data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "The first term of a sequence is \\( a_{1} = 1 \\), and each subsequent term is defined by\n\n\\[ a_{n+1} = 1 + \\frac{n}{a_{n}}, \\quad n = 1, 2, 3, \\ldots \\]\n\nDoes the following limit exist? If it exists, determine it.\n\n\\[ \\lim_{n \\rightarrow \\infty} \\left(a_{n} - \... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first term of an arithmetic series of consecutive integers is $k^2 + 1$. The sum of $2k + 1$ terms of this series may be expressed as:"
}
] | math | {
"style": "rule",
"ground_truth": "$k^3 + (k + 1)^3$"
} | {
"index": 32401,
"solution": "To find the sum of the first $2k+1$ terms of an arithmetic series where the first term is $k^2 + 1$ and the common difference is 1 (since the series consists of consecutive integers), we use the formula for the sum of an arithmetic series:\n\n\\[ S_n = \\frac{n}{2} \\times (\\text{fir... |
DeepScaleR | [
{
"role": "user",
"content": "The first three numbers of a sequence are \\(1, 7, 8\\). Every subsequent number is the remainder obtained when the sum of the previous three numbers is divided by 4. Find the sum of the first 2011 numbers in this sequence."
}
] | math | {
"style": "rule",
"ground_truth": "3028"
} | {
"index": 32402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangeme... | math | {
"style": "rule",
"ground_truth": "751"
} | {
"index": 32403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangemen... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 32404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first three stages of a pattern are shown below, where each line segment represents a straw. If the pattern continues such that at each successive stage, four straws are added to the previous arrangement, how many straws are necessary to create the arrangement for the 100th... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 32405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first three terms of a geometric progression are $\\sqrt 3$, $\\sqrt[3]3$, and $\\sqrt[6]3$. What is the fourth term?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32406,
"solution": "1. **Identify the pattern in the terms**: The given terms of the geometric progression are $\\sqrt{3}$, $\\sqrt[3]{3}$, and $\\sqrt[6]{3}$. We can rewrite these terms using exponents:\n - $\\sqrt{3} = 3^{1/2} = 3^{3/6}$\n - $\\sqrt[3]{3} = 3^{1/3} = 3^{2/6}$\n - $\\sqrt[6]{3} = ... |
DeepScaleR | [
{
"role": "user",
"content": "The first three terms of a geometric progression are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. Find the fourth term."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32407,
"solution": "1. **Identify the sequence and rewrite terms with fractional exponents:**\n The given terms are $\\sqrt{2}, \\sqrt[3]{2}, \\sqrt[6]{2}$. We can rewrite these terms using fractional exponents:\n \\[\n \\sqrt{2} = 2^{1/2}, \\quad \\sqrt[3]{2} = 2^{1/3}, \\quad \\sqrt[6]{2} = 2^{1/... |
DeepScaleR | [
{
"role": "user",
"content": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 32408,
"solution": "1. **Identify the common difference**: \n Given the arithmetic progression (AP) terms are $x - 1$, $x + 1$, and $2x + 3$. In an AP, the difference between consecutive terms is constant. Therefore, the common difference $d$ can be calculated as:\n \\[\n d = (x + 1) - (x - 1)\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The first three terms of an arithmetic sequence are $2x - 3$, $5x - 11$, and $3x + 1$ respectively. The $n$th term of the sequence is $2009$. What is $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "502"
} | {
"index": 32409,
"solution": "1. **Identify the common difference**: \n Since the sequence is arithmetic, the difference between consecutive terms must be constant. Therefore, we set up the following equations based on the given terms:\n \\[\n (5x - 11) - (2x - 3) = (3x + 1) - (5x - 11)\n \\]\n\n2. **Simpl... |
DeepScaleR | [
{
"role": "user",
"content": "The first three terms of an arithmetic sequence are 1, 10 and 19, respectively. What is the value of the 21st term?"
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 32410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first tourist travels for 1.5 hours on a bicycle at a speed of 16 km/h, then takes a break for 1.5 hours, and then continues at the initial speed. Four hours after the first tourist starts, a second tourist starts chasing the first tourist on a motorcycle at a speed of 56 k... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 32411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first two digits of a natural four-digit number are either both less than 5 or both greater than 5. The same condition applies to the last two digits. How many such numbers are there in total? Justify your answer."
}
] | math | {
"style": "rule",
"ground_truth": "1476"
} | {
"index": 32412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The first two terms of a given sequence are 1 and 1 respectively, and each successive term is the sum of the two preceding terms. What is the value of the first term which exceeds 1000?"
}
] | math | {
"style": "rule",
"ground_truth": "1597"
} | {
"index": 32413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The five books \"Poetry,\" \"Documents,\" \"Rites,\" \"Changes,\" and \"Spring and Autumn\" all have different numbers of pages. The differences in the number of pages between the books are as follows:\n1. \"Poetry\" and \"Documents\" differ by 24 pages.\n2. \"Documents\" and \... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 32414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The five integers $2, 5, 6, 9, 14$ are arranged into a different order. In the new arrangement, the sum of the first three integers is equal to the sum of the last three integers. What is the middle number in the new arrangement?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The five solutions to the equation\\[(z-1)(z^2+2z+4)(z^2+4z+6)=0\\] may be written in the form $x_k+y_ki$ for $1\\le k\\le 5,$ where $x_k$ and $y_k$ are real. Let $\\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 32416,
"solution": "1. **Factorize the given equation**:\n \\[(z-1)(z^2+2z+4)(z^2+4z+6)=0.\\]\n We complete the square for the quadratic terms:\n \\[(z-1)\\left((z+1)^2+3\\right)\\left((z+2)^2+2\\right)=0.\\]\n\n2. **Find the roots**:\n - From $(z-1)=0$, we get $z=1$.\n - From $(z+1)^2+3=0$, we... |
DeepScaleR | [
{
"role": "user",
"content": "The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used?"
}
] | math | {
"style": "rule",
"ground_truth": "24000"
} | {
"index": 32417,
"solution": "1. **Total Miles Driven by All Tires**: The car has five tires, but only four tires are used at any given time. Therefore, over the course of $30,000$ miles, the total number of tire-miles (the sum of the miles driven by each individual tire) is calculated by multiplying the total mil... |
DeepScaleR | [
{
"role": "user",
"content": "The five-digit number $246\\underline{\\hspace{5pt}}8$ is divisible by 9. What is the missing digit?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 32418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The flea Kuzya can make a jump on the plane in any direction for a distance of exactly 17 mm. Its task is to get from point \\( A \\) to point \\( B \\) on the plane, the distance between which is 1947 cm. What is the minimum number of jumps it must make?"
}
] | math | {
"style": "rule",
"ground_truth": "1146"
} | {
"index": 32419,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The floor plan of a castle wall is a regular pentagon with a side length of \\( a (= 100 \\text{ m}) \\). The castle is patrolled by 3 guards along paths from which every point can see the base shape of the castle wall at angles of \\( 90^\\circ, 60^\\circ, \\) and \\( 54^\\cir... | math | {
"style": "rule",
"ground_truth": "5.83"
} | {
"index": 32420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The foci of the ellipse $\\frac{x^2}{16} + \\frac{y^2}{b^2} = 1$ and the foci of the hyperbola\n\\[\\frac{x^2}{144} - \\frac{y^2}{81} = \\frac{1}{25}\\]coincide. Find $b^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 32421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The foci of the ellipse \\(\\frac{x^2}{25} + \\frac{y^2}{b^2} = 1\\) and the foci of the hyperbola\n\\[\\frac{x^2}{196} - \\frac{y^2}{121} = \\frac{1}{49}\\] coincide. Find \\(b^2\\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{908}{49}"
} | {
"index": 32422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\\omega$ is needed to rotate the vessel around its axis so that the water ... | math | {
"style": "rule",
"ground_truth": "9.9"
} | {
"index": 32423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The focus of the parabola $y^{2}=4x$ is $F$, and the equation of the line $l$ is $x=ty+7$. Line $l$ intersects the parabola at points $M$ and $N$, and $\\overrightarrow{MF}⋅\\overrightarrow{NF}=0$. The tangents to the parabola at points $M$ and $N$ intersect at point $P$. Find ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 32424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+line... | math | {
"style": "rule",
"ground_truth": "608"
} | {
"index": 32425,
"solution": "This is more of a solution sketch and lacks rigorous proof for interim steps, but illustrates some key observations that lead to a simple solution.\n Note that one can visualize this problem as walking on a $N \\times N$ grid where the edges warp. Your goal is to have a single path ac... |
DeepScaleR | [
{
"role": "user",
"content": "The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 32426,
"solution": "\nTo find the median length of the names represented in the bar graph, we need to determine the position of the median in a sorted list of the name lengths. Since there are 19 names, the median will be the length of the 10th name when the names are arranged in increasing order of thei... |
DeepScaleR | [
{
"role": "user",
"content": "The following construction is used for training astronauts:\nA circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls... | math | {
"style": "rule",
"ground_truth": "n - 1"
} | {
"index": 32427,
"solution": "\nConsider the problem of determining the number of revolutions an astronaut performs relative to the ground while being fastened to a circle \\( C_3 \\) of radius \\( R \\). This circle rolls inside another circle \\( C_2 \\) of radius \\( 2R \\), which in turn rolls along the inside... |
DeepScaleR | [
{
"role": "user",
"content": "The following diagram shows a square where each side has four dots that divide the side into three equal segments. The shaded region has area 105. Find the area of the original square.\n\n[center][/center]"
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 32428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following diagram shows equilateral triangle $\\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\\vartriangle ABC$ a distance $\\frac18 AB$ along a side of $\\vartriangle ABC$ in the direction... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 32429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following grid represents a mountain range; the number in each cell represents the height of the mountain located there. Moving from a mountain of height \\( a \\) to a mountain of height \\( b \\) takes \\( (b-a)^{2} \\) time. Suppose that you start on the mountain of heig... | math | {
"style": "rule",
"ground_truth": "212"
} | {
"index": 32430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following image is 1024 pixels by 1024 pixels, and each pixel is either black or white. The border defines the boundaries of the image, but is not part of the image. Let $a$ be the proportion of pixels that are black. Estimate $A=\\lfloor 10000 a\\rfloor$. An estimate of $E... | math | {
"style": "rule",
"ground_truth": "3633"
} | {
"index": 32431,
"solution": "This is an area estimation problem. A good place to start is to focus on the jacket. The hair adds about as much area as the hand takes away; the jacket seems to occupy about $\\frac{2}{3}$ of the width of the square and $\\frac{1}{2}$ of the height. A crude estimate of $\\frac{1}{3} ... |
DeepScaleR | [
{
"role": "user",
"content": "The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.\n\nCalculate $({-15})÷({\\frac{1}{3}-3-\\frac{3}{2}})×6$.\n\nSolution: Original expression $=({-15})÷({-\\frac{{25}}{6}})×6\\ldots \\ldots $ First step $=\\left(-15\... | math | {
"style": "rule",
"ground_truth": "\\frac{108}{5}"
} | {
"index": 32432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following is Xiaoying's process of solving a linear equation. Please read carefully and answer the questions.\n\n解方程:$\\frac{{2x+1}}{3}-\\frac{{5x-1}}{6}=1$ \nSolution: \nTo eliminate the denominators, we get $2\\left(2x+1\\right)-\\left(5x-1\\right)=1$ ... Step 1 \nExpa... | math | {
"style": "rule",
"ground_truth": "x = -3"
} | {
"index": 32433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following is the process of simplifying fractions by Xiaobai. Please read carefully and complete the corresponding tasks.\n\nSolution: $(\\frac{3x+4}{x^2-1}-\\frac{2}{x-1})÷\\frac{x+2}{x^2-2x+1}$\n\n$=[\\frac{3x+4}{(x+1)(x-1)}-\\frac{2}{x-1}]÷\\frac{x+2}{(x-1)^2}\\ldots \\l... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 32434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The following operation is allowed on a finite graph: Choose an arbitrary cycle of length 4 (if there is any), choose an arbitrary edge in that cycle, and delete it from the graph. For a fixed integer ${n\\ge 4}$, find the least number of edges of a graph that can be obtained b... | math | {
"style": "rule",
"ground_truth": "n"
} | {
"index": 32435,
"solution": "\nConsider the complete graph \\( K_n \\) on \\( n \\) vertices, where \\( n \\geq 4 \\). The graph initially contains \\(\\binom{n}{2} = \\frac{n(n-1)}{2}\\) edges. We want to find the least number of edges that can be left in the graph by repeatedly applying the following operation:... |
DeepScaleR | [
{
"role": "user",
"content": "The force needed to loosen a bolt varies inversely with the length of the handle of the wrench used. A wrench with a handle length of 9 inches requires 375 pounds of force to loosen a certain bolt. A wrench of 15 inches will require how many pounds of force to loosen the same b... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 32436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The forecast predicts an 80 percent chance of rain for each day of a three-day festival. If it doesn't rain, there is a 50% chance it will be sunny and a 50% chance it will be cloudy. Mina and John want exactly one sunny day during the festival for their outdoor activities. Wha... | math | {
"style": "rule",
"ground_truth": "0.243"
} | {
"index": 32437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The formula $N=8 \\times 10^{8} \\times x^{-3/2}$ gives, for a certain group, the number of individuals whose income exceeds $x$ dollars.\nThe lowest income, in dollars, of the wealthiest $800$ individuals is at least:"
}
] | math | {
"style": "rule",
"ground_truth": "10^4"
} | {
"index": 32438,
"solution": "1. **Identify the given formula and what it represents**: The formula $N = 8 \\times 10^8 \\times x^{-3/2}$ represents the number of individuals whose income exceeds $x$ dollars.\n\n2. **Set up the equation for the specific case**: We are given that there are 800 individuals whose inc... |
DeepScaleR | [
{
"role": "user",
"content": "The formula expressing the relationship between $x$ and $y$ in the table is:\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline x & 2 & 3 & 4 & 5 & 6\\\n\\hline y & 0 & 2 & 6 & 12 & 20\\\n\\hline\n\\end{tabular}"
}
] | math | {
"style": "rule",
"ground_truth": "$y = x^{2}-3x+2$"
} | {
"index": 32439,
"solution": "To find the correct formula that expresses the relationship between $x$ and $y$ in the given table, we will substitute each pair $(x, y)$ into the formulas provided in the options and check for consistency.\n\nThe table provided is:\n\\[\\begin{tabular}{|c|c|c|c|c|c|}\\hline x & 2 & 3... |
DeepScaleR | [
{
"role": "user",
"content": "The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the ne... | math | {
"style": "rule",
"ground_truth": "539"
} | {
"index": 32440,
"solution": "Examine $F - 32$ modulo 9.\nIf $F - 32 \\equiv 0 \\pmod{9}$, then we can define $9x = F - 32$. This shows that $F = \\left[\\frac{9}{5}\\left[\\frac{5}{9}(F-32)\\right] + 32\\right] \\Longrightarrow F = \\left[\\frac{9}{5}(5x) + 32\\right] \\Longrightarrow F = 9x + 32$. This case work... |
DeepScaleR | [
{
"role": "user",
"content": "The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the ne... | math | {
"style": "rule",
"ground_truth": "539"
} | {
"index": 32441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The formula for the total surface area of a cylinder is $SA = 2\\pi r^2 + 2\\pi rh,$ where $r$ is the radius and $h$ is the height. A particular solid right cylinder of radius 2 feet has a total surface area of $12\\pi$ square feet. What is the height of this cylinder?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 32442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is:\n\\[\\begin{tabular}[t]{|c|c|c|c|c|c|}\\hline x&0&1&2&3&4\\\\\\hline y&100&90&70&40&0\\\\\\hline\\end{tabular}\\]"
}
] | math | {
"style": "rule",
"ground_truth": "$y=100-5x-5x^{2}$"
} | {
"index": 32443,
"solution": "To find the correct formula that expresses the relationship between $x$ and $y$, we will substitute the given points into each of the proposed formulas and check which one satisfies all the points.\n\n#### Step 1: Test the point $(0, 100)$\n- **Option A:** $y = 100 - 10x \\implies y =... |
DeepScaleR | [
{
"role": "user",
"content": "The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 32444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four complex roots of\n\\[2z^4 + 8iz^3 + (-9 + 9i)z^2 + (-18 - 2i)z + (3 - 12i) = 0,\\]when plotted in the complex plane, form a rhombus. Find the area of the rhombus."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 32445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four consecutive digits $a$, $b$, $c$ and $d$ are used to form the four-digit numbers $abcd$ and $dcba$. What is the greatest common divisor of all numbers of the form $abcd+dcba$?"
}
] | math | {
"style": "rule",
"ground_truth": "1111"
} | {
"index": 32446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the f... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 32447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four hydrogen atoms in the methane molecule $\\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 32448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four partners in a business decide to split the profits of their company in the ratio 2:3:3:5. If the profit one year is $\\$26,\\!000$ what is the largest number of dollars received by any of the four partners?"
}
] | math | {
"style": "rule",
"ground_truth": "10,\\!000"
} | {
"index": 32449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P ."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 32450,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four points $A(-4,0), B(0,-4), X(0,8),$ and $Y(14,k)$ are grouped on the Cartesian plane. If segment $AB$ is parallel to segment $XY$ what is the value of $k$?"
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 32451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four positive integers $a,$ $b,$ $c,$ $d$ satisfy\n\\[a \\times b \\times c \\times d = 10!.\\]Find the smallest possible value of $a + b + c + d.$"
}
] | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 32452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four zeros of the polynomial $x^4 + jx^2 + kx + 256$ are distinct real numbers in arithmetic progression. Compute the value of $j$."
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"index": 32453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four zeros of the polynomial \\(x^4 + px^2 + qx - 144\\) are distinct real numbers in arithmetic progression. Compute the value of \\(p.\\)"
}
] | math | {
"style": "rule",
"ground_truth": "-40"
} | {
"index": 32454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four-corner codes for the characters \"华\", \"杯\", and \"赛\" are $2440$, $4199$, and $3088$, respectively. By concatenating these, the encoded value for \"华杯赛\" is $244041993088$. If the digits in the odd positions remain unchanged and the digits in the even positions are r... | math | {
"style": "rule",
"ground_truth": "254948903981"
} | {
"index": 32455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four-digit numeral $3AA1$ is divisible by 9. What digit does $A$ represent?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 32456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The four-member math team at Pecanridge Middle School is chosen from the math club, which has three girls and five boys. How many different teams made up of two girls and two boys could be chosen?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 32457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fourth power of $\\sqrt{1+\\sqrt{1+\\sqrt{1}}}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3+2\\sqrt{2}"
} | {
"index": 32458,
"solution": "Let's denote the expression $\\sqrt{1+\\sqrt{1+\\sqrt{1}}}$ as $x$. Thus, we have:\n\\[ x = \\sqrt{1+\\sqrt{1+\\sqrt{1}}} \\]\n\n1. **Simplify the innermost square root:**\n \\[ \\sqrt{1} = 1 \\]\n So, the expression becomes:\n \\[ x = \\sqrt{1+\\sqrt{1+1}} = \\sqrt{1+\\sqrt{2}}... |
DeepScaleR | [
{
"role": "user",
"content": "The fourth term of a geometric sequence is 512, and the 9th term is 8. Determine the positive, real value for the 6th term."
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 32459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fraction\n\\[\\frac{\\left(3^{2008}\\right)^2-\\left(3^{2006}\\right)^2}{\\left(3^{2007}\\right)^2-\\left(3^{2005}\\right)^2}\\]\nsimplifies to what?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 32460,
"solution": "1. **Rewrite the expression using the difference of squares formula**:\n \\[\n \\frac{(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2} = \\frac{(3^{2008} + 3^{2006})(3^{2008} - 3^{2006})}{(3^{2007} + 3^{2005})(3^{2007} - 3^{2005})}\n \\]\n\n2. **Simplify each term**:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac {5x - 11}{2x^2 + x - 6}$ was obtained by adding the two fractions $\\frac {A}{x + 2}$ and $\\frac {B}{2x - 3}$. The values of $A$ and $B$ must be, respectively:"
}
] | math | {
"style": "rule",
"ground_truth": "3, -1"
} | {
"index": 32461,
"solution": "To find the values of $A$ and $B$ in the expression $\\frac{5x - 11}{2x^2 + x - 6}$, we first factorize the denominator:\n\n\\[ 2x^2 + x - 6 = (2x - 3)(x + 2) \\]\n\nGiven that the fraction can be expressed as the sum of two simpler fractions, we write:\n\n\\[ \\frac{5x - 11}{(2x - 3)... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{1}{2015}$ has a unique \"(restricted) partial fraction decomposition\" of the form $\\frac{1}{2015}=\\frac{a}{5}+\\frac{b}{13}+\\frac{c}{31}$ where $a, b, c$ are integers with $0 \\leq a<5$ and $0 \\leq b<13$. Find $a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 32462,
"solution": "This is equivalent to $1=13 \\cdot 31 a+5 \\cdot 31 b+5 \\cdot 13 c$. Taking modulo 5 gives $1 \\equiv 3 \\cdot 1 a (\\bmod 5)$, so $a \\equiv 2(\\bmod 5)$. Taking modulo 13 gives $1 \\equiv 5 \\cdot 5 b=25 b \\equiv-b(\\bmod 13)$, so $b \\equiv 12 (\\bmod 13)$. The size constraints o... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{1}{5}$ is written as an infinite binary fraction. How many ones are there among the first 2022 digits after the binary point in this representation? (12 points)"
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"index": 32463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{2(\\sqrt2+\\sqrt6)}{3\\sqrt{2+\\sqrt3}}$ is equal to"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac43"
} | {
"index": 32464,
"solution": "1. **Simplify the given expression**: We start with the fraction\n \\[\n \\frac{2(\\sqrt{2}+\\sqrt{6})}{3\\sqrt{2+\\sqrt{3}}}.\n \\]\n To simplify, we first square the entire expression to eliminate the square roots:\n \\[\n \\left(\\frac{2(\\sqrt{2}+\\sqrt{6})}{3\\sqrt{2+... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{\\sqrt{a^2+x^2}-\\frac{x^2-a^2}{\\sqrt{a^2+x^2}}}{a^2+x^2}$ reduces to:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2a^2}{(a^2+x^2)^{\\frac{3}{2}}}"
} | {
"index": 32465,
"solution": "1. **Simplify the Numerator**: Start by simplifying the numerator of the given fraction:\n \\[\n \\sqrt{a^2+x^2} - \\frac{x^2-a^2}{\\sqrt{a^2+x^2}}\n \\]\n Multiply the second term by $\\frac{\\sqrt{a^2+x^2}}{\\sqrt{a^2+x^2}}$ to get a common denominator:\n \\[\n \\sqrt{a^... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to:"
}
] | math | {
"style": "rule",
"ground_truth": "a^{-2}+b^{-2}"
} | {
"index": 32466,
"solution": "1. **Rewrite the numerator using the difference of squares formula**: \n The difference of squares formula states that $x^2 - y^2 = (x+y)(x-y)$. Applying this to $a^{-4} - b^{-4}$, where $x = a^{-2}$ and $y = b^{-2}$, we get:\n \\[\n a^{-4} - b^{-4} = (a^{-2} - b^{-2})(a^{-2} + ... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction $\\frac{a}{a+27}$, where $a$ is a positive integer, equals $0.865$. What is the value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "173"
} | {
"index": 32467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fraction \\(\\frac{1}{221}\\) is represented as a repeating decimal. Find the length of the repeating sequence. (For example, the length of the repeating sequence for the fraction \\(\\frac{25687}{99900} = 0.25712712712 \\ldots = 0.25(712)\\) is 3.)"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 32468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fraction \\(\\frac{1}{99^2}=0.\\overline{b_{n-1}b_{n-2}\\ldots b_2b_1b_0},\\) where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\\cdots+b_{n-1}$?"
}
] | math | {
"style": "rule",
"ground_truth": "883"
} | {
"index": 32469,
"solution": "To solve the problem, we first need to understand the decimal expansion of $\\frac{1}{99^2}$ and then find the sum of the digits in one period of the repeating decimal.\n\n1. **Calculate $\\frac{1}{99}$:**\n \\[\n \\frac{1}{99} = 0.\\overline{01}\n \\]\n This is because $99 \\... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction \\(\\frac{p}{q}\\) is in its simplest form. If \\(\\frac{7}{10} < \\frac{p}{q} < \\frac{11}{15}\\), where \\(q\\) is the smallest possible positive integer and \\(c = pq\\), find the value of \\(c\\)."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 32470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fraction halfway between $\\frac{1}{5}$ and $\\frac{1}{3}$ (on the number line) is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 32471,
"solution": "To find the fraction halfway between $\\frac{1}{5}$ and $\\frac{1}{3}$, we calculate their average. The average of two numbers $a$ and $b$ is given by $\\frac{a+b}{2}$.\n\n1. **Convert fractions to a common denominator**:\n \\[\n \\frac{1}{5} = \\frac{3}{15} \\quad \\text{and} \\q... |
DeepScaleR | [
{
"role": "user",
"content": "The fraction of the area of rectangle P Q R S that is shaded must be calculated."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 32472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fractional equation $\\dfrac{x-5}{x+2}=\\dfrac{m}{x+2}$ has a root, determine the value of $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"index": 32473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The fractional part of a positive number, its integer part, and the number itself form an increasing geometric progression. Find all such numbers."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} + 1}{2}"
} | {
"index": 32474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When is it advisable to swap the tires so that they wear out equally? (Assume that the tires are swapped only once, although in practice drivers do this more frequently.)"
}
] | math | {
"style": "rule",
"ground_truth": "9375"
} | {
"index": 32475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The front tires of a car wear out after 25,000 km, and the rear tires wear out after 15,000 km. When should the tires be swapped so that they wear out at the same time?"
}
] | math | {
"style": "rule",
"ground_truth": "9375"
} | {
"index": 32476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The front view of a cone is an equilateral triangle with a side length of 4. Find the surface area of the cone."
}
] | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"index": 32477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The front wheel of Georgina's bicycle has a diameter of 0.75 metres. She cycled for 6 minutes at a speed of 24 kilometres per hour. The number of complete rotations that the wheel made during this time is closest to:"
}
] | math | {
"style": "rule",
"ground_truth": "1020"
} | {
"index": 32478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function\n$$\n\\begin{aligned}\ny= & |x-1|+|2x-1|+|3x-1|+ \\\\\n& |4x-1|+|5x-1|\n\\end{aligned}\n$$\n\nachieves its minimum value when the variable $x$ equals ______."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{3}$"
} | {
"index": 32479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function\n\\[f(z) = \\frac{(-1 + i \\sqrt{3}) z + (-2 \\sqrt{3} - 18i)}{2}\\]represents a rotation around some complex number $c$. Find $c$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 5i"
} | {
"index": 32480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ satisfies the condition $$ f (x + 1) = \\frac{1 + f (x)}{1 - f (x)} $$ for all real $x$ , for which the function is defined. Determine $f(2012)$ , if we known that $f(1000)=2012$ ."
}
] | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"index": 32481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$ has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine $f(8)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"index": 32482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f: N \\to N_0$ is such that $f (2) = 0, f (3)> 0, f (6042) = 2014$ and $f (m + n)- f (m) - f (n) \\in\\{0,1\\}$ for all $m,n \\in N$ . Determine $f (2014)$ . $N_0=\\{0,1,2,...\\}$"
}
] | math | {
"style": "rule",
"ground_truth": "671"
} | {
"index": 32483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$ . Find the greatest real number $m$ , for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true f... | math | {
"style": "rule",
"ground_truth": "7/3"
} | {
"index": 32484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f_n (x)\\ (n=1,2,\\cdots)$ is defined as follows.\r\n\r\n\\[f_1 (x)=x,\\ f_{n+1}(x)=2x^{n+1}-x^n+\\frac{1}{2}\\int_0^1 f_n(t)\\ dt\\ \\ (n=1,2,\\cdots)\\]\r\n\r\nEvaluate \r\n\r\n\\[\\lim_{n\\to\\infty} f_n \\left(1+\\frac{1}{2n}\\right)\\]"
}
] | math | {
"style": "rule",
"ground_truth": "e^{1/2}"
} | {
"index": 32485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $\\lfloor x\\rfloor$ is defined as the largest integer less than or equal to $x$. For example, $\\lfloor 5.67\\rfloor = 5$, $\\lfloor -\\tfrac 14\\rfloor = -1$, and $\\lfloor 8\\rfloor = 8$.\n\nWhat is the range of the function $$f(x) = \\lfloor x\\rfloor - x~?$$Ex... | math | {
"style": "rule",
"ground_truth": "(-1,0]"
} | {
"index": 32486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $\\mathbf{y}=f(x)$ satisfies the following conditions:\n\na) $f(4)=2$;\n\nb) $f(n+1)=\\frac{1}{f(0)+f(1)}+\\frac{1}{f(1)+f(2)}+\\ldots+\\frac{1}{f(n)+f(n+1)}, n \\geq 0$.\n\nFind the value of $f(2022)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2022}"
} | {
"index": 32487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f : \\mathbb{R} \\to \\mathbb{R}$ satisfies\n\\[f(x) + 2f(1 - x) = 3x^2\\]for all $x.$ Find $f(4).$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ defined by $f(x)= \\frac{ax+b}{cx+d}$, where $a$,$b$,$c$ and $d$ are nonzero real numbers, has the properties $f(19)=19$, $f(97)=97$ and $f(f(x))=x$ for all values except $\\frac{-d}{c}$. Find the unique number that is not in the range of $f$."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 32489,
"solution": "Because there are no other special numbers other than $19$ and $97$, take the average to get $\\boxed{58}$. (Note I solved this problem the solution one way but noticed this and this probably generalizes to all $f(x)=x, f(y)=y$ questions like these)"
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ has the property that for each real number $x$ in its domain, $1/x$ is also in its domain and \\[\nf(x) + f\\left(\\frac{1}{x}\\right) = x.\n\\]What is the largest set of real numbers that can be in the domain of $f$?\n\n(a) ${\\{x\\mid x\\ne0\\}}$\n\n(b) ${\\{... | math | {
"style": "rule",
"ground_truth": "E"
} | {
"index": 32490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is defined on positive integers as follows:\n\\[f(n) = \\left\\{\n\\begin{array}{cl}\nn + 10 & \\text{if $n < 10$}, \\\\\nf(n - 5) & \\text{if $n \\ge 10$}.\n\\end{array}\n\\right.\\]Find the maximum value of the function."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 32491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is defined on positive integers as follows:\n\\[f(n) = \\left\\{\n\\begin{array}{cl}\nn + 15 & \\text{if $n < 15$}, \\\\\nf(n - 7) & \\text{if $n \\ge 15$}.\n\\end{array}\n\\right.\\]\nFind the maximum value of the function."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 32492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is defined on the set of integers and satisfies \\[f(n)= \\begin{cases} n-3 & \\mbox{if }n\\ge 1000 \\\\ f(f(n+5)) & \\mbox{if }n<1000. \\end{cases}\\]Find $f(84)$."
}
] | math | {
"style": "rule",
"ground_truth": "997"
} | {
"index": 32493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is given by the table\n\\[\\begin{tabular}{|c||c|c|c|c|c|} \\hline x & 1 & 2 & 3 & 4 & 5 \\ \\hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \\hline \\end{tabular}\\]\nIf $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \\ge 0$, find $u_{2002}$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 32494,
"solution": "1. **Identify the sequence**: We are given a function $f$ defined by a table and a sequence defined recursively by $u_{n+1} = f(u_n)$ with $u_0 = 4$.\n\n2. **Calculate the first few terms**:\n - $u_1 = f(u_0) = f(4) = 5$\n - $u_2 = f(u_1) = f(5) = 2$\n - $u_3 = f(u_2) = f(2) = 1... |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is graphed below. Each small box has width and height 1.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownle... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 32495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is linear and satisfies $f(d+1)-f(d) = 3$ for all real numbers $d$. What is $f(3)-f(5)$?"
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 32496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ is not defined for $x = 0,$ but for all non-zero real numbers $x,$\n\\[f(x) + 2f \\left( \\frac{1}{x} \\right) = 3x.\\]Find the real solutions to $f(x) = f(-x).$ Enter the real solutions, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2},-\\sqrt{2}"
} | {
"index": 32497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ satisfies\n\\[\nf(x) + f(3x+y) + 7xy = f(4x - y) + 3x^2 + 2y + 3\n\\]\nfor all real numbers $x, y$. Determine the value of $f(10)$."
}
] | math | {
"style": "rule",
"ground_truth": "-37"
} | {
"index": 32498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "The function $f$ satisfies $f(x)+f(2 x+y)+5 x y=f(3 x-y)+2 x^{2}+1$ for all real numbers $x, y$. Determine the value of $f(10)$."
}
] | math | {
"style": "rule",
"ground_truth": "-49"
} | {
"index": 32499,
"solution": "Setting $x=10$ and $y=5$ gives $f(10)+f(25)+250=f(25)+200+1$, from which we get $f(10)=-49$. Remark: By setting $y=\\frac{x}{2}$, we see that the function is $f(x)=-\\frac{1}{2} x^{2}+1$, and it can be checked that this function indeed satisfies the given equation."
} |
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