data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\sqrt{10}$, $CP=60\\sqrt{5}$, $DP=120\\sqrt{2}$, and $GP=36\\sqrt{7}$. Find $AP.$ Let's think step b... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 5000,
"solution": "Let $E$ be the vertex of the cube such that $ABED$ is a square. By the British Flag Theorem, we can easily we can show that \\[PA^2 + PE^2 = PB^2 + PD^2\\] and \\[PA^2 + PG^2 = PC^2 + PE^2\\] Hence, adding the two equations together, we get $2PA^2 + PG^2 = PB^2 + PC... |
deepscaler | [
{
"role": "user",
"content": "A rock is dropped off a cliff of height $ h $ As it falls, a camera takes several photographs, at random intervals. At each picture, I measure the distance the rock has fallen. Let the average (expected value) of all of these distances be $ kh $ . If the number of photograph... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 5001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\\overline{BC}$, $\\overline{CD}$, $\\overline{EF}$, and $\\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{27}"
} | {
"split": "train",
"index": 5002,
"solution": "1. **Identify the Geometry of the Hexagon and Triangles**:\n - Hexagon $ABCDEF$ is regular, meaning all sides are equal and all internal angles are $120^\\circ$.\n - Connect $F$ to $C$ to form line segment $FC$. Drop a perpendicular from $E$ to $FC$ at point $P$... |
deepscaler | [
{
"role": "user",
"content": "You are trying to cross a 6 foot wide river. You can jump at most 4 feet, but you have one stone you can throw into the river; after it is placed, you may jump to that stone and, if possible, from there to the other side of the river. However, you are not very accurate and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5003,
"solution": "To be able to cross, the stone must land between 2 and 4 feet from the river bank you are standing on. Therefore the probability is $\\frac{2}{6}=\\frac{1}{3}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 9x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 5004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that there are two alloys with different percentages of copper, with alloy A weighing 40 kg and alloy B weighing 60 kg, a piece of equal weight is cut from each of these two alloys, and each cut piece is then melted together with the remaining part of the other alloy, det... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 5005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $40\\%$ of students initially answered \"Yes\", $40\\%$ answered \"No\", and $20\\%$ were \"Undecided\", and $60\\%$ answered \"Yes\" after a semester, $30\\%$ answered \"No\", and $10\\%$ remained \"Undecided\", determine the difference between the maximum and minim... | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"split": "train",
"index": 5006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be the endpoints of a semicircular arc of radius $3$. The arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\\dots$, $C_8$. All chords of the form $\\overline {AC_i}$ or $\\overline {BC_i}$ are drawn. Find the product of the l... | math | {
"style": "rule",
"ground_truth": "387420489"
} | {
"split": "train",
"index": 5007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points $(0,0),(1,2),(2,1),(2,2)$ in the plane are colored red while the points $(1,0),(2,0),(0,1),(0,2)$ are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest po... | math | {
"style": "rule",
"ground_truth": "305"
} | {
"split": "train",
"index": 5008,
"solution": "If $(2,2)$ is connected to $(0,1)$ or $(1,0)$, then the other 6 points can be connected with segments of total length 3, which is minimal. This leads to a total length of $3+\\sqrt{5}$. On the other hand, if $(2,2)$ is connected to $(0,2)$ or $(0,2)$, then connectin... |
deepscaler | [
{
"role": "user",
"content": "Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \\( L \\) cm. What is the value of \\( L \\)? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 5009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the \"Nine Chapters on the Mathematical Art,\" a tetrahedron with all four faces being right-angled triangles is referred to as a \"turtle's knee.\" Given that in the turtle's knee $M-ABC$, $MA \\perp$ plane $ABC$, and $MA=AB=BC=2$, the sum of the surface areas of the circum... | math | {
"style": "rule",
"ground_truth": "24\\pi-8\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 5010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $DEF$ where leg $DE = 30$ and leg $EF = 40$, determine the number of line segments with integer length that can be drawn from vertex $E$ to a point on hypotenuse $\\overline{DF}$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 5011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $4^{-1} \\pmod{35}$, as a residue modulo 35. (Give an answer between 0 and 34, inclusive.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 5012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ninety-four bricks, each measuring $4''\\times10''\\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\\,$ or $10''\\,$ or $19''\\,$ to the total height of the tower. How many different tower hei... | math | {
"style": "rule",
"ground_truth": "465"
} | {
"split": "train",
"index": 5013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If five points are given on a plane, then by considering all possible triples of these points, 30 angles can be formed. Denote the smallest of these angles by $\\alpha$. Find the maximum value of $\\alpha$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 5014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system \\(xOy\\), the equation of the ellipse \\(C\\) is given by the parametric form:\n\\[\n\\begin{cases}\nx=5\\cos\\varphi \\\\\ny=3\\sin\\varphi\n\\end{cases}\n\\]\nwhere \\(\\varphi\\) is the parameter.\n(I) Find the general equation of the stra... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence consists of $2010$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2010$ terms is $5307$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? Let's think step ... | math | {
"style": "rule",
"ground_truth": "2151"
} | {
"split": "train",
"index": 5016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius $r$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8$. Find the value of $r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 5017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 5105^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11.430"
} | {
"split": "train",
"index": 5018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In square ABCD, an isosceles triangle AEF is inscribed; point E lies on side BC, point F lies on side CD, and AE = AF. The tangent of angle AEF is 3. Find the cosine of angle FAD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 5019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $(x^2 + 3x + 2)(x^2 + 7x + 12) + (x^2 + 5x - 6)$ as the product of two non-constant polynomials. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(x^2 + 5x + 2)(x^2 + 5x + 9)"
} | {
"split": "train",
"index": 5020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( ABC \\) with sides \\( AB = 8 \\), \\( AC = 4 \\), \\( BC = 6 \\), the angle bisector \\( AK \\) is drawn, and on the side \\( AC \\) a point \\( M \\) is marked such that \\( AM : CM = 3 : 1 \\). Point \\( N \\) is the intersection point of \\( AK \\) and \\( B... | math | {
"style": "rule",
"ground_truth": "\\frac{18\\sqrt{6}}{11}"
} | {
"split": "train",
"index": 5021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What are the last two digits in the sum of factorials of the first 100 positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 5022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A basketball championship has been played in a round-robin format, with each pair of teams playing twice and no ties (overtime is played until one team wins). The winner of a match receives 2 points, and the loser receives 1 point. At the end of the championship, the sum of the... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 5023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle \\( \\triangle ABC \\), point \\( P \\) is an internal point such that \\( \\angle PBC = \\angle PCB = 24^\\circ \\). If \\( \\angle ABP = 30^\\circ \\) and \\( \\angle ACP = 54^\\circ \\), find the measure of \\( \\angle BAP \\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the measure of the angle\n\n$$\n\\delta=\\arccos \\left(\\left(\\sin 2903^{\\circ}+\\sin 2904^{\\circ}+\\cdots+\\sin 6503^{\\circ}\\right)^{\\cos 2880^{\\circ}+\\cos 2881^{\\circ}+\\cdots+\\cos 6480^{\\circ}}\\right)\n$$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 5025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, angle CAB is 30 degrees, and angle ABC is 80 degrees. The point M lies inside the triangle such that angle MAC is 10 degrees and angle MCA is 30 degrees. Find angle BMC in degrees. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 5026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs $(a,b)$ of complex numbers such that\n\\[a^4 b^6 = a^8 b^3 = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 5027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the Horner's method to calculate the value of the polynomial $f(x) = 5x^5 + 2x^4 + 3.5x^3 - 2.6x^2 + 1.7x - 0.8$ when $x=1$ and find the value of $v_3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.9"
} | {
"split": "train",
"index": 5028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the hexagonal pyramid $(P-ABCDEF)$, the base is a regular hexagon with side length $\\sqrt{2}$, $PA=2$ and is perpendicular to the base. Find the volume of the circumscribed sphere of the hexagonal pyramid. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}\\pi"
} | {
"split": "train",
"index": 5029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $\\omega_1$, $\\omega_2$, and $\\omega_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1$, $P_2$, and $P_3$ lie on $\\omega_1$, $\\omega_2$, and $\\omega_3$ respectively such that $P_1P_2=P_2P_3=P... | math | {
"style": "rule",
"ground_truth": "552"
} | {
"split": "train",
"index": 5030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of subsets $S$ of $\\{1,2, \\ldots 63\\}$ the sum of whose elements is 2008. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 5031,
"solution": "Note that $1+2+\\cdots+63=2016$. So the problem is equivalent to finding the number of subsets of $\\{1,2, \\cdots 63\\}$ whose sum of elements is 8. We can count this by hand: $\\{8\\},\\{1,7\\},\\{2,6\\}$, $\\{3,5\\},\\{1,2,5\\},\\{1,3,4\\}$."
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\(a\\), \\(b\\), and \\(c\\) are real numbers such that:\n\\[\n\\frac{ac}{a + b} + \\frac{ba}{b + c} + \\frac{cb}{c + a} = -12\n\\]\nand\n\\[\n\\frac{bc}{a + b} + \\frac{ca}{b + c} + \\frac{ab}{c + a} = 15.\n\\]\nCompute the value of:\n\\[\n\\frac{a}{a + b} + \\frac{b}... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 5032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain country, there are exactly 2019 cities and between any two of them, there is exactly one direct flight operated by an airline company, that is, given cities $A$ and $B$, there is either a flight from $A$ to $B$ or a flight from $B$ to $A$. Find the minimum number o... | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"split": "train",
"index": 5033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her sib... | math | {
"style": "rule",
"ground_truth": "441"
} | {
"split": "train",
"index": 5034,
"solution": "1. **Count the total number of twins and triplets:**\n - There are $9$ sets of twins, with $2$ twins per set, giving a total of $9 \\times 2 = 18$ twins.\n - There are $6$ sets of triplets, with $3$ triplets per set, giving a total of $6 \\times 3 = 18$ triplets... |
deepscaler | [
{
"role": "user",
"content": "The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In five years Jack will be twice as old as Bill will be then. What is the difference in their current ages? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 5035,
"solution": "1. **Define the ages**: Let Jack's age be represented as $\\overline{ab} = 10a + b$ where $a$ and $b$ are the tens and units digits respectively. Similarly, Bill's age is $\\overline{ba} = 10b + a$.\n\n2. **Future ages**: In five years, Jack's age will be $10a + b +... |
deepscaler | [
{
"role": "user",
"content": "Positive integer $n$ cannot be divided by $2$ and $3$ , there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$ . Find the minimum value of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 5036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 5037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac {π}{2}＜α＜ \\frac {3π}{2}$, points A, B, and C are in the same plane rectangular coordinate system with coordinates A(3, 0), B(0, 3), and C(cosα, sinα) respectively.\n(1) If $| \\overrightarrow {AC}|=| \\overrightarrow {BC}|$, find the value of angle α;\n(2) W... | math | {
"style": "rule",
"ground_truth": "- \\frac {5}{9}"
} | {
"split": "train",
"index": 5038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the val... | math | {
"style": "rule",
"ground_truth": "\\frac{8080}{199}"
} | {
"split": "train",
"index": 5039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain math competition, there are 10 multiple-choice questions. Each correct answer earns 4 points, no answer earns 0 points, and each wrong answer deducts 1 point. If the total score becomes negative, the grading system automatically sets the total score to zero. How ma... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 5040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 18 teams participating in the opening ceremony of a competition. When entering, the 1st team has 27 members, the 2nd team has 26 members, and the 18th team has 10 members. If they enter in a single file, and all 18 teams' members are assigned numbers from 1 to 333 in ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the largest integer that is the product of exactly 3 distinct prime numbers $d$, $e$, and $10d+e$, where $d$ and $e$ are single digits. What is the sum of the digits of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5042,
"solution": "1. **Identify the possible values for $d$ and $e$:**\n - Since $d$ and $e$ are single-digit prime numbers, the possible values for $d$ are $\\{2, 3, 5, 7\\}$.\n - Since $e$ is also a single-digit prime number and must be the units digit of the prime number $10d ... |
deepscaler | [
{
"role": "user",
"content": "If $(x + y)^2 = 25$ and $xy = 6$, what is the value of $x^2 + y^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 5043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 27 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in feet? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jolene and Tia are playing a two-player game at a carnival. In one bin, there are five red balls numbered 5, 10, 15, 20, and 25. In another bin, there are 25 green balls numbered 1 through 25. In the first stage of the game, Jolene chooses one of the red balls at random. Next, ... | math | {
"style": "rule",
"ground_truth": "13/40"
} | {
"split": "train",
"index": 5045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function satisfying the following conditions:\n(1) $f(1)=1$;\n(2) $\\forall n\\in \\mathbb{N}$, $3f(n) f(2n+1) =f(2n) ( 1+3f(n) )$;\n(3) $\\forall n\\in \\mathbb{N}$, $f(2n) < 6 f(n)$.\nFind all solutions of equation $f(k) +f(l... | math | {
"style": "rule",
"ground_truth": "(5, 47), (7, 45), (13, 39), (15, 37)"
} | {
"split": "train",
"index": 5046,
"solution": "\nLet \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) be a function satisfying the following conditions:\n1. \\( f(1) = 1 \\);\n2. For all \\( n \\in \\mathbb{N} \\), \\( 3f(n) f(2n+1) = f(2n) (1 + 3f(n)) \\);\n3. For all \\( n \\in \\mathbb{N} \\), \\( f(2n) < 6 f(... |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^{2}=4x$, a line $l$ passing through its focus $F$ intersects the parabola at points $A$ and $B$ (with point $A$ in the first quadrant), such that $\\overrightarrow{AF}=3\\overrightarrow{FB}$. A line passing through the midpoint of $AB$ and perpendicular to... | math | {
"style": "rule",
"ground_truth": "\\frac{32\\sqrt{3}}{9}"
} | {
"split": "train",
"index": 5047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a zoo, there were 200 parrots. One day, they each made a statement in turn. Starting from the second parrot, all statements were: \"Among the previous statements, more than 70% are false.\" How many false statements did the parrots make in total? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 5048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ann made a $3$-step staircase using $18$ toothpicks as shown in the figure. How many toothpicks does she need to add to complete a $5$-step staircase?\n[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\npath h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45);\nfor(in... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 5049,
"solution": "To solve this problem, we first need to understand the pattern in the number of toothpicks required to build each step of the staircase.\n\n1. **Identify the pattern:**\n - A $1$-step staircase requires $4$ toothpicks.\n - A $2$-step staircase requires $10$ toot... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{10a^3}{55a^2}$ when $a=3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6}{11}"
} | {
"split": "train",
"index": 5050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $2\\sin^2 \\frac{A+B}{2}-\\cos 2C=1$, and the radius of the circumcircle $R=2$. \n$(1)$ Find $C$; \n$(2)$ Find the maximum value of $S_{\\triangle ABC}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 5051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean (average) of 5 consecutive integers is 9. What is the smallest of these 5 integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 5052,
"solution": "Since the mean of five consecutive integers is 9, then the middle of these five integers is 9. Therefore, the integers are $7,8,9,10,11$, and so the smallest of the five integers is 7."
} |
deepscaler | [
{
"role": "user",
"content": "The cost of purchasing a car is 150,000 yuan, and the annual expenses for insurance, tolls, and gasoline are about 15,000 yuan. The maintenance cost for the first year is 3,000 yuan, which increases by 3,000 yuan each year thereafter. Determine the best scrap year limit for thi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5054,
"solution": "1. **Factor the first polynomial**: \n The polynomial $x^2 - 3x + 2$ can be factored as follows:\n \\[\n x^2 - 3x + 2 = (x - 1)(x - 2)\n \\]\n This implies that the roots of $x^2 - 3x + 2$ are $x = 1$ and $x = 2$.\n\n2. **Determine the value of $k$ for eac... |
deepscaler | [
{
"role": "user",
"content": "There are 5 balls numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$ and 5 boxes numbered $(1)$, $(2)$, $(3)$, $(4)$, $(5)$. Each box contains one ball. The number of ways in which at most two balls have the same number as their respective boxes is $\\_\\_\\_\\_\\_\\_$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 5055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dylan has a \\( 100 \\times 100 \\) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "9999"
} | {
"split": "train",
"index": 5056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 5057,
"solution": "1. **Identify the dimensions of the rectangle**: Given that the circle is inscribed in the rectangle, the diameter of the circle is equal to the width of the rectangle. Since the radius of the circle is $5$, the diameter is $2 \\times 5 = 10$. Therefore, the width o... |
deepscaler | [
{
"role": "user",
"content": "The numbers \\(2, 3, 12, 14, 15, 20, 21\\) may be divided into two sets so that the product of the numbers in each set is the same.\nWhat is this product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 5058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (-3, 0),$ $B=(-2,1),$ $C=(2,1),$ and $D=(3,0).$ Suppose that point $P$ satisfies \\[PA + PD = PB + PC = 8.\\]Then the $y-$coordinate of $P,$ when simplified, can be expressed in the form $\\frac{-a + b \\sqrt{c}}{d},$ where $a,$ $b,$ $c,$ $d$ are positive integers. Fi... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 5059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\\frac mn$, where $m$ and $n$ are relatively p... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 5060,
"solution": "1. **Identify Coordinates of Rectangle $ABCD$**:\n Given $B=(0,0)$, we can determine the coordinates of the other vertices based on the side lengths:\n - $A = (0, 3)$ (since $AB = 3$ and is vertical),\n - $C = (11, 0)$ (since $BC = 11$ and is horizontal),\n ... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\lg 2 = 0.3010$, determine the number of digits in the integer $2^{2015}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "607"
} | {
"split": "train",
"index": 5061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}_{n\\geq 0}$ of real numbers satisfies the relation:\n\\[ a_{m+n} + a_{m-n} - m + n -1 = \\frac12 (a_{2m} + a_{2n}) \\]\nfor all non-negative integers $m$ and $n$, $m \\ge n$. If $a_1 = 3$ find $a_{2004}$. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "4018021"
} | {
"split": "train",
"index": 5062,
"solution": "\nWe are given the sequence \\( \\{a_n\\}_{n \\geq 0} \\) which satisfies the relation:\n\n\\[\na_{m+n} + a_{m-n} - m + n - 1 = \\frac{1}{2} (a_{2m} + a_{2n})\n\\]\n\nfor all non-negative integers \\( m \\) and \\( n \\) with \\( m \\ge n \\). We are also given that... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $a^{2}$, $b^{2}$, $c^{2}$ form an arithmetic sequence. Calculate the maximum value of $\\sin B$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\dfrac{ \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 5063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$ ,\n$(*)$ $\\frac{S_{m+n}}{m+n}=\\frac{S_m}{m}\\frac{S_n}{n}$ \nfor $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,... | math | {
"style": "rule",
"ground_truth": "(m, n) = (5, 2), (2, 5), (3, 2), (2, 3)"
} | {
"split": "train",
"index": 5064,
"solution": "Claim Both $m,n$ can not be even.\nProof $x+y+z=0$ , $\\implies x=-(y+z)$ .\nSince $\\frac{S_{m+n}}{m+n} = \\frac{S_m S_n}{mn}$ ,\nby equating cofficient of $y^{m+n}$ on LHS and RHS ,get\n$\\frac{2}{m+n}=\\frac{4}{mn}$ .\n$\\implies \\frac{m}{2} + \\frac {n}{2} = \... |
deepscaler | [
{
"role": "user",
"content": "From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without stopping headed towards point $B$ and caught ... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 5065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a triangle is 600 square feet. Find the altitude, in feet, of the triangle if the length of the corresponding base is 30 feet. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 5066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sqrt{x}+\\frac{1}{\\sqrt{x}}=3$, determine the value of $\\frac{x}{x^{2}+2018 x+1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2025}"
} | {
"split": "train",
"index": 5067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M = 123456789101112\\dots5354$ be the number that results from writing the integers from $1$ to $54$ consecutively. What is the remainder when $M$ is divided by $55$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 5068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular quadrilateral pyramid $P-ABCD$, if $\\angle APC = 60^{\\circ}$, find the cosine of the dihedral angle between planes $A-PB-C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{7}"
} | {
"split": "train",
"index": 5069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the simplified value of the sum: $-1^{2004} + (-1)^{2005} + 1^{2006} -1^{2007}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 5070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to partition $7$ students into the groups of $2$ or $3$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 5071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and line $l$ passes through points $(a, 0)$ and $(0, b)$. The sum of the distances from $F_1$ and $F_2$ to line $l$ is $\\frac{4c}{5}$. Determine th... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{21}}{21}"
} | {
"split": "train",
"index": 5072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\sqrt[4]{16} \\cdot \\sqrt[3]{8} \\cdot \\sqrt{4}$ expressed as a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag has 4 red marbles, 5 white marbles, and 6 blue marbles. Three marbles are drawn from the bag (without replacement). What is the probability that they are all the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{34}{455}"
} | {
"split": "train",
"index": 5074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the positive difference between the median and the mode of the data given in the stem and leaf plot below? In this plot $5|8$ represents $58.$\n\n\\begin{tabular}{|c|c|}\\hline\n\\textbf{Tens} & \\textbf{Units} \\\\ \\hline\n1 & $2 \\hspace{2mm} 3 \\hspace{2mm} 4 \\hspa... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 5075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "4. Suppose that $\\overline{A2021B}$ is a six-digit integer divisible by $9$ . Find the maximum possible value of $A \\cdot B$ .\n\n5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of int... | math | {
"style": "rule",
"ground_truth": "143"
} | {
"split": "train",
"index": 5076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ctibor marked a square land plot on a map with a scale of 1:50000 and calculated that its side corresponds to $1 \\mathrm{~km}$ in reality. He then resized the map on a copier such that the marked square had an area $1.44 \\mathrm{~cm}^{2}$ smaller than the original.\n\nWhat wa... | math | {
"style": "rule",
"ground_truth": "1:62500"
} | {
"split": "train",
"index": 5077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct primes, each greater than 20, are multiplied. What is the least possible product of these two primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"split": "train",
"index": 5078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of $\\cos x + 2 \\sin x,$ over all angles $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 5079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine which of the following expressions has the largest value: $4^2$, $4 \\times 2$, $4 - 2$, $\\frac{4}{2}$, or $4 + 2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5080,
"solution": "We evaluate each of the five choices: $4^{2}=16$, $4 \\times 2=8$, $4-2=2$, $\\frac{4}{2}=2$, $4+2=6$. Of these, the largest is $4^{2}=16$."
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the least positive base ten number which requires six digits for its binary representation? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 5081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\begin{pmatrix} 1 & -4 \\\\ 3 & -2 \\end{pmatrix} = \\begin{pmatrix} -16 & -6 \\\\ 7 & 2 \\end{pmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 5 & -7 \\\\ -2 & 3 \\end{pmatrix}"
} | {
"split": "train",
"index": 5082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain entertainment unit, each member can sing or dance at least one of the two. It is known that there are 4 people who can sing and 5 people who can dance. Now, 2 people are selected from them to participate in a social charity performance. Let $\\xi$ be the number of ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"split": "train",
"index": 5083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(Q\\) be a point chosen uniformly at random inside the unit square with vertices at \\((0,0), (1,0), (1,1)\\), and \\((0,1)\\). Calculate the probability that the slope of the line determined by \\(Q\\) and the point \\(\\left(\\frac{1}{4}, \\frac{3}{4}\\right)\\) is grea... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 5084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair standard four-sided die is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one \"3\" is rolled?\nA) $\\frac{1}{12}$\nB) $\\frac{1}{6}$\nC) $\\frac{3}{10}$\nD) $\\frac{1}{4}$\nE) $\\frac{1}{2}$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 5085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On square $ABCD$, point $E$ lies on side $AD$ and point $F$ lies on side $BC$, so that $BE=EF=FD=30$. Find the area of the square $ABCD$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "810"
} | {
"split": "train",
"index": 5086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest number of positive, consecutive integers whose sum is 105? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-congruent squares can be drawn, such that their vertices are lattice points on the 5 by 5 grid of lattice points shown? [asy]\ndot((0,0));dot((1,0));dot((2,0));dot((3,0));dot((4,0));\ndot((0,1));dot((1,1));dot((2,1));dot((3,1));dot((4,1));\ndot((0,2));dot((1,2));do... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the following expression: \\[2x+3x^2+1-(6-2x-3x^2).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6x^2+4x-5"
} | {
"split": "train",
"index": 5089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\( x \\) and \\( y \\) are integers such that \\( xy + 3x + 2y = -4 \\). Find the greatest possible value of \\( y \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two identical cylindrical vessels are connected by a small tube with a valve at the bottom. Initially, the valve is closed, and water is poured into the first vessel while oil is poured into the second vessel, such that the liquid levels are equal and are $h=40$ cm. At what lev... | math | {
"style": "rule",
"ground_truth": "32.94"
} | {
"split": "train",
"index": 5091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $\\mathcal{C}_{1}$ and $\\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6)$, and the product of the radii is $68$. The x-axis and the line $y = mx$, where $m > 0$, are tangent to both circles. It is given that $m$ can be written in the form $a\\sqrt {b}/c... | math | {
"style": "rule",
"ground_truth": "282"
} | {
"split": "train",
"index": 5092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two spheres touch the plane of triangle \\(ABC\\) at points \\(B\\) and \\(C\\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 12, and the distance between their centers is \\(4 \\sqrt{29}\\). The center of a third sphere with radius 8... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 5093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $(x + 1)^{2010}$ is divided by $x^2 + x + 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 5094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nConsider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"split": "train",
"index": 5095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $\\angle Y = 90^\\circ$, $YZ = 4$, and $XY = 5$. What is $\\tan X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 5096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the probability that Margo is paired with her best friend, Irma? Express your answer as a common fraction. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{29}"
} | {
"split": "train",
"index": 5097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers \\( n \\) exist such that both \\(\\frac{n+1}{3}\\) and \\(3n+1\\) are three-digit integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 5099,
"solution": "The equation given can be rewritten as:\n$40|x| = - y^2 - 2xy + 400$\n We can split the equation into a piecewise equation by breaking up the absolute value:\n$40x = -y^2 - 2xy + 400\\quad\\quad x\\ge 0$\n$40x = y^2 + 2xy - 400 \\quad \\quad x < 0$\n Factoring the f... |
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