data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with legs $AB$ and $AC$. Points $X$ and $Y$ lie on legs $AB$ and $AC$, respectively, so that $AX:XB = AY:YC = 1:2$. If $BY = 16$ units, and $CX = 28$ units, what is the length of hypotenuse $BC$? Express your answer in simplest radical form."
... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{26}"
} | {
"index": 36000,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is an equilateral triangle with each side of length 9. Points $D$, $E$, and $F$ are the trisection points of sides $AB$, $BC$, and $CA$ respectively, such that $D$ is closer to $A$, $E$ is closer to $B$, and $F$ is closer to $C$. Point $G$ is the midpoint of segm... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 36001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is an isosceles right triangle with the measure of angle $A$ equal to 90 degrees. The length of segment $AC$ is 6 cm. What is the area of triangle $ABC$, in square centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is an isosceles triangle with $AB=BC$. Point $D$ is the midpoint of both $\\overline{BC}$ and $\\overline{AE}$, and $\\overline{CE}$ is 11 units long. What is the length of $\\overline{BD}$? Express your answer as a decimal to the nearest tenth.\n\n[asy]\ndraw((... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"index": 36003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is an isosceles triangle with side lengths of 25, 25 and 48 centimeters. What is the area of triangle $ABC$, in square centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 36004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"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,... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 36005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is equilateral with $AB=1$. Points $E$ and $G$ are on $\\overline{AC}$ and points $D$ and $F$ are on $\\overline{AB}$ such that both $\\overline{DE}$ and $\\overline{FG}$ are parallel to $\\overline{BC}$. Furthermore, triangle $ADE$ and trapezoids $DFGE$ and $FBC... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{13}"
} | {
"index": 36006,
"solution": "1. **Assign Variables:**\n Let $AD = x$ and $AG = y$. Since $DE$ and $FG$ are parallel to $BC$, and $ABC$ is equilateral, $DE$ and $FG$ are also equal to $x$ and $y$ respectively. We need to find $DE + FG = x + y$.\n\n2. **Use Perimeter Conditions:**\n Given that $\\triangle ADE$,... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$?"
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 36007,
"solution": "1. **Identify the Triangle and Circle**: Triangle $ABC$ is equilateral with side length $6$. We need to find the area of the circle passing through points $A$, $O$ (the incenter of $\\triangle ABC$), and $C$.\n\n2. **Properties of the Incenter**: In an equilateral triangle, the incent... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in a circle of radius 2 with $\\angle ABC \\geq 90^\\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$."
}
] | math | {
"style": "rule",
"ground_truth": "$x = -\\frac{1}{2} (\\sqrt6 \\pm \\sqrt 2)$"
} | {
"index": 36008,
"solution": "\nTo solve the given problem, we will begin by analyzing the geometric aspect of the triangle \\( ABC \\), its circumcircle, and the properties relevant to the polynomial equation.\n\n### Step 1: Understanding the Geometry\n\nThe triangle \\( ABC \\) is inscribed in a circle with radi... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in a circle, and $\\angle B = \\angle C = 4\\angle A$. If $B$ and $C$ are adjacent vertices of a regular polygon of $n$ sides inscribed in this circle, then $n=$"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 36009,
"solution": "1. **Identify the relationship between angles**: Given that $\\angle B = \\angle C = 4\\angle A$, we can use the fact that the sum of angles in a triangle is $180^\\circ$. Therefore, we write:\n \\[\n \\angle B + \\angle C + \\angle A = 180^\\circ\n \\]\n Substituting $\\angle... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$ and $CA$ are, respectively, $x+75^{\\circ} , 2x+25^{\\circ},3x-22^{\\circ}$. Then one interior angle of the triangle is:"
}
] | math | {
"style": "rule",
"ground_truth": "$61^{\\circ}$"
} | {
"index": 36010,
"solution": "1. **Identify the relationship between the arcs and the angles**: Since triangle $ABC$ is inscribed in a circle, the measure of each interior angle of the triangle is half the measure of its corresponding arc. This is due to the Inscribed Angle Theorem, which states that an angle insc... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in circle $\\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\\overline{BC}$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diameter $\\overline{DE}$. Circles $\\omega$ and $\\gamma$ ... | math | {
"style": "rule",
"ground_truth": "919"
} | {
"index": 36011,
"solution": "Use the angle bisector theorem to find $CD=\\tfrac{21}{8}$, $BD=\\tfrac{35}{8}$, and use Stewart's Theorem to find $AD=\\tfrac{15}{8}$. Use Power of Point $D$ to find $DE=\\tfrac{49}{8}$, and so $AE=8$. Then use the Extended Law of Sine to find that the length of the circumradius of $... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in circle $\\omega$. Points $P$ and $Q$ are on side $\\overline{AB}$ with $AP<AQ$. Rays $CP$ and $CQ$ meet $\\omega$ again at $S$ and $T$ (other than $C$), respectively. If $AP=4,PQ=3,QB=6,BT=5,$ and $AS=7$, then $ST=\\frac{m}{n}$, where $m$ and $n$ ... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 36012,
"solution": "By Ptolemy's Theorem applied to quadrilateral $ASTB$, we find \\[AS\\cdot BT+AB\\cdot ST=AT\\cdot BS.\\] Projecting through $C$ we have \\[\\frac{AQ \\cdot PB}{PQ \\cdot AB} = (A,Q; P,B)\\stackrel{C}{=}(A,T; S,B)=\\frac{AT \\cdot BS}{ST \\cdot AB}.\\] Therefore \\[AT \\cdot BS = \\fra... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AB + AC$ and $BC = 65$ cm. $P$ is a point on $\\overline{BC}$ such that the perpendicular distances from $P$ to $\\overline{AB}$ and $\\overline{AC}$ are $24$ cm and $36$ cm, respectively. The area of $\\triangle ABC$, in cm $^2$, is"
}
] | math | {
"style": "rule",
"ground_truth": "2535"
} | {
"index": 36013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AB = AC = 2$ and $BC = 1.5$. Points $E$ and $G$ are on segment $\\overline{AC}$, and points $D$ and $F$ are on segment $\\overline{AB}$ such that both $\\overline{DE}$ and $\\overline{FG}$ are parallel to $\\overline{BC}$. Furthermore, triangle... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{6}"
} | {
"index": 36014,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$"
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 36015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$\n[asy] pointpen = black; pathpen = black+linewidt... | math | {
"style": "rule",
"ground_truth": "83^\\circ"
} | {
"index": 36016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AC = BC$ and $\\angle ACB = 106^\\circ.$ Point $M$ is in the interior of the triangle so that $\\angle MAC = 7^\\circ$ and $\\angle MCA = 23^\\circ.$ Find the number of degrees in $\\angle CMB.$\n[asy] pointpen = black; pathpen = black+linewidt... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 36017,
"solution": "Without loss of generality, let $AC = BC = 1$. Then, using the Law of Sines in triangle $AMC$, we get $\\frac {1}{\\sin 150} = \\frac {MC}{\\sin 7}$, and using the sine addition formula to evaluate $\\sin 150 = \\sin (90 + 60)$, we get $MC = 2 \\sin 7$.\n Then, using the Law of Cosine... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 36018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with angle $B$ congruent to angle $C$. The measure of angle $C$ is four times the measure of angle $A$. What is the number of degrees in the measure of angle $B$?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 36019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles, with $AB=AC$ and altitude $AM=11.$ Suppose that there is a point $D$ on $\\overline{AM}$ with $AD=10$ and $\\angle BDC=3\\angle BAC.$ Then the perimeter of $\\triangle ABC$ may be written in the form $a+\\sqrt{b},$ where $a$ and $b$ are integers. Fi... | math | {
"style": "rule",
"ground_truth": "616"
} | {
"index": 36020,
"solution": "Let $x=\\angle CAM$, so $3x=\\angle CDM$. Then, $\\frac{\\tan 3x}{\\tan x}=\\frac{CM/1}{CM/11}=11$. Expanding $\\tan 3x$ using the angle sum identity gives \\[\\tan 3x=\\tan(2x+x)=\\frac{3\\tan x-\\tan^3x}{1-3\\tan^2x}.\\] Thus, $\\frac{3-\\tan^2x}{1-3\\tan^2x}=11$. Solving, we get $\... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ lies in the cartesian plane and has an area of $70$. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20),$ respectively, and the coordinates of $A$ are $(p,q).$ The line containing the median to side $BC$ has slope $-5.$ Find the largest possible value of $... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 36021,
"solution": "Using the equation of the median from above, we can write the coordinates of $A$ as $(p,\\ -5p + 107)$. The equation of $\\overline{BC}$ is $\\frac{20 - 19}{23 - 12} = \\frac{y - 19}{x - 12}$, so $x - 12 = 11y - 209$. In general form, the line is $x - 11y + 197 = 0$. Use the equation ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ obeys $AB=2AC$ and $\\angle BAC=120^{\\circ}$. Points $P$ and $Q$ lie on segment $BC$ such that $$\\begin{aligned} AB^{2}+BC \\cdot CP & =BC^{2} \\\\ 3AC^{2}+2BC \\cdot CQ & =BC^{2} \\end{aligned}$$ Find $\\angle PAQ$ in degrees."
}
] | math | {
"style": "rule",
"ground_truth": "40^{\\circ}"
} | {
"index": 36022,
"solution": "We have $AB^{2}=BC(BC-CP)=BC \\cdot BP$, so triangle $ABC$ is similar to triangle $PBA$. Also, $AB^{2}=BC(BC-2CQ)+AC^{2}=(BC-CQ)^{2}-CQ^{2}+AC^{2}$, which rewrites as $AB^{2}+CQ^{2}=$ $BQ^{2}+AC^{2}$. We deduce that $Q$ is the foot of the altitude from $A$. Thus, $\\angle PAQ=90^{\\ci... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\\overline{AB}$, and let $E$ be the midpoint of $\\overline{AC}$. The angle bisector of $\\angle BAC$ intersects $\\overline{DE}$ and $\\overline{BC}$ at $F$ and $G$, respectively. What is the a... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 36023,
"solution": "1. **Identify the areas of sub-triangles**: \n - The area of $\\triangle ABC$ is given as $120$.\n - Since $D$ and $E$ are midpoints of $AB$ and $AC$ respectively, $AD = DB = 25$ and $AE = EC = 5$.\n - The area of $\\triangle ADE$ can be calculated using the formula for the area... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ with right angle at $C$, $\\angle BAC < 45^\\circ$ and $AB = 4$. Point $P$ on $\\overline{AB}$ is chosen such that $\\angle APC = 2\\angle ACP$ and $CP = 1$. The ratio $\\frac{AP}{BP}$ can be represented in the form $p + q\\sqrt{r}$, where $p$, $q$, $r$ are posit... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 36024,
"solution": "Let $\\angle{ACP}$ be equal to $x$. Then by Law of Sines, $PB = -\\frac{\\cos{x}}{\\cos{3x}}$ and $AP = \\frac{\\sin{x}}{\\sin{3x}}$. We then obtain $\\cos{3x} = 4\\cos^3{x} - 3\\cos{x}$ and $\\sin{3x} = 3\\sin{x} - 4\\sin^3{x}$. Solving, we determine that $\\sin^2{x} = \\frac{4 \\pm ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ with vertices $A(-2, 0)$, $B(1, 4)$ and $C(-3, 2)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $C$ to $C'$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ with vertices $A(1, -3)$, $B(-2, 0)$ and $C(4, 3)$ is reflected over the $y$-axis to form triangle $A'B'C'$. What is the length of a segment drawn from $A$ to $A'$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 36026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$ with vertices of $A(6,2)$, $B(2,5)$, and $C(2,2)$ is reflected over the x-axis to triangle $A'B'C'$. This triangle is reflected over the y-axis to triangle $A''B''C''$. What are the coordinates of point $C''$?"
}
] | math | {
"style": "rule",
"ground_truth": "(-2, -2)"
} | {
"index": 36027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$, $ADE$, and $EFG$ are all equilateral. Points $D$ and $G$ are midpoints of $\\overline{AC}$ and $\\overline{AE}$, respectively. If $AB=4$, what is the perimeter of figure $ABCDEFG$? [asy]\n/* AMC8 2000 #15 Problem */\ndraw((0,0)--(4,0)--(5,2)--(5.5,1)--(4.5,1));\... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 36028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{95}"
} | {
"index": 36029,
"solution": "1. **Assigning Coordinates**: Assume without loss of generality that vertex $A$ is on the $x$-axis, vertex $B$ is on the $y$-axis, and vertex $C$ is on the $z$-axis. Let the coordinates of $A$, $B$, and $C$ be $(a,0,0)$, $(0,b,0)$, and $(0,0,c)$ respectively.\n\n2. **Using Triangle Si... |
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": 36030,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $ABC_0$ has a right angle at $C_0$. Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$. Let $C_1$ be the foot of the altitude to $\\overline{AB}$, and for $n \\geq 2$, let $C_n$ be the foot of the altitude to $\\overline{C_{n-2}B... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 36031,
"solution": "For this problem, first notice that its an infinite geometric series of $6(a+b+c)=\\frac{ab}{c-b}$ if $c$ is the hypotenuse. WLOG $a<b$, we can generalize a pythagorean triple of $x^2-y^2, 2xy, x^2+y^2$. Let $b=2xy$, then this generalization gives $6(a+b+c)(c-b)=ab$ \\[(x^2-y^2)2xy=6(... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\\overline{AB_{n-1}}$ and $\\overline{AC_{n-1}}$, respectively, creating three similar triangles $\\triangle AB_nC_n \\sim \\tr... | math | {
"style": "rule",
"ground_truth": "961"
} | {
"index": 36032,
"solution": "Note that every $B_nC_n$ is parallel to each other for any nonnegative $n$. Also, the area we seek is simply the ratio $k=\\frac{B_0B_1C_1}{B_0B_1C_1+C_1C_0B_0}$, because it repeats in smaller and smaller units. Note that the area of the triangle, by Heron's formula, is 90.\n For ease... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $AHI$ is equilateral. We know $\\overline{BC}$, $\\overline{DE}$ and $\\overline{FG}$ are all parallel to $\\overline{HI}$ and $AB = BD = DF = FH$. What is the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$? Express your answer as a common fraction... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 36033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\epsilon$, the area of $WXYZ$ can be expres... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\omega$, the area of $WXYZ$ can be expresse... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36035,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\omega$, the area of $WXYZ$ can be expresse... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 39$, and $FD = 36$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\theta$, the area of $WXYZ$ can be expresse... | math | {
"style": "rule",
"ground_truth": "229"
} | {
"index": 36037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}-10"
} | {
"index": 36038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ is isosceles with angle $E$ congruent to angle $F$. The measure of angle $F$ is three times the measure of angle $D$. What is the number of degrees in the measure of angle $E$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{540}{7}"
} | {
"index": 36039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $DEF$ is similar to triangle $ABC$. If $DE=6$, $EF=12$, and $BC=18$ units, what is the length of segment $AB$?\n\n[asy]draw((0,0)--(7,0));\ndraw((0,0)--(0,4));\ndraw((0,4)--(7,0));\nlabel(\"E\",(0,0),W);\nlabel(\"F\",(7,0),E);\nlabel(\"D\",(0,4),W);\ndraw((15,0)--(25.... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 36040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $GHI$ has sides of length 7, 24, and 25 units, and triangle $JKL$ has sides of length 9, 40, and 41 units. Both triangles have an altitude to the hypotenuse such that for $GHI$, the altitude splits the triangle into two triangles whose areas have a ratio of 2:3. For $J... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{15}"
} | {
"index": 36041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\\angle ABO=90^\\circ$ and $\\angle AOB=30^\\circ$. Suppose that $OA$ is rotated $90^\\circ$ counterclockwise about $O$. What are the coordinates of the image of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\left( - \\frac {5}{3}\\sqrt {3},5\\right)$"
} | {
"index": 36042,
"solution": "1. **Identify the Coordinates of Point $A$:**\n Given that $\\triangle OAB$ has $O = (0,0)$, $B = (5,0)$, and $\\angle ABO = 90^\\circ$, point $A$ must lie on the line perpendicular to $OB$ at $B$. Since $\\angle AOB = 30^\\circ$, we can use trigonometric relationships to find the c... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PAB$ is formed by three tangents to circle $O$ and $\\angle APB = 40^\\circ$. Find $\\angle AOB$.\n\n[asy]\nimport graph;\n\nunitsize(1.5 cm);\n\npair A, B, O, P, R, S, T;\n\nR = dir(115);\nS = dir(230);\nT = dir(270);\nP = extension(R, R + rotate(90)*(R), T, T + rot... | math | {
"style": "rule",
"ground_truth": "70^\\circ"
} | {
"index": 36043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has $PQ = 28.$ The incircle of the triangle evenly trisects the median $PS.$ If the area of the triangle is $p \\sqrt{q}$ where $p$ and $q$ are integers, and $q$ is prime, find $p+q.$"
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 36044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has positive integer side lengths with $PQ = PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ = 10$. Find the smallest possible perimeter of $\\triangle PQR$."
}
] | math | {
"style": "rule",
"ground_truth": "1818"
} | {
"index": 36045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\\triangle PQR$."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 36046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has side lengths $PQ=160, QR=300$, and $PR=240$. Lines $m_P, m_Q$, and $m_R$ are drawn parallel to $\\overline{QR}, \\overline{RP}$, and $\\overline{PQ}$, respectively, such that the intersections of $m_P, m_Q$, and $m_R$ with the interior of $\\triangle PQR$ are... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"index": 36047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has side lengths $PQ=6$, $QR=8$, and $PR=9$. Two bugs start simultaneously from $P$ and crawl along the perimeter of the triangle in opposite directions at the same speed. They meet at point $S$. What is $QS$?"
}
] | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"index": 36048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has side-lengths $PQ = 20, QR = 40,$ and $PR = 30.$ The line through the incenter of $\\triangle PQR$ parallel to $\\overline{QR}$ intersects $\\overline{PQ}$ at $X$ and $\\overline{PR}$ at $Y.$ What is the perimeter of $\\triangle PXY?$"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 36049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has sides $\\overline{PQ}$, $\\overline{QR}$, and $\\overline{RP}$ of length 47, 14, and 50, respectively. Let $\\omega$ be the circle circumscribed around $\\triangle PQR$ and let $S$ be the intersection of $\\omega$ and the perpendicular bisector of $\\overline... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 36050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has sides of length 7, 24, and 25 units, and triangle $XYZ$ has sides of length 9, 40, and 41 units. Calculate the ratio of the area of triangle $PQR$ to the area of triangle $XYZ$, and also find the ratio of their perimeters."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{28}{45}"
} | {
"index": 36051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices $P = (4,0)$, $Q = (0,4)$, and $R$, where $R$ is on the line $x + y = 8$ and also on the line $y = 2x$. Find the area of $\\triangle PQR$.\nA) $\\frac{4}{3}$\nB) $\\frac{6}{3}$\nC) $\\frac{8}{3}$\nD) $\\frac{10}{3}$\nE) $\\frac{12}{3}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{3}"
} | {
"index": 36052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices $P(0, 10)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ bisects the area of $\\triangle PQR$. Find the sum of the slope and $y$-intercept of this line."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-20}{3}"
} | {
"index": 36053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices $P(0, 6)$, $Q(3, 0)$, $R(9, 0)$. A line through $Q$ cuts the area of $\\triangle PQR$ in half. Find the sum of the slope and $y$-intercept of this line."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 36054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ has vertices at $P(7, 5)$, $Q(1, -3)$, and $R(4, 4)$. The point $S$ with coordinates $(x, y)$ is chosen inside the triangle so that the three smaller triangles $PQS$, $PRS$, and $QRS$ all have equal areas. What is the value of $12x + 3y$?"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 36055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $PQR$ is a right triangle with legs $PQ$ and $PR$. Points $U$ and $V$ are on legs $PQ$ and $PR$, respectively so that $PU:UQ = PV:VR = 1:3$. If $QU = 18$ units, and $RV = 45$ units, what is the length of hypotenuse $PQ$? Express your answer in simplest radical form."
... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{29}"
} | {
"index": 36056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $XYZ$ has vertices $X(1, 9)$, $Y(3, 1)$, and $Z(9, 1)$. A line through $Y$ cuts the area of $\\triangle XYZ$ in half. Find the sum of the slope and the $y$-intercept of this line."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 36057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $XYZ$ is a right, isosceles triangle. Angle $X$ measures 45 degrees. What is the number of degrees in the measure of angle $Y$?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 36058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\triangle ABC$ has a right angle at $C$, $\\angle A = 45^\\circ$, and $AC=12$. Find the radius of the incircle of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "6 - 3\\sqrt{2}"
} | {
"index": 36059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\triangle ABC$ has a right angle at $C$, $\\angle A = 60^\\circ$, and $AC=10$. Find the radius of the incircle of $\\triangle ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "5(\\sqrt{3}-1)"
} | {
"index": 36060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\triangle ABC$ in the figure has area $10$. Points $D, E$ and $F$, all distinct from $A, B$ and $C$, \nare on sides $AB, BC$ and $CA$ respectively, and $AD = 2, DB = 3$. If triangle $\\triangle ABE$ and quadrilateral $DBEF$ \nhave equal areas, then that area is"
}
... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36061,
"solution": "1. **Identify Key Relationships**: Given that $AD = 2$, $DB = 3$, and $[ABE] = [DBEF]$, where $[X]$ denotes the area of figure $X$.\n\n2. **Area of $\\triangle ABC$**: The area of $\\triangle ABC$ is given as $10$.\n\n3. **Ratio of Segments on $AB$**: Since $AD = 2$ and $DB = 3$, the ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\triangle DEF$ has a right angle at $F$, $\\angle D = 60^\\circ$, and $DF = 6$. Find the radius of the incircle and the circumference of the circumscribed circle around $\\triangle DEF$."
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 36062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\triangle DEF$ has a right angle at $F$, $\\angle D = 60^\\circ$, and $DF=12$. Find the radius of the incircle of $\\triangle DEF$."
}
] | math | {
"style": "rule",
"ground_truth": "6(\\sqrt{3}-1)"
} | {
"index": 36063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle $\\triangle P Q R$, with $P Q=P R=5$ and $Q R=6$, is inscribed in circle $\\omega$. Compute the radius of the circle with center on $\\overline{Q R}$ which is tangent to both $\\omega$ and $\\overline{P Q}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{9}"
} | {
"index": 36064,
"solution": "Solution 1: Denote the second circle by $\\gamma$. Let $T$ and $r$ be the center and radius of $\\gamma$, respectively, and let $X$ and $H$ be the tangency points of $\\gamma$ with $\\omega$ and $\\overline{P Q}$, respectively. Let $O$ be the center of $\\omega$, and let $M$ be the mi... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is $4\\pi$ sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.\n\nNote: The area of a circle with radius $r$... | math | {
"style": "rule",
"ground_truth": "12\\sqrt 3"
} | {
"index": 36065,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle PQR is a right triangle with PQ = 6, QR = 8, and PR = 10. Point S is on PR, and QS bisects the right angle at Q. The inscribed circles of triangles PQS and QRS have radii rp and rq, respectively. Find rp/rq."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{28}\\left(10-\\sqrt{2}\\right)"
} | {
"index": 36066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\( ABC \\) has \\( AB=24 \\), \\( AC=26 \\), and \\( BC=22 \\). Points \\( D \\) and \\( E \\) are located on \\( \\overline{AB} \\) and \\( \\overline{AC} \\), respectively, so that \\( \\overline{DE} \\) is parallel to \\( \\overline{BC} \\) and contains the center ... | math | {
"style": "rule",
"ground_truth": "\\frac{275}{18}"
} | {
"index": 36067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\( ABC \\) has a right angle at \\( B \\). Point \\( D \\) lies on side \\( BC \\) such that \\( 3 \\angle BAD = \\angle BAC \\). Given \\( AC = 2 \\) and \\( CD = 1 \\), compute \\( BD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 36068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\( ABC \\) is isosceles, and \\( \\angle ABC = x^\\circ \\). If the sum of the possible measures of \\( \\angle BAC \\) is \\( 240^\\circ \\), find \\( x \\)."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36069,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(A B C\\) has side lengths \\(A B = 65\\), \\(B C = 33\\), and \\(A C = 56\\). Find the radius of the circle tangent to sides \\(A C\\) and \\(B C\\) and to the circumcircle of triangle \\(A B C\\)."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(ABC\\) has \\(AB = 10\\) and \\(BC:AC = 35:36\\). What is the largest area that this triangle can have?"
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"index": 36071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(ABC\\) has \\(\\angle A = 90^\\circ\\), side \\(BC = 25\\), \\(AB > AC\\), and area 150. Circle \\(\\omega\\) is inscribed in \\(ABC\\), with \\(M\\) as its point of tangency on \\(AC\\). Line \\(BM\\) meets \\(\\omega\\) a second time at point \\(L\\). Find the len... | math | {
"style": "rule",
"ground_truth": "\\frac{45\\sqrt{17}}{17}"
} | {
"index": 36072,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(ABC\\) has a right angle at \\(B\\), with \\(AB = 3\\) and \\(BC = 4\\). If \\(D\\) and \\(E\\) are points on \\(AC\\) and \\(BC\\), respectively, such that \\(CD = DE = \\frac{5}{3}\\), find the perimeter of quadrilateral \\(ABED\\)."
}
] | math | {
"style": "rule",
"ground_truth": "28/3"
} | {
"index": 36073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(ABC\\) has sides \\(AB = 14\\), \\(BC = 13\\), and \\(CA = 15\\). It is inscribed in circle \\(\\Gamma\\), which has center \\(O\\). Let \\(M\\) be the midpoint of \\(AB\\), let \\(B'\\) be the point on \\(\\Gamma\\) diametrically opposite \\(B\\), and let \\(X\\) b... | math | {
"style": "rule",
"ground_truth": "\\frac{65}{12}"
} | {
"index": 36074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(ABC\\) is isosceles with \\(AB = AC\\) and \\(BC = 65 \\, \\text{cm}\\). \\(P\\) is a point on \\(BC\\) such that the perpendicular distances from \\(P\\) to \\(AB\\) and \\(AC\\) are \\(24 \\, \\text{cm}\\) and \\(36 \\, \\text{cm}\\), respectively. Find the area o... | math | {
"style": "rule",
"ground_truth": "2535"
} | {
"index": 36075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(ABC\\) is isosceles with \\(AB = AC\\) and \\(BC = 65 \\, \\text{cm}\\). \\(P\\) is a point on \\(BC\\) such that the perpendicular distances from \\(P\\) to \\(AB\\) and \\(AC\\) are \\(24 \\, \\text{cm}\\) and \\(36 \\, \\text{cm}\\), respectively. The area of \\(... | math | {
"style": "rule",
"ground_truth": "2535"
} | {
"index": 36076,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(\\triangle A B C\\) has \\(A B=21, B C=55\\), and \\(C A=56\\). There are two points \\(P\\) in the plane of \\(\\triangle A B C\\) for which \\(\\angle B A P=\\angle C A P\\) and \\(\\angle B P C=90^{\\circ}\\). Find the distance between them."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} \\sqrt{409}"
} | {
"index": 36077,
"solution": "Let \\(P_{1}\\) and \\(P_{2}\\) be the two possible points \\(P\\), with \\(A P_{1}<A P_{2}\\). Both lie on the \\(\\angle A\\)-bisector and the circle \\(\\gamma\\) with diameter \\(B C\\). Let \\(D\\) be the point where the \\(\\angle A\\)-bisector intersects \\(B C\\), let \\(M\\) ... |
DeepScaleR | [
{
"role": "user",
"content": "Triangle \\(\\triangle P N R\\) has side lengths \\(P N=20, N R=18\\), and \\(P R=19\\). Consider a point \\(A\\) on \\(P N\\). \\(\\triangle N R A\\) is rotated about \\(R\\) to \\(\\triangle N^{\\prime} R A^{\\prime}\\) so that \\(R, N^{\\prime}\\), and \\(P\\) lie on the sam... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{18}"
} | {
"index": 36078,
"solution": "Denote the intersection of \\(P R\\) and \\(A A^{\\prime}\\) be \\(D\\). Note \\(R A^{\\prime}=R A\\), so \\(D\\), being the altitude of an isosceles triangle, is the midpoint of \\(A A^{\\prime}\\). Thus, \\(\\angle A R D=\\angle A^{\\prime} R D=\\angle N R A\\) so \\(R A\\) is the a... |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside $\\triangle ABC$, $\\angle ABC = 40^\\circ$, and $\\angle ADC = 140^\\circ$. What is the degree measure of $\\angle\nBAD$?"
}
] | math | {
"style": "rule",
"ground_truth": "50^{\\circ}"
} | {
"index": 36079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $ABC$ and $ADE$ have areas $2007$ and $7002,$ respectively, with $B=(0,0), C=(223,0), D=(680,380),$ and $E=(689,389).$ What is the sum of all possible $x$-coordinates of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"index": 36080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "2400"
} | {
"index": 36081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $ABC$ and $AEF$ are such that $B$ is the midpoint of $\\overline{EF}.$ Also, $AB = EF = 1,$ $BC = 6,$ $CA = \\sqrt{33},$ and\n\\[\\overrightarrow{AB} \\cdot \\overrightarrow{AE} + \\overrightarrow{AC} \\cdot \\overrightarrow{AF} = 2.\\]Find the cosine of the angle be... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 36082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $ABC$ and $AFG$ have areas $3009$ and $9003,$ respectively, with $B=(0,0), C=(331,0), F=(800,450),$ and $G=(813,463).$ What is the sum of all possible $x$-coordinates of $A$?"
}
] | math | {
"style": "rule",
"ground_truth": "1400"
} | {
"index": 36083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $ABC$ and $AFG$ have areas $3012$ and $10004$, respectively, with $B=(0,0),$ $C=(335,0),$ $F=(1020, 570),$ and $G=(1030, 580).$ Find the sum of all possible $x$-coordinates of $A$."
}
] | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"index": 36084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\\triangle ABC$ to $\... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 36085,
"solution": "After sketching, it is clear a $90^{\\circ}$ rotation is done about $(x,y)$. Looking between $A$ and $A'$, $x+y=18$ and $x-y=24$. Solving gives $(x,y)\\implies(21,-3)$. Thus $90+21-3=\\boxed{108}$. ~mn28407"
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $\\triangle ABC$ and $\\triangle DEC$ share side $BC$. Given that $AB = 7\\ \\text{cm}$, $AC = 15\\ \\text{cm}$, $EC = 9\\ \\text{cm}$, and $BD = 26\\ \\text{cm}$, what is the least possible integral number of centimeters in $BC$?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,15)$, $E'(30,15)$, $F'(20,5)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $\... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 36087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are in the coordinate plane with vertices $D(2,2)$, $E(2,14)$, $F(18,2)$, $D'(32,26)$, $E'(44,26)$, $F'(32,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $... | math | {
"style": "rule",
"ground_truth": "124"
} | {
"index": 36088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are positioned in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(14,0)$, $D'(20,20)$, $E'(30,20)$, $F'(20,8)$. Determine the angle of rotation $n$ degrees clockwise around the point $(p,q)$ where $0<n<180$, that tra... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 36089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ lie in the coordinate plane with vertices $D(0,0)$, $E(0,10)$, $F(15,0)$, $D'(15,25)$, $E'(25,25)$, $F'(15,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"index": 36090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Triangles \\(ABC\\) and \\(ABD\\) are inscribed in a semicircle with diameter \\(AB = 5\\). A perpendicular from \\(D\\) to \\(AB\\) intersects segment \\(AC\\) at point \\(Q\\), ray \\(BC\\) at point \\(R\\), and segment \\(AB\\) at point \\(P\\). It is known that \\(PR = \\fr... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 36091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he disco... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 36092,
"solution": "Let's denote the amount of money Fox has at the beginning as $x$ coins. We will analyze the changes in the amount of money Fox has after each crossing and payment.\n\n1. **First Crossing:**\n - Before crossing: Fox has $x$ coins.\n - After doubling: $2x$ coins.\n - After paying ... |
DeepScaleR | [
{
"role": "user",
"content": "Trodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds t... | math | {
"style": "rule",
"ground_truth": "1920"
} | {
"index": 36093,
"solution": "We look at the number of cottages after each wave of peasants. Let $A_{n}$ be the number of cottages remaining after $8 n$ minutes. During each 8 minute interval, Trodgor burns a total of $480 / 5=96$ peasants and cottages. Trodgor first burns $A_{n}$ peasants and spends the remaining... |
DeepScaleR | [
{
"role": "user",
"content": "Try to divide the set $\\{1,2,\\cdots, 1989\\}$ into 117 mutually disjoint subsets $A_{i}, i = 1,2,\\cdots, 117$, such that\n(1) Each $A_{i}$ contains 17 elements;\n(2) The sum of the elements in each $A_{i}$ is the same."
}
] | math | {
"style": "rule",
"ground_truth": "16915"
} | {
"index": 36094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the t... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 36095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the l... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 36096,
"solution": "\nTo solve this problem, we will analyze the board's structure and derive a strategy for Turbo to ensure he reaches the last row in a guaranteed number of attempts. We'll consider the distribution of monsters and Turbo's possible paths.\n\nGiven:\n- The board has 2024 rows and 2023 co... |
DeepScaleR | [
{
"role": "user",
"content": "Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \\dots$, Turbo successively crawls distances $c_1, c_2, c_3, \\dots$ around the circle, each time choosing to crawl either clockwise or counte... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 36097,
"solution": "\nTo find the largest constant \\( C > 0 \\) with the given property, we first need to understand the problem setup. Turbo starts at a point on a circle with a circumference of 1 and moves according to the sequence of positive real numbers \\( c_1, c_2, c_3, \\ldots \\). At each step,... |
DeepScaleR | [
{
"role": "user",
"content": "Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, \"So far, we've lied once.\" A second then says, \"Now it's twice.\" A third exclaims, \"Three times now,\" and so on, up to the twelfth who claims that before him, they lied t... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Twelve chairs are evenly spaced around a round table and numbered clockwise from $1$ through $12$. Six married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse or next to someone of the same profe... | math | {
"style": "rule",
"ground_truth": "2880"
} | {
"index": 36099,
"solution": ""
} |
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