"index": 35964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has a perimeter of 2007 units. The sides have lengths that are all integer values with $AB< BC \\leq AC$. What is the smallest possible value of $BC - AB$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 35965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has a right angle at $B$, $AB=1$, and $BC=2$. The bisector of $\\angle BAC$ meets $\\overline{BC}$ at $D$. What is $BD$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 35966,
"solution": "1. **Identify the given information and the goal:** \n Triangle $ABC$ is a right triangle with $\\angle B = 90^\\circ$, $AB = 1$, and $BC = 2$. We need to find the length of $BD$, where $D$ is the point on $BC$ such that $AD$ is the angle bisector of $\\angle BAC$.\n\n2. **Calculat... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has a right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$.\n\n[asy]\nunitsize(0.2 cm);\n\npair A, B, C, P;\n\nA = (0,14);\nB = (0,0);\nC = (21*sqrt(3),0);\nP = intersectionpoint(arc(... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 35967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=3$, and $DC=4$. What is the area of $\\triangle ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "$7\\sqrt{3}$"
} | {
"index": 35968,
"solution": "1. **Identify the relationship between segments in the triangle**: Given that $ABC$ is a right triangle with the right angle at $B$, and $D$ is the foot of the altitude from $B$ to $AC$, we know that $AD = 3$ and $DC = 4$. \n\n2. **Use the geometric property of right triangles**: In a... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has an area 1. Points $E,F,G$ lie, respectively, on sides $BC$, $CA$, $AB$ such that $AE$ bisects $BF$ at point $R$, $BF$ bisects $CG$ at point $S$, and $CG$ bisects $AE$ at point $T$. Find the area of the triangle $RST$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7 - 3 \\sqrt{5}}{4}"
} | {
"index": 35969,
"solution": "Choose $r,s,t$ so that $EC = rBC, FA = sCA, GB = tCB$, and let $[XYZ]$ denote the area of triangle $XYZ$. Then $[ABE] = [AFE]$ since the triangles have the same altitude and base. Also $[ABE] = (BE/BC) [ABC] = 1-r$, and $[ECF] = (EC/BC)(CF/CA)[ABC] = r(1-s)$ (e.g., by the law of sines... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has an inradius of $5$ and a circumradius of $16$. If $2\\cos{B} = \\cos{A} + \\cos{C}$, then the area of triangle $ABC$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is... | math | {
"style": "rule",
"ground_truth": "141"
} | {
"index": 35970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has its vertices $A$, $B$, and $C$ on the sides of a rectangle 4 units by 5 units as shown. What is the area of triangle $ABC$ in square units?\n\n[asy]\n\nfill((0,1)--(4,0)--(2,5)--cycle,lightgray);\n\nfor(int i=1; i < 5; ++i){\nfor(int k=1; k < 4; ++k){\ndraw((... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 35971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\\angle B$ and $\\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 35972,
"solution": "Let $M$ be midpoint $BC, BM = x, AB = y, \\angle IBM = \\alpha.$\n$BI$ is the bisector of $\\angle ABM$ in $\\triangle ABM.$ $BI = \\frac {2 xy \\cos \\alpha}{x+y} = 8, \\cos \\alpha = \\frac {x}{8} \\implies \\frac {x^2 y}{x+y} = 32.$ \\[y = \\frac {32 x} {x^2 - 32}.\\] $BC = 2x$ is ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\\angle APB = \\angle BPC = \\angle CPA$. Find $PC$.\n[asy] unitsize(0.2 cm); pair A, B, C, P; A = (0,14); B = (0,0); C = (21*sqrt(3),0); P = intersectionpoint(arc(B,6,0,180),arc... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 35973,
"solution": "Let $PC = x$. Since $\\angle APB = \\angle BPC = \\angle CPA$, each of them is equal to $120^\\circ$. By the Law of Cosines applied to triangles $\\triangle APB$, $\\triangle BPC$ and $\\triangle CPA$ at their respective angles $P$, remembering that $\\cos 120^\\circ = -\\frac12$, we ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 11, BC=24$, and $CA = 20$. The bisector of $\\angle{BAC}$ intersects $\\overline{BC}$ in point $D$, and intersects the circumcircle of $\\triangle{ABC}$ in point $E \\ne A$. The circumcircle of $\\triangle{BED}$ intersects the line $AB$ in ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 35974,
"solution": "1. **Identify the triangle and given lengths**: We are given a triangle $ABC$ with side lengths $AB = 11$, $BC = 24$, and $CA = 20$. \n\n2. **Angle bisector and point definitions**: The bisector of $\\angle BAC$ intersects $BC$ at point $D$ and the circumcircle of $\\triangle ABC$ at ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ = \\omega$, the area of $PQRS$ can be expresse... | math | {
"style": "rule",
"ground_truth": "161"
} | {
"index": 35975,
"solution": "Proceed as in solution 1. When $\\omega$ is equal to zero, $\\alpha - \\beta\\omega=\\alpha$ is equal to the altitude. This means that $25\\beta$ is equal to $\\frac{36}{5}$, so $\\beta = \\frac{36}{125}$, yielding $\\boxed{161}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\\overline{AB}$, vertex $Q$ on $\\overline{AC}$, and vertices $R$ and $S$ on $\\overline{BC}$. In terms of the side length $PQ = \\omega$, the area of $PQRS$ can be expresse... | math | {
"style": "rule",
"ground_truth": "161"
} | {
"index": 35976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB = 9$, $BC =$ $5\\sqrt{3}$, and $AC = 12$. Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ between $P_{k-1}$ and $P_{k+1}$ for $k = 1, 2, ..., 2449$, and points $A = Q_{0}, Q_{1}, Q_{2}, ... , ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35977,
"solution": "Let $T_1$ stand for $AP_1Q_1$, and $T_k = AP_kQ_k$. All triangles $T$ are similar by AA. Let the area of $T_1$ be $x$. The next trapezoid will also have an area of $x$, as given. Therefore, $T_k$ has an area of $kx$. The ratio of the areas is equal to the square of the scale factor fo... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the interior of ... | math | {
"style": "rule",
"ground_truth": "715"
} | {
"index": 35978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the interior of ... | math | {
"style": "rule",
"ground_truth": "715"
} | {
"index": 35979,
"solution": "Notation shown on diagram. By similar triangles we have \\[k_1 = \\frac{EF}{BC} = \\frac{AE}{AB} = \\frac {AF}{AC} = \\frac {1}{4},\\] \\[k_2 = \\frac{F''E''}{AC} = \\frac {BF''}{AB} = \\frac{1}{4},\\] \\[k_3 = \\frac{E'F'}{AB} = \\frac{E'C }{AC} = \\frac{1}{8}.\\] So, \\[\\frac{ZE}{B... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=19, BC=20$, and $CA=21$. Points $X$ and $Y$ are selected on sides $AB$ and $AC$, respectively, such that $AY=XY$ and $XY$ is tangent to the incircle of $\\triangle ABC$. If the length of segment $AX$ can be written as $\\frac{a}{b}$, where $a... | math | {
"style": "rule",
"ground_truth": "6710"
} | {
"index": 35980,
"solution": "Note that the incircle of $\\triangle ABC$ is the $A$-excenter of $\\triangle AXY$. Let $r$ be the radius of this circle. We can compute the area of $\\triangle AXY$ in two ways: $$\\begin{aligned} K_{AXY} & =\\frac{1}{2} \\cdot AX \\cdot AY \\sin A \\\\ & =r \\cdot(AX+AY-XY) / 2 \\\\... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expresse... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 35981,
"solution": "Let $P=AE\\cap CF$. Let $CP=5x$ and $BP=5y$; from $\\triangle{CBP}\\sim\\triangle{EFP}$ we have $EP=7x$ and $FP=7y$. From $\\triangle{CAP}\\sim\\triangle{DFP}$ we have $\\frac{6}{4+5y}=\\frac{2}{7y}$ giving $y=\\frac{1}{4}$. So $BP=\\frac{5}{4}$ and $FP=\\frac{7}{4}$. These similar tr... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=5$, $BC=6$, and $AC=7$. Two bugs start simultaneously from $A$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $D$. What is $BD$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 35982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not e... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 35983,
"solution": "By the definition of $K$, it is the spiral center mapping $BA\\to AC$, which means that it is the midpoint of the $A$-symmedian chord. In particular, if $M$ is the midpoint of $BC$ and $M'$ is the reflection of $A$ across $K$, we have $\\triangle ABM'\\sim\\triangle AMC$. By Stewart's... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\\triangle ABC$ parallel to $\\overline{BC}$ intersects $\\overline{AB}$ at $M$ and $\\overline{AC}$ at $N.$ What is the perimeter of $\\triangle AMN?$\n$\\textbf{(A)}\\ 27 \\qq... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 35984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has sides $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$ of length 43, 13, and 48, respectively. Let $\\omega$ be the circle circumscribed around $\\triangle ABC$ and let $D$ be the intersection of $\\omega$ and the perpendicular bisector of $\\overline... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 35985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has sides of $6$ units, $8$ units, and $10$ units. The width of a rectangle, whose area is equal to the area of the triangle, is $4$ units. What is the perimeter of this rectangle, in units?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has sides of length 5, 12 and 13 units, and triangle $DEF$ has sides of length 8, 15 and 17 units. What is the ratio of the area of triangle $ABC$ to the area of triangle $DEF$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"index": 35987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\\triangle ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 35988,
"solution": "\n#### Step 1: Understanding the problem\nWe are given a triangle $ABC$ with vertices $A = (3,0)$, $B = (0,3)$, and $C$ on the line $x + y = 7$. We need to find the area of $\\triangle ABC$.\n\n#### Step 2: Analyzing the line $x + y = 7$\nThe line $x + y = 7$ is parallel to the line c... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(-2, 10)$, $B(3, 0)$, $C(10, 0)$. A line through $B$ cuts the area of $\\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"index": 35989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(0, 10)$, $B(3, 0)$, $C(9, 0)$. A horizontal line with equation $y=s$ intersects line segment $\\overline{AB}$ at $P$ and line segment $\\overline{AC}$ at $Q$, forming $\\triangle APQ$ with area 18. Compute $s$."
}
] | math | {
"style": "rule",
"ground_truth": "10 - 2\\sqrt{15}"
} | {
"index": 35990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A horizontal line with equation $y=t$ intersects line segment $ \\overline{AB} $ at $T$ and line segment $ \\overline{AC} $ at $U$, forming $\\triangle ATU$ with area 13.5. Compute $t$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 35991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(0, 8)$, $B(2, 0)$, $C(8, 0)$. A line through $B$ cuts the area of $\\triangle ABC$ in half; find the sum of the slope and $y$-intercept of this line."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 35992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 35993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices at $A(5,8)$, $B(3,-2)$, and $C(6,1)$. The point $D$ with coordinates $(m,n)$ is chosen inside the triangle so that the three small triangles $ABD$, $ACD$ and $BCD$ all have equal areas. What is the value of $10m + n$?"
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 35994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices with coordinates $A(2,3),$ $B(7,8),$ and $C(-4,6)$. The triangle is reflected about line $L$. The image points are $A'(2,-5),$ $B'(7,-10),$ and $C'(-4,-8)$. What is the equation of line $L$?"
}
] | math | {
"style": "rule",
"ground_truth": "y = -1"
} | {
"index": 35995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is a right isosceles triangle. Points $D$, $E$ and $F$ are the midpoints of the sides of the triangle. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"index": 35996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with $AB = 6$, $BC = 8$, and $AC = 10$. Point $D$ is on line $\\overline{BC}$ such that $\\overline{AD}$ bisects angle $BAC$. The inscribed circles of $\\triangle ADB$ and $\\triangle ADC$ have radii $r_1$ and $r_2$, respectively. What is $r_1... | math | {
"style": "rule",
"ground_truth": "\\frac{24}{35}"
} | {
"index": 35997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $AD = BD = 15.$ Given that the area of triangle $CDM$ may be expressed as $\\frac {m\\sqrt {n}}{p},$ w... | math | {
"style": "rule",
"ground_truth": "578"
} | {
"index": 35998,
"solution": "We use the Pythagorean Theorem on $ABC$ to determine that $AB=25.$\n Let $N$ be the orthogonal projection from $C$ to $AB.$ Thus, $[CDM]=\\frac{(DM)(MN)} {2}$, $MN=AM-AN$, and $[ABC]=\\frac{24 \\cdot 7} {2} =\\frac{25 \\cdot (CN)} {2}.$\n From the third equation, we get $CN=\\frac{168... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with $\\angle ACB$ as its right angle, $m\\angle ABC = 60^\\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\\overline{BP}$ to meet $\\overline{AC}$ at $D$. What is the probability that $BD > 5\\sqrt2$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt3}{3}"
} | {
"index": 35999,
"solution": "1. **Identify the lengths of sides in triangle $ABC$**:\n Given that $\\angle ACB = 90^\\circ$ and $\\angle ABC = 60^\\circ$, triangle $ABC$ is a 30-60-90 triangle. In such triangles, the sides are in the ratio $1:\\sqrt{3}:2$. Since $AB = 10$ (hypotenuse), the other sides are:\n ... |
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