data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A school organizes a social practice activity during the summer vacation and needs to allocate 8 grade 10 students to company A and company B evenly. Among these students, two students with excellent English scores cannot be allocated to the same company, and neither can the th... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 3300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school organizes a table tennis competition. Class A has 5 male students and 3 female students registered; Class B has 6 male students and 2 female students registered. If 2 students are selected from each of Class A and Class B, there are a total of $\\boxed{345}$ different ... | math | {
"style": "rule",
"ground_truth": "345"
} | {
"index": 3301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school plans to purchase two brands of soccer balls, brand A and brand B. It is known that the unit price of brand A soccer balls is $30 less than the unit price of brand B soccer balls. The quantity of brand A soccer balls that can be purchased with $1000 is the same as the ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 3302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school plans to purchase volleyball and basketball in one go. The price of each basketball is $30$ yuan more expensive than a volleyball. It costs a total of $340$ yuan to buy $2$ volleyballs and $3$ basketballs.<br/>$(1)$ Find the price of each volleyball and basketball.<br/... | math | {
"style": "rule",
"ground_truth": "3660"
} | {
"index": 3303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school program will randomly start between 8:30AM and 9:30AM and will randomly end between 7:00PM and 9:00PM. What is the probability that the program lasts for at least 11 hours and starts before 9:00AM?"
}
] | math | {
"style": "rule",
"ground_truth": "5/16"
} | {
"index": 3304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where A and B cannot go together, and A and C can only go together or not go at all. The total number of different dispatch plans is \\_\\_\\_\\_\\_\\_ (answer in numbers)."
... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school selects 4 teachers from 8 to teach in 4 remote areas, with one teacher per area. Among them, A and B cannot go together, and A and C must either both go or both not go. Derive the total number of different dispatch plans."
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 3307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school table tennis championship was held using the Olympic system. The winner won 6 matches. How many participants in the championship won more matches than they lost? (In the first round of the championship, conducted using the Olympic system, participants are divided into ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 3308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school uses a systematic sampling method to conduct a vision test on 50 out of the 800 students in the first year. The 800 students are numbered from 1 to 800 and are evenly divided into 50 groups in ascending order of their numbers, with group numbers from 1 to 50. It is kno... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"index": 3309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school wants to understand the psychological state of its senior high school students regarding their studies. They decide to use a systematic sampling method to select 40 students out of 800 for a certain test. The students are randomly numbered from 1 to 800. After grouping... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A school wishes to understand the psychological state of learning among its senior students and adopts a systematic sampling method to select 40 students out of 800 for a test. The students are randomly assigned numbers from 1 to 800 and then grouped. In the first group, number... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 3311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A science student is asked to find the coefficient of the $x^2$ term in the expansion of $(x^2-3x+2)^4$. The coefficient is \\_\\_\\_\\_\\_\\_. (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "248"
} | {
"index": 3312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 3313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes ... | math | {
"style": "rule",
"ground_truth": "24.2"
} | {
"index": 3314,
"solution": "1. **Identify the relationship between the trees' heights:** Each tree is either twice as tall or half as tall as the one to its right. This means that for any tree $i$ and tree $i+1$, the height of tree $i$ is either $2 \\times \\text{height of tree } i+1$ or $\\frac{1}{2} \\times \\t... |
DeepScaleR | [
{
"role": "user",
"content": "A scout troop buys $1000$ candy bars at a price of five for $2$ dollars. They sell all the candy bars at the price of two for $1$ dollar. What was their profit, in dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 3315,
"solution": "1. **Calculate the cost price of the candy bars:**\n The troop buys candy bars at a rate of 5 bars for $2 dollars. Therefore, the cost per bar is:\n \\[\n \\frac{2 \\text{ dollars}}{5 \\text{ bars}} = 0.4 \\text{ dollars per bar}\n \\]\n Since they buy 1000 bars, the total co... |
DeepScaleR | [
{
"role": "user",
"content": "A seafood wholesaler purchased 1000 kg of crucian carp at a price of 10 yuan/kg and cultured them in a lake (assuming that the weight of each crucian carp remains unchanged during the culture period). According to market estimates, the market price of crucian carp after lake cu... | math | {
"style": "rule",
"ground_truth": "6000"
} | {
"index": 3316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \\(1 \\mathrm{~cm}\\) to a cylinder of radius \\(3 \\mathrm{~cm}\\). When the bottle is right side up, the height of the water inside is \\(20 \\mathrm{~cm}\\). When the bottle is upsi... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 3317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sealed bottle, which contains water, has been constructed by attaching a cylinder of radius \\(1 \\text{ cm}\\) to a cylinder of radius \\(3 \\text{ cm}\\). When the bottle is right side up, the height of the water inside is \\(20 \\text{ cm}\\). When the bottle is upside dow... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 3318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A secret agent is trying to decipher a passcode. So far, he has obtained the following information:\n\n- It is a four-digit number.\n- It is not divisible by seven.\n- The digit in the tens place is the sum of the digit in the units place and the digit in the hundreds place.\n-... | math | {
"style": "rule",
"ground_truth": "4583"
} | {
"index": 3319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A secret facility is a rectangle measuring $200 \\times 300$ meters. Outside the facility, there is one guard at each of the four corners. An intruder approached the perimeter of the facility from the outside, and all the guards ran towards the intruder using the shortest paths... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A secret facility is a rectangle measuring $200 \\times 300$ meters. There is one guard at each of the four corners of the facility. An intruder approaches the perimeter of the facility from the outside, and all the guards run towards the intruder by the shortest routes along t... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A secret facility is in the shape of a rectangle measuring $200 \\times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by t... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 3322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A section is cut out of a circular piece of paper having radius four inches, as shown. Points A and B are then glued together to form a right circular cone. What is the circumference of the base of the resulting cone? Express your answer in terms of $\\pi$. (The $270^\\circ$ s... | math | {
"style": "rule",
"ground_truth": "6 \\pi"
} | {
"index": 3323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A section of a book fell out. The first page of this section is numbered 143, and the number of the last page consists of the same digits but in a different order. How many pages fell out of the book?"
}
] | math | {
"style": "rule",
"ground_truth": "172"
} | {
"index": 3324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sector with a central angle of 135° has an area of $S_1$, and the total surface area of the cone formed by it is $S_2$. Find the value of $\\frac{S_{1}}{S_{2}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{11}"
} | {
"index": 3325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sector with acute central angle $\\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is\n$\\textbf{(A)}\\ 3\\cos\\theta \\qquad \\textbf{(B)}\\ 3\\sec\\theta \\qquad \\textbf{(C)}\\ 3 \\cos \\frac12 \\theta \\qquad \\textbf{(D)}\... | math | {
"style": "rule",
"ground_truth": "3 \\sec \\frac{1}{2} \\theta"
} | {
"index": 3326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sector with central angle $60^\\circ$ is cut from a circle of radius 10. Determine both the radius of the circle circumscribed about the sector and the inradius of the triangle formed by the sector."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{3}}{3}"
} | {
"index": 3327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A segment \\( AB \\) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \\( \\pi / 3 \\) to the diameter \\( CD \\) of this sphere. The distance from the end \\( C \\) of the diameter to the nearest end \\( A \\) of the chord \\( AB \\) is ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 3328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the length of the common chord, given that the radii of the circles are in the ratio \\(4: 3\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{23}"
} | {
"index": 3329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is:"
}
] | math | {
"style": "rule",
"ground_truth": "x < \\frac{1}{2}"
} | {
"index": 3330,
"solution": "To determine the conditions under which four segments can form a quadrilateral, we need to consider the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For four sides to form a quadr... |
DeepScaleR | [
{
"role": "user",
"content": "A segment with endpoints at $A(2, -2)$ and $B(14, 4)$ is extended through $B$ to point $C$. If $BC = \\frac{1}{3} \\cdot AB$, what are the coordinates for point $C$? Express your answer as an ordered pair."
}
] | math | {
"style": "rule",
"ground_truth": "(18,6)"
} | {
"index": 3331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A self-employed individual plans to distribute two types of products, A and B. According to a survey, when the investment amount is $x$ (where $x \\geq 0$) in ten thousand yuan, the profits obtained from distributing products A and B are $f(x)$ and $g(x)$ in ten thousand yuan, ... | math | {
"style": "rule",
"ground_truth": "12.6"
} | {
"index": 3332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A self-employed person plans to distribute two types of goods, A and B. According to a survey, when the investment amount is $x(x\\geqslant 0)$ ten thousand yuan, the profits obtained from distributing goods A and B are $f(x)$ ten thousand yuan and $g(x)$ ten thousand yuan, res... | math | {
"style": "rule",
"ground_truth": "12.6"
} | {
"index": 3333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{5}"
} | {
"index": 3334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle has diameter $XY$ . A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$ , $U$ on the semicircle, and $V$ on $XY$ . What is the area of $STUV$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 3335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle is inscribed in a quarter circle. What fraction of the quarter circle is shaded?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{2}{3}$"
} | {
"index": 3336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{120}{17}"
} | {
"index": 3337,
"solution": "1. **Identify the Triangle and its Properties**: \n Let's denote the isosceles triangle as $\\triangle ABD$ where $AB = AD$ and $BD = 16$. The height from $A$ to $BD$ is given as $15$. Let $M$ be the midpoint of $BD$. \n\n2. **Divide the Triangle**:\n Draw a line from $A$ to $M$. T... |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\\textit{lune}$. Determine the area of this lune. Express your answer in terms of $\\pi$ and in simpl... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4} - \\frac{1}{24}\\pi"
} | {
"index": 3338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle of diameter 3 sits at the top of a semicircle of diameter 4, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a $\\textit{lune}$. Determine the area of this lune. Express your answer in terms of $\\pi$ and in simpl... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{24}\\pi"
} | {
"index": 3339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 3340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \\sqrt{n},$ find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "544"
} | {
"index": 3341,
"solution": "It is easy after getting the image, after drawing labeling the lengths of those segments, assume the radius is $x$, we can see $x=\\sqrt{2}(8-x)$ and we get $2x=32-\\sqrt{512}$ and we have the answer $\\boxed{544}$ ~bluesoul"
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\\angle AOC < \\angle AOD = 90^{\\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $... | math | {
"style": "rule",
"ground_truth": "673"
} | {
"index": 3342,
"solution": "We are given $$m \\angle EOC = m \\angle COD$$ and $$m \\angle AOC + m \\angle COD = 2m \\angle EOC + m \\angle COD = 90^{\\circ}$$ So $m \\angle EOC = 30^{\\circ}$ and $m \\angle AOC = 60^{\\circ}$. Letting the radius of the semicircle be $R$, we have $$(R-r) \\sin \\angle AOC = r \\R... |
DeepScaleR | [
{
"role": "user",
"content": "A semicircular sheet of iron with a radius of 6 is rolled into the lateral surface of a cone. The volume of this cone is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "9\\sqrt{3}\\pi"
} | {
"index": 3343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semicircular sponge with a diameter of $20 \\text{ cm}$ is used to wipe a corner of a room's floor such that the ends of the diameter continuously touch the two walls forming a right angle. What area does the sponge wipe?"
}
] | math | {
"style": "rule",
"ground_truth": "100\\pi"
} | {
"index": 3344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A semipro baseball league has teams with $21$ players each. League rules state that a player must be paid at least $15,000$ dollars, and that the total of all players' salaries for each team cannot exceed $700,000$ dollars. What is the maximum possible salary, in dollars, for a... | math | {
"style": "rule",
"ground_truth": "400000"
} | {
"index": 3345,
"solution": "1. **Understand the constraints**: Each team has $21$ players. Each player must be paid at least $15,000$ dollars. The total salary for all players on a team cannot exceed $700,000$ dollars.\n\n2. **Set up the equation**: To find the maximum possible salary for a single player, assume ... |
DeepScaleR | [
{
"role": "user",
"content": "A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least $15,000 and that the total of all players' salaries for each team cannot exceed $700,000. What is the maximum possible salary, in dollars, for a single player?"
}
... | math | {
"style": "rule",
"ground_truth": "400,000"
} | {
"index": 3346,
"solution": "1. **Identify the constraints**: Each team has 21 players, and each player must be paid at least $15,000. The total salary for all players on a team cannot exceed $700,000.\n\n2. **Calculate the minimum total salary for 20 players**: If each of the 20 players receives the minimum salar... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $ (S_n), n \\geq 1$ of sets of natural numbers with $ S_1 = \\{1\\}, S_2 = \\{2\\}$ and\r\n\\[{ S_{n + 1} = \\{k \\in }\\mathbb{N}|k - 1 \\in S_n \\text{ XOR } k \\in S_{n - 1}\\}.\r\n\\]\r\nDetermine $ S_{1024}.$"
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 3347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $(a_n)$ is defined as follows: $a_1 = 1,$ $a_2 = \\frac{1}{2},$ and\n\\[a_n = \\frac{1 - a_{n - 1}}{2a_{n - 2}}\\]for all $n \\ge 3.$ Find $a_{120}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{20}{41}"
} | {
"index": 3348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\\geq1$, we have\n$$a_{n+1}=\\frac{n-1}{n+1}a_n-\\frac{n-2}{n^2+n}a_{n-1}.$$\nCalculate the value of $\\frac{a_1}{a_2}-\\frac{a_2}{a_3}+\\frac{a_3}{a_4}-\\frac{a_4}{a_5}+\\ldots+\\frac{a_{2013}}{... | math | {
"style": "rule",
"ground_truth": "3021"
} | {
"index": 3349,
"solution": "\nWe start by examining the sequence \\((a_n)\\) given by the recurrence relations \\(a_0 = 1\\) and \\(a_1 = 2015\\), with the following recursive formula for \\(n \\geq 1\\):\n\n\\[\na_{n+1} = \\frac{n-1}{n+1}a_n - \\frac{n-2}{n^2+n}a_{n-1}.\n\\]\n\nThe goal is to evaluate the expres... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $(c_n)$ is defined as follows: $c_1 = 1$, $c_2 = \\frac{1}{3}$, and\n\\[c_n = \\frac{2 - c_{n-1}}{3c_{n-2}}\\] for all $n \\ge 3$. Find $c_{100}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 3350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $\\left\\{a_{n}\\right\\}_{n \\geq 1}$ of positive reals is defined by the rule $a_{n+1} a_{n-1}^{5}=a_{n}^{4} a_{n-2}^{2}$ for integers $n>2$ together with the initial values $a_{1}=8$ and $a_{2}=64$ and $a_{3}=1024$. Compute $$\\sqrt{a_{1}+\\sqrt{a_{2}+\\sqrt{a_{3}... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"index": 3351,
"solution": "Taking the base-2 $\\log$ of the sequence $\\left\\{a_{n}\\right\\}$ converts the multiplicative rule to a more familiar additive rule: $\\log _{2}\\left(a_{n+1}\\right)-4 \\log _{2}\\left(a_{n}\\right)+5 \\log _{2}\\left(a_{n-1}\\right)-2 \\log _{2}\\left(a_{n-2}\\right)=0$. The chara... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $a_1$, $a_2$, $\\ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\\geq1$. If $a_1=999$, $a_2<999$, and $a_{2006}=1$, how many different values of $a_2$ are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 3352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \\ge 3,$\n\\[a_k = \\frac{1}{3} a_{k - 1} + \\frac{1}{4} a_{k - 2}.\\]Evaluate $a_1 + a_2 + a_3 + \\dotsb.$"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 3353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of positive reals satisfies $a_{n+1}=\\sqrt{\\frac{1+a_{n}}{2}}$. Determine all $a_{1}$ such that $a_{i}=\\frac{\\sqrt{6}+\\sqrt{2}}{4}$ for some positive integer $i$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}+\\sqrt{6}}{2}, \\frac{\\sqrt{3}}{2}, \\frac{1}{2}"
} | {
"index": 3354,
"solution": "Clearly $a_{1}<1$, or else $1 \\leq a_{1} \\leq a_{2} \\leq a_{3} \\leq \\ldots$ We can therefore write $a_{1}=\\cos \\theta$ for some $0<\\theta<90^{\\circ}$. Note that $\\cos \\frac{\\theta}{2}=\\sqrt{\\frac{1+\\cos \\theta}{2}}$, and $\\cos 15^{\\circ}=$ $\\frac{\\sqrt{6}+\\sqrt{2}}... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $b_1, b_2, b_3, \\dots,$ is defined recursively by $b_1 = 3,$ $b_2 = 2,$ and for $k \\ge 3,$\n\\[ b_k = \\frac{1}{4} b_{k-1} + \\frac{2}{5} b_{k-2}. \\]\nEvaluate $b_1 + b_2 + b_3 + \\dotsb.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{85}{7}"
} | {
"index": 3355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $s_{0}, s_{1}, s_{2}, s_{3}, \\ldots$ is defined by $s_{0}=s_{1}=1$ and, for every positive integer $n, s_{2 n}=s_{n}, s_{4 n+1}=s_{2 n+1}, s_{4 n-1}=s_{2 n-1}+s_{2 n-1}^{2} / s_{n-1}$. What is the value of $s_{1000}$?"
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 3356,
"solution": "720 Some experimentation with small values may suggest that $s_{n}=k$!, where $k$ is the number of ones in the binary representation of $n$, and this formula is in fact provable by a straightforward induction. Since $1000_{10}=1111101000_{2}$, with six ones, $s_{1000}=6!=720$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence $y_1,y_2,\\dots,y_k$ of real numbers is called \\emph{zigzag} if $k=1$, or if $y_2-y_1, y_3-y_2, \\dots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1,X_2,\\dots,X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a(X_1,X_2,\\dots,X... | math | {
"style": "rule",
"ground_truth": "\\frac{2n+2}{3}"
} | {
"index": 3357,
"solution": "The expected value is $\\frac{2n+2}{3}$. Divide the sequence $X_1,\\dots,X_n$ into alternating increasing and decreasing segments, with $N$ segments in all. Note that removing one term cannot increase $N$: if the removed term is interior to some segment then the number remains unchange... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence \\( b_1, b_2, b_3, \\dots \\) is defined recursively by \\( b_1 = 2, b_2 = 2, \\) and for \\( k \\ge 3, \\)\n\\[\nb_k = \\frac{1}{2} b_{k - 1} + \\frac{1}{3} b_{k - 2}.\n\\]\nEvaluate \\( b_1 + b_2 + b_3 + \\dotsb. \\)"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 3358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence \\( b_1, b_2, b_3, \\ldots \\) is defined recursively by \\( b_1 = 2 \\), \\( b_2 = 3 \\), and for \\( k \\geq 3 \\),\n\\[ b_k = \\frac{1}{2} b_{k-1} + \\frac{1}{3} b_{k-2}. \\]\nEvaluate \\( b_1 + b_2 + b_3 + \\dotsb. \\)"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 3359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence \\(a_1\\), \\(a_2\\), \\(\\ldots\\) of non-negative integers is defined by the rule \\(a_{n+2}=|a_{n+1}-a_n|\\) for \\(n\\geq1\\). If \\(a_1=1010\\), \\(a_2<1010\\), and \\(a_{2023}=0\\), how many different values of \\(a_2\\) are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "399"
} | {
"index": 3360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence begins with 3, and each subsequent term is triple the sum of all preceding terms. Determine the first term in the sequence that exceeds 15000."
}
] | math | {
"style": "rule",
"ground_truth": "36864"
} | {
"index": 3361,
"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?"
}
] | math | {
"style": "rule",
"ground_truth": "2151"
} | {
"index": 3362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence consists of $2020$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2020$ terms is $5410$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?"
}
] | math | {
"style": "rule",
"ground_truth": "2200"
} | {
"index": 3363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence consists of $3000$ terms. Each term after the first is $3$ larger than the previous term. The sum of the $3000$ terms is $12000$. Calculate the sum when every third term is added up, starting with the first term and ending with the third to last term."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 3364,
"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. What is the sum when every second term is added up, starting with the first term and ending with the second last term?"
}
] | math | {
"style": "rule",
"ground_truth": "2151"
} | {
"index": 3365,
"solution": "We label the terms $x_{1}, x_{2}, x_{3}, \\ldots, x_{2009}, x_{2010}$. Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, $S=x_{1}+x_{3}+x_{5}+\\cdots+x_{2007}+x_{2009}$. We know that the sum of all of the terms is 5307; that is, $x_{1}+x_{2}+x_{3}+\\cdots+... |
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?"
}
] | math | {
"style": "rule",
"ground_truth": "2151"
} | {
"index": 3366,
"solution": "We label the terms $x_{1}, x_{2}, x_{3}, \\ldots, x_{2009}, x_{2010}$. Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, $S=x_{1}+x_{3}+x_{5}+\\cdots+x_{2007}+x_{2009}$. We know that the sum of all of the terms is 5307; that is, $x_{1}+x_{2}+x_{3}+\\cdots+... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence consists of the digits $122333444455555 \\ldots$ such that each positive integer $n$ is repeated $n$ times, in increasing order. Find the sum of the 4501st and 4052nd digits of this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 3367,
"solution": "Note that $n$ contributes $n \\cdot d(n)$ digits, where $d(n)$ is the number of digits of $n$. Then because $1+\\cdots+99=4950$, we know that the digits of interest appear amongst copies of two digit numbers. Now for $10 \\leq n \\leq 99$, the number of digits in the subsequence up to ... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence has 101 terms, each of which is a positive integer. If a term, $n$, is even, the next term is equal to $\\frac{1}{2}n+1$. If a term, $n$, is odd, the next term is equal to $\\frac{1}{2}(n+1)$. If the first term is 16, what is the 101st term?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 3368,
"solution": "The 1st term is 16. Since 16 is even, the 2nd term is $\\frac{1}{2} \\cdot 16+1=9$. Since 9 is odd, the 3rd term is $\\frac{1}{2}(9+1)=5$. Since 5 is odd, the 4th term is $\\frac{1}{2}(5+1)=3$. Since 3 is odd, the 5th term is $\\frac{1}{2}(3+1)=2$. Since 2 is even, the 6th term is $\\f... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence has terms $a_{1}, a_{2}, a_{3}, \\ldots$. The first term is $a_{1}=x$ and the third term is $a_{3}=y$. The terms of the sequence have the property that every term after the first term is equal to 1 less than the sum of the terms immediately before and after it. What ... | math | {
"style": "rule",
"ground_truth": "2x+y+2015"
} | {
"index": 3369,
"solution": "Substituting $n=1$ into the equation $a_{n+1}=a_{n}+a_{n+2}-1$ gives $a_{2}=a_{1}+a_{3}-1$. Since $a_{1}=x$ and $a_{3}=y$, then $a_{2}=x+y-1$. Rearranging the given equation, we obtain $a_{n+2}=a_{n+1}-a_{n}+1$ for each $n \\geq 1$. Thus, $a_{4}=a_{3}-a_{2}+1=y-(x+y-1)+1=2-x$, $a_{5}=a... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined as follows $a_1=a_2=a_3=1,$ and, for all positive integers $n, a_{n+3}=a_{n+2}+a_{n+1}+a_n.$ Given that $a_{28}=6090307, a_{29}=11201821,$ and $a_{30}=20603361,$ find the remainder when $\\sum^{28}_{k=1} a_k$ is divided by 1000."
}
] | math | {
"style": "rule",
"ground_truth": "834"
} | {
"index": 3370,
"solution": "Since the problem only asks for the first 28 terms and we only need to calculate mod 1000, we simply bash the first 28 terms:\n$a_{1}\\equiv 1 \\pmod {1000} \\newline a_{2}\\equiv 1 \\pmod {1000} \\newline a_{3}\\equiv 1 \\pmod {1000} \\newline a_{4}\\equiv 3 \\pmod {1000} \\newline a_... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined as follows: $a_1=a_2=a_3=1$, and, for all positive integers $n$, $a_{n+3}=a_{n+2}+a_{n+1}+a_n$. Given that $a_{28}= 6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\\displaystyle \\sum_{k=1}^{28}a_k$ is divided by 1000."
}
] | math | {
"style": "rule",
"ground_truth": "834"
} | {
"index": 3371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined by $A_{0}=0, A_{1}=1, A_{2}=2$, and, for integers $n \\geq 3$, $$A_{n}=\\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\\frac{1}{n^{4}-n^{2}}$$ Compute $\\lim _{N \\rightarrow \\infty} A_{N}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{6}-\\frac{\\pi^{2}}{12}"
} | {
"index": 3372,
"solution": "If we sum the given equation for $n=3,4,5, \\ldots, N$, we obtain $$\\sum_{n=3}^{N} A_{n}=\\sum_{n=3}^{N} \\frac{A_{n-1}+A_{n-2}+A_{n-3}}{3}+\\frac{1}{n^{4}-n^{2}}$$ This reduces dramatically to $$\\begin{equation*} A_{N}+\\frac{2 A_{N-1}}{3}+\\frac{A_{N-2}}{3}=A_{2}+\\frac{2 A_{1}}{3}... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined by $a_0 = \\frac{1}{2}$ and $a_n = 1 + (a_{n - 1} - 1)^2.$ Compute\n\\[a_0 a_1 a_2 \\dotsm.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 3373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined by $a_{0}=1$ and $a_{n}=2^{a_{n-1}}$ for $n \\geq 1$. What is the last digit (in base 10) of $a_{15}$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 3374,
"solution": "6. Certainly $a_{13} \\geq 2$, so $a_{14}$ is divisible by $2^{2}=4$. Writing $a_{14}=4 k$, we have $a_{15}=2^{4 k}=16^{k}$. But every power of 16 ends in 6, so this is the answer."
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined over non-negative integral indexes in the following way: $a_{0}=a_{1}=3$, $a_{n+1}a_{n-1}=a_{n}^{2}+2007$.\n Find the greatest integer that does not exceed $\\frac{a_{2006}^{2}+a_{2007}^{2}}{a_{2006}a_{2007}}.$"
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"index": 3375,
"solution": "We are given that\n$a_{n+1}a_{n-1}= a_{n}^{2}+2007$, $a_{n-1}^{2}+2007 = a_{n}a_{n-2}$.\n Add these two equations to get\n$a_{n-1}(a_{n-1}+a_{n+1}) = a_{n}(a_{n}+a_{n-2})$\n$\\frac{a_{n+1}+a_{n-1}}{a_{n}}= \\frac{a_{n}+a_{n-2}}{a_{n-1}}$.\n This is an invariant. Defining $b_{i}= \\frac... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is defined recursively as follows: \\( t_{1} = 1 \\), and for \\( n > 1 \\):\n\n- If \\( n \\) is even, \\( t_{n} = 1 + t_{\\frac{n}{2}} \\).\n- If \\( n \\) is odd, \\( t_{n} = \\frac{1}{t_{n-1}} \\).\n\nGiven that \\( t_{n} = \\frac{19}{87} \\), find the sum of the... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 3376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence is formed by arranging the following arrays into a row: $$( \\frac {1}{1}), ( \\frac {2}{1}, \\frac {1}{2}), ( \\frac {3}{1}, \\frac {2}{2}, \\frac {1}{3}), ( \\frac {4}{1}, \\frac {3}{2}, \\frac {2}{3}, \\frac {1}{4}), ( \\frac {5}{1}, \\frac {4}{2}, \\frac {3}{3}, ... | math | {
"style": "rule",
"ground_truth": "\\frac {5}{59}"
} | {
"index": 3377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of 11 positive real numbers, $a_{1}, a_{2}, a_{3}, \\ldots, a_{11}$, satisfies $a_{1}=4$ and $a_{11}=1024$ and $a_{n}+a_{n-1}=\\frac{5}{2} \\sqrt{a_{n} \\cdot a_{n-1}}$ for every integer $n$ with $2 \\leq n \\leq 11$. For example when $n=7, a_{7}+a_{6}=\\frac{5}{2} \... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3378,
"solution": "Suppose that, for some integer $n \\geq 2$, we have $a_{n}=x$ and $a_{n-1}=y$. The equation $a_{n}+a_{n-1}=\\frac{5}{2} \\sqrt{a_{n} \\cdot a_{n-1}}$ can be re-written as $x+y=\\frac{5}{2} \\sqrt{x y}$. Since $x>0$ and $y>0$, squaring both sides of the equation gives an equivalent equa... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of 2020 natural numbers is written in a row. Each of them, starting from the third number, is divisible by the previous one and by the sum of the two preceding ones.\n\nWhat is the smallest possible value for the last number in the sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "2019!"
} | {
"index": 3379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of distinct circles $\\omega_1, \\omega_2, \\cdots$ is inscribed in the parabola $y=x^2$ so that $\\omega_n$ and $\\omega_{n+1}$ are tangent for all $n$ . If $\\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$ , find the diameter of $\\om... | math | {
"style": "rule",
"ground_truth": "3995"
} | {
"index": 3380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of figures is formed using tiles. Each tile is an equilateral triangle with side length 7 cm. The first figure consists of 1 tile. Each figure after the first is formed by adding 1 tile to the previous figure. How many tiles are used to form the figure in the sequenc... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 3381,
"solution": "The first figure consists of one tile with perimeter $3 \\times 7 \\mathrm{~cm} = 21 \\mathrm{~cm}$. Each time an additional tile is added, the perimeter of the figure increases by 7 cm (one side length of a tile), because one side length of the previous figure is 'covered up' and two ... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of functions $\\, \\{f_n(x) \\} \\,$ is defined recursively as follows: \\begin{align*} f_1(x) &= \\sqrt {x^2 + 48}, \\quad \\text{and} \\\\ f_{n + 1}(x) &= \\sqrt {x^2 + 6f_n(x)} \\quad \\text{for } n \\geq 1. \\end{align*} (Recall that $\\sqrt {\\makebox[5mm]{}}$ ... | math | {
"style": "rule",
"ground_truth": "\\[ x = 4 \\]"
} | {
"index": 3382,
"solution": "We define $f_0(x) = 8$ . Then the recursive relation holds for $n=0$ , as well.\nSince $f_n (x) \\ge 0$ for all nonnegative integers $n$ , it suffices to consider nonnegative values of $x$ .\nWe claim that the following set of relations hold true for all natural numbers $n$ and nonneg... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492?"
}
] | math | {
"style": "rule",
"ground_truth": "986"
} | {
"index": 3383,
"solution": "The problem gives us a sequence defined by a recursion, so let's calculate a few values to get a feel for how it acts. We aren't given initial values, so let $a_1 = a$ and $a_2 = b$. Then $a_3 = b - a$, $a_4 = (b - a) - b = -a$, $a_5 = -a - (b - a) = -b$, $a_6 = -b - (-a) = a - b$, $a_... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of integers $a_1, a_2, a_3, \\ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \\ge 3.$ What is the sum of the first $2001$ terms of this sequence if the sum of the first $1492$ terms is $1985,$ and the sum of the first $1985$ terms is $1492$?"
}
... | math | {
"style": "rule",
"ground_truth": "986"
} | {
"index": 3384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of integers has a mode of 32, a mean of 22, a smallest number of 10, and a median of \\( m \\). If \\( m \\) is replaced by \\( m+10 \\), the new sequence has a mean of 24 and a median of \\( m+10 \\). If \\( m \\) is replaced by \\( m-8 \\), the new sequence has a m... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 3385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of integers is defined as follows: $a_i = i$ for $1 \\le i \\le 5,$ and\n\\[a_i = a_1 a_2 \\dotsm a_{i - 1} - 1\\]for $i > 5.$ Evaluate $a_1 a_2 \\dotsm a_{2011} - \\sum_{i = 1}^{2011} a_i^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "-1941"
} | {
"index": 3386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of length 15 consisting of the letters $A$ and $B$ satisfies the following conditions:\n\nFor any two consecutive letters, $AA$ appears 5 times, $AB$, $BA$, and $BB$ each appear 3 times. How many such sequences are there?\n\nFor example, in $AA B B A A A A B A A B B ... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"index": 3387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of natural numbers $\\left\\{x_{n}\\right\\}$ is constructed according to the following rules:\n$$\nx_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \\text{ for } n \\geq 1.\n$$\n\nIt is known that some term in the sequence is 1000. What is the smallest possible value of $a+... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers $t_{1}, t_{2}, t_{3}, \\ldots$ has its terms defined by $t_{n}=\\frac{1}{n}-\\frac{1}{n+2}$ for every integer $n \\geq 1$. What is the largest positive integer $k$ for which the sum of the first $k$ terms is less than 1.499?"
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 3389,
"solution": "We note that $t_{1}=\\frac{1}{1}-\\frac{1}{3}=\\frac{2}{3} \\approx 0.67$, $t_{1}+t_{2}=\\left(\\frac{1}{1}-\\frac{1}{3}\\right)+\\left(\\frac{1}{2}-\\frac{1}{4}\\right)=\\frac{2}{3}+\\frac{1}{4}=\\frac{11}{12} \\approx 0.92$, $t_{1}+t_{2}+t_{3}=\\left(\\frac{1}{1}-\\frac{1}{3}\\right)... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime posi... | math | {
"style": "rule",
"ground_truth": "173"
} | {
"index": 3390,
"solution": "Let the sum of all of the terms in the sequence be $\\mathbb{S}$. Then for each integer $k$, $x_k = \\mathbb{S}-x_k-k \\Longrightarrow \\mathbb{S} - 2x_k = k$. Summing this up for all $k$ from $1, 2, \\ldots, 100$,\n\\begin{align*}100\\mathbb{S}-2(x_1 + x_2 + \\cdots + x_{100}) &= 1 + ... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers \\(a_1, a_2, \\cdots, a_n, \\cdots\\) is defined. Let \\(S(a_i)\\) be the sum of all the digits of \\(a_i\\). For example, \\(S(22) = 2 + 2 = 4\\). If \\(a_1 = 2017\\), \\(a_2 = 22\\), and \\(a_n = S(a_{n-1}) + S(a_{n-2})\\), what is the value of \\(a_{201... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 3391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers is arranged in the following pattern: \\(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \\cdots\\). Starting from the first number on the left, find the sum of the first 99 numbers."
}
] | math | {
"style": "rule",
"ground_truth": "1782"
} | {
"index": 3392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers is defined by $D_0=0,D_1=0,D_2=1$ and $D_n=D_{n-1}+D_{n-3}$ for $n\\ge 3$. What are the parities (evenness or oddness) of the triple of numbers $(D_{2021},D_{2022},D_{2023})$, where $E$ denotes even and $O$ denotes odd?"
}
] | math | {
"style": "rule",
"ground_truth": "(E,O,E)"
} | {
"index": 3393,
"solution": "1. **Define the sequence and initial conditions**: The sequence $\\{D_n\\}$ is defined by $D_0 = 0$, $D_1 = 0$, $D_2 = 1$, and for $n \\geq 3$, $D_n = D_{n-1} + D_{n-3}$.\n\n2. **Compute initial terms**: We calculate the first few terms of the sequence to observe any patterns, especial... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \\frac{3}{7}$, and\n\\[a_n=\\frac{a_{n-2} \\cdot a_{n-1}}{2a_{n-2} - a_{n-1}}\\]for all $n \\geq 3$. Then $a_{2019}$ can be written as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Wha... | math | {
"style": "rule",
"ground_truth": "8078"
} | {
"index": 3394,
"solution": "1. **Base Case Verification**: \n - For $n=1$, $a_1 = 1 = \\frac{3}{3}$, which matches the formula $\\frac{3}{4n-1}$ when $n=1$.\n - For $n=2$, $a_2 = \\frac{3}{7}$, which matches the formula $\\frac{3}{4n-1}$ when $n=2$.\n\n2. **Inductive Hypothesis**: \n - Assume that for some ... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of numbers is written on the blackboard: \\(1, 2, 3, \\cdots, 50\\). Each time, the first 4 numbers are erased, and the sum of these 4 erased numbers is written at the end of the sequence, creating a new sequence. This operation is repeated until there are fewer than... | math | {
"style": "rule",
"ground_truth": "755"
} | {
"index": 3395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of one hundred natural numbers $x, x+1, x+2, \\cdots, x+99$ has a sum $a$. If the sum of the digits of $a$ is 50, what is the smallest possible value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "99950"
} | {
"index": 3396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers $a_1,a_2,\\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$ , then $a_m$ is a divisor of $a_n$ and $a_m<a_n$ . Find the least possible value of $a_{2000}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 3397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers $a_{1}, a_{2}, \\ldots, a_{2017}$ has the property that for all integers $m$ where $1 \\leq m \\leq 2017,3\\left(\\sum_{i=1}^{m} a_{i}\\right)^{2}=\\sum_{i=1}^{m} a_{i}^{3}$. Compute $a_{1337}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ a_{1337} = 4011 \\]"
} | {
"index": 3398,
"solution": "I claim that $a_{i}=3 i$ for all $i$. We can conjecture that the sequence should just be the positive multiples of three because the natural numbers satisfy the property that the square of their sum is the sum of their cubes, and prove this by induction. At $i=1$, we have that $3 a_{i}... |
DeepScaleR | [
{
"role": "user",
"content": "A sequence of positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ satisfies $$a_{n+1}=n\\left\\lfloor\\frac{a_{n}}{n}\\right\\rfloor+1$$ for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_{1}$ take? (For a real number $x$, $\\lfloor x\\rfloor$ denotes t... | math | {
"style": "rule",
"ground_truth": "274"
} | {
"index": 3399,
"solution": "It is straightforward to show that if $a_{1}=1$, then $a_{n}=n$ for all $n$. Since $a_{n+1}$ is an increasing function in $a_{n}$, it follows that the set of possible $a_{1}$ is of the form $\\{1,2, \\ldots, m\\}$ for some $m$, which will be the answer to the problem. Consider the sequ... |
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