conversations listlengths 2 2 | system stringclasses 1
value | score float64 0.08 1 |
|---|---|---|
[
{
"role": "user",
"content": "In how many ways can 7 persons sit around a round table such that no two arrangements have the same neighbors for all individuals?"
},
{
"role": "assistant",
"content": "To determine the number of ways 7 persons can sit around a round table such that no two arrangem... | You are a helpful assistant. | 1 |
[
{
"role": "user",
"content": "In how many ways can \\( n \\) people stand to form a ring?"
},
{
"role": "assistant",
"content": "To determine the number of ways \\( n \\) people can stand to form a ring, we need to consider circular permutations. In a circular arrangement, rotations of the same ... | You are a helpful assistant. | 0.958108 |
[
{
"role": "user",
"content": "The sum of some positive integers is $2018$. What is the maximum possible value of their product?"
},
{
"role": "assistant",
"content": "To find the maximum possible value of the product of positive integers that sum to 2018, we use the strategy of breaking the numb... | You are a helpful assistant. | 0.830571 |
[
{
"role": "user",
"content": "Determine the minimum number of cards needed to guarantee the existence of a set in the game of Set."
},
{
"role": "assistant",
"content": "To determine the minimum number of cards needed to guarantee the existence of a set in the game of Set, we need to delve into ... | You are a helpful assistant. | 0.747453 |
[
{
"role": "user",
"content": "Given a positive integer in its base-10 representation, if you repeatedly add it to its reverse, will you always eventually reach a palindrome? If so, provide a bound on the number of iterations required. If not, find or describe a number that never palindromizes."
},
{
... | You are a helpful assistant. | 0.707355 |
[
{
"role": "user",
"content": "How can you add eight 8's to get the number 1000?"
},
{
"role": "assistant",
"content": "To solve the problem of adding eight 8's to get the number 1000, we need to creatively combine the digits into multi-digit numbers. Here is the clean and detailed solution:\n\n1... | You are a helpful assistant. | 0.656721 |
[
{
"role": "user",
"content": "We have $2n$ batteries $(n>2)$, with $n$ full batteries and $n$ empty ones. What is the smallest number of tries needed to ensure that we have at least 2 full batteries to make a lamp work?"
},
{
"role": "assistant",
"content": "To determine the smallest number of t... | You are a helpful assistant. | 0.536622 |
[
{
"role": "user",
"content": "For a N-player symmetric game with S possible strategies, how many distinct strategy profiles are there?"
},
{
"role": "assistant",
"content": "To determine the number of distinct strategy profiles in an \\( N \\)-player symmetric game with \\( S \\) possible strate... | You are a helpful assistant. | 0.524463 |
[
{
"role": "user",
"content": "Given 5 towns, what is the minimum amount of road needed to connect each town to every other town?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the minimum number of roads required to connect 5 towns such that each town is connected to ever... | You are a helpful assistant. | 0.506571 |
[
{
"role": "user",
"content": "Find all pairs \\((x, y)\\) of positive consecutive integers."
},
{
"role": "assistant",
"content": "To find all pairs \\((x, y)\\) of positive consecutive integers, we start by defining what it means for two integers to be consecutive. Consecutive integers are numb... | You are a helpful assistant. | 0.49041 |
[
{
"role": "user",
"content": "How do you graph the polar point \\((2, \\frac{3\\pi}{4})\\)?"
},
{
"role": "assistant",
"content": "To graph the polar point \\((2, \\frac{3\\pi}{4})\\):\n\n1. **Identify the angle**: The angle \\(\\frac{3\\pi}{4}\\) radians corresponds to \\(135^\\circ\\), which i... | You are a helpful assistant. | 0.479269 |
[
{
"role": "user",
"content": "In a singles tennis tournament with 10 players, in how many ways can the players be organised into 5 pairs?"
},
{
"role": "assistant",
"content": "To determine the number of ways to organize 10 players into 5 pairs, we can use the concept of combinations and adjust ... | You are a helpful assistant. | 0.445285 |
[
{
"role": "user",
"content": "If \\( f(x) \\) belongs to the interval \\([-2, 3]\\), determine the interval to which \\( f(x-2) \\) belongs."
},
{
"role": "assistant",
"content": "To determine the interval to which \\( f(x-2) \\) belongs, given that \\( f(x) \\) belongs to the interval \\([-2, 3... | You are a helpful assistant. | 0.443645 |
[
{
"role": "user",
"content": "A standard deck of cards is shuffled and 3 cards are selected. What is the probability that the 3rd card is either a black face card or a queen, if the queen of hearts and queen of spades are the first two cards selected?"
},
{
"role": "assistant",
"content": "To so... | You are a helpful assistant. | 0.442 |
[
{
"role": "user",
"content": "At a birthday party, everyone who gets an invitation card can come alone or bring a friend. Everyone invited was present at the event. The ratio of the number of invitation cards to the number of guests attending is $3:5$. What is the ratio of the number of invitation cards to ... | You are a helpful assistant. | 0.43792 |
[
{
"role": "user",
"content": "Given that \\( f(1) = 2 \\) and \\( f'(1) = 7 \\), what is \\( f'(1^5)^2 \\)?"
},
{
"role": "assistant",
"content": "To solve the problem of finding \\( f'(1^5)^2 \\), we follow these steps:\n\n1. Simplify the expression inside the derivative: \\( 1^5 = 1 \\). There... | You are a helpful assistant. | 0.428306 |
[
{
"role": "user",
"content": "Find all limit points of the sequence \\( x_n = \\cos(n) \\)."
},
{
"role": "assistant",
"content": "To find all limit points of the sequence \\( x_n = \\cos(n) \\), we need to analyze the behavior of the sequence \\( \\cos(n) \\) where \\( n \\) is an integer.\n\n#... | You are a helpful assistant. | 0.426615 |
[
{
"role": "user",
"content": "No two students in Mrs. Vale's 26-student mathematics class have the same two initials. Each student's first name and last name begin with the same letter. If the letter ``Y'' is considered a vowel, what is the probability of randomly picking a student whose initials are vowels... | You are a helpful assistant. | 0.413063 |
[
{
"role": "user",
"content": "How many distinguishable ways are there to construct a regular octahedron using eight congruent equilateral triangles, each of a different color? Two colored octahedrons are considered distinguishable if neither can be rotated to look just like the other."
},
{
"role": ... | You are a helpful assistant. | 0.400272 |
[
{
"role": "user",
"content": "Find the limit \\(\\lim_{x \\to -\\infty} \\left(\\frac{x e^x}{1 - e^x}\\right)\\)."
},
{
"role": "assistant",
"content": "To find the limit \\(\\lim_{x \\to -\\infty} \\left(\\frac{x e^x}{1 - e^x}\\right)\\), we analyze the behavior of the numerator and the denomin... | You are a helpful assistant. | 0.392457 |
[
{
"role": "user",
"content": "Form the integer 24 using the numbers 2, 4, 5, 7. Only addition, subtraction, division, and multiplication can be used. Each number can only be used once."
},
{
"role": "assistant",
"content": "To form the integer 24 using the numbers 2, 4, 5, and 7 with the operati... | You are a helpful assistant. | 0.391696 |
[
{
"role": "user",
"content": "Eight eligible bachelors and seven beautiful models randomly purchase single seats in a 15-seat row of a theatre. On average, how many pairs of adjacent seats are occupied by a bachelor and a model?"
},
{
"role": "assistant",
"content": "To solve the problem of find... | You are a helpful assistant. | 0.382834 |
[
{
"role": "user",
"content": "What is the smallest digit that is never found in the units place of an even number?"
},
{
"role": "assistant",
"content": "To determine the smallest digit that never appears in the units place of an even number, we first need to identify which digits can appear in ... | You are a helpful assistant. | 0.380894 |
[
{
"role": "user",
"content": "Evaluate $\\lim_{x\\rightarrow 7}\\sqrt{x-10}$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{x \\to 7} \\sqrt{x - 10}\\), we need to consider the domain of the function \\(\\sqrt{x - 10}\\). The square root function \\(\\sqrt{y}\\) is defin... | You are a helpful assistant. | 0.37621 |
[
{
"role": "user",
"content": "Harry has 3 sisters and 5 brothers. His sister Harriet has $X$ sisters and $Y$ brothers. What is the product of $X$ and $Y$?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the number of sisters and brothers Harriet has, given that ... | You are a helpful assistant. | 0.375939 |
[
{
"role": "user",
"content": "Find the maximal dimension of a subspace of \\( M_n(K) \\) consisting only of nilpotent matrices, where \\( K = \\mathbb{R} \\) or \\( \\mathbb{C} \\)."
},
{
"role": "assistant",
"content": "To find the maximal dimension of a subspace of \\( M_n(K) \\) consisting on... | You are a helpful assistant. | 0.3735 |
[
{
"role": "user",
"content": "Determine whether \\( H = A_4 \\) is a normal subgroup of \\( G = S_4 \\). If it is, write out a Cayley table for the factor group \\( G/H \\)."
},
{
"role": "assistant",
"content": "To determine whether \\( H = A_4 \\) is a normal subgroup of \\( G = S_4 \\) and to... | You are a helpful assistant. | 0.367801 |
[
{
"role": "user",
"content": "A gambler plays a game \\( n \\) times, with the probability of winning each game being \\( p \\). If it is known that the gambler did not win \\( x \\) times, what is the expected number of times the gambler won?"
},
{
"role": "assistant",
"content": "To find the e... | You are a helpful assistant. | 0.363619 |
[
{
"role": "user",
"content": "Given the equation \\((x-7)^2 + (y + 4)^2 = 9\\), how do we determine whether to leave it as a circle equation or to divide by the constant to fit the ellipse equation?"
},
{
"role": "assistant",
"content": "Given the equation \\((x-7)^2 + (y+4)^2 = 9\\), we need to... | You are a helpful assistant. | 0.360337 |
[
{
"role": "user",
"content": "A group of 5 is to be chosen from 6 men and 4 women. In how many ways can the group be chosen if it contains a particular man X but excludes a particular woman Y?"
},
{
"role": "assistant",
"content": "To solve the problem of choosing a group of 5 from 6 men and 4 w... | You are a helpful assistant. | 0.358067 |
[
{
"role": "user",
"content": "Find $\\ln{-1}$."
},
{
"role": "assistant",
"content": "To find \\(\\ln(-1)\\), we need to use the properties of complex logarithms. The natural logarithm function \\(\\ln(x)\\) is not defined for negative real numbers in the real number system, but it can be extend... | You are a helpful assistant. | 0.348357 |
[
{
"role": "user",
"content": "Jordan and his three best friends are on a relay team. His relay team will run a race, where the first runner runs a lap, then the second, then the third, then the fourth.\n\nJordan is the fastest so he will run the fourth lap. How many different orders can the four team memb... | You are a helpful assistant. | 0.348259 |
[
{
"role": "user",
"content": "If $c$ is the total cost (in million rupiah) for producing $q$ units of a product, then the average cost per unit is given by\n\\[\\overline{c} = \\frac{5000 + 6q}{q}.\\]\na. If producers continuously increase their production, what is the limit value of this average cost?\nb. ... | You are a helpful assistant. | 0.340616 |
[
{
"role": "user",
"content": "In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?"
},
{
"role": "assistant",
"content": "To determine the total number of people in the line, we need to use the... | You are a helpful assistant. | 0.334692 |
[
{
"role": "user",
"content": "If $a\\equiv 18\\pmod{42}$ and $b\\equiv 73\\pmod{42}$, then for what integer $n$ in the set $\\{100,101,102,\\ldots,140,141\\}$ is it true that $$a-b\\equiv n\\pmod{42}~?$$"
},
{
"role": "assistant",
"content": "To solve for \\( n \\) such that \\( a - b \\equiv n ... | You are a helpful assistant. | 0.329889 |
[
{
"role": "user",
"content": "Let \\( K \\) be a commutative field. Find all prime ideals of the polynomial ring \\( K[X] \\)."
},
{
"role": "assistant",
"content": "To find all prime ideals of the polynomial ring \\( K[X] \\) where \\( K \\) is a commutative field, we proceed as follows:\n\n1. ... | You are a helpful assistant. | 0.327402 |
[
{
"role": "user",
"content": "What is the probability of winning all three games, given that a team has three games to play and each game can result in either a win (W) or a loss (L)?"
},
{
"role": "assistant",
"content": "To find the probability of a team winning all three games, we need to con... | You are a helpful assistant. | 0.3252 |
[
{
"role": "user",
"content": "Given the sequence defined by \\( u_0 = \\frac{1}{3} \\) and \\( u_{n+1} = 2u_n^2 - 1 \\), find a closed form for \\( u_n \\)."
},
{
"role": "assistant",
"content": "To find a closed-form expression for the sequence defined by \\( u_0 = \\frac{1}{3} \\) and \\( u_{n... | You are a helpful assistant. | 0.324587 |
[
{
"role": "user",
"content": "In a plane, there are \\( n \\) points out of which 4 are collinear. How many distinct triangles can be drawn?"
},
{
"role": "assistant",
"content": "To determine the number of distinct triangles that can be formed from \\( n \\) points in a plane, where 4 of these ... | You are a helpful assistant. | 0.321193 |
[
{
"role": "user",
"content": "Find the set of values of \\( x \\) for which \\( f(x) = |f(x)| \\) for the function \\( f(x) = 2 - (x+1)^{\\frac{1}{3}} \\)."
},
{
"role": "assistant",
"content": "To find the set of values of \\( x \\) for which \\( f(x) = |f(x)| \\) for the function \\( f(x) = 2 ... | You are a helpful assistant. | 0.315729 |
[
{
"role": "user",
"content": "A circle is inscribed in a triangle with sides of lengths 8, 13, and 17. Let the segments of the side of length 8 made by a point of tangency be r and s, with r < s. Find the ratio of r:s."
},
{
"role": "assistant",
"content": "To find the ratio of the segments \\( ... | You are a helpful assistant. | 0.310069 |
[
{
"role": "user",
"content": "Contractor Steve agreed to complete a job in 30 days. After 6 days he found that the 8 people assigned to the work had already done $\\frac{1}{3}$ of the job. If everyone works at the same rate, what is the least number of people he must keep on the job to ensure that the job w... | You are a helpful assistant. | 0.305951 |
[
{
"role": "user",
"content": "What is the sum of the digits of the decimal representation of $2^{2005} \\times 5^{2007} \\times 3$?"
},
{
"role": "assistant",
"content": "To find the sum of the digits of the decimal representation of \\(2^{2005} \\times 5^{2007} \\times 3\\), we start by simplif... | You are a helpful assistant. | 0.30463 |
[
{
"role": "user",
"content": "Let $A$, $B$, and $C$ represent the complex numbers $z_1$, $z_2$, and $z_3$ respectively on the complex plane. If the circumcenter of the triangle $ABC$ lies at the origin, what is the complex number representing the orthocenter?"
},
{
"role": "assistant",
"content"... | You are a helpful assistant. | 0.30457 |
[
{
"role": "user",
"content": "Find the parametric equations for the horizontal line and vertical line passing through the point $(-4, 5)$."
},
{
"role": "assistant",
"content": "To find the parametric equations for the horizontal and vertical lines passing through the point \\((-4, 5)\\), we pro... | You are a helpful assistant. | 0.295496 |
[
{
"role": "user",
"content": "When does \\( x^2 \\) reach its minimum value?"
},
{
"role": "assistant",
"content": "To determine when \\( x^2 \\) reaches its minimum value, we can use both algebraic and calculus-based approaches.\n\nFirst, observe that for any real number \\( x \\), \\( x^2 \\ge... | You are a helpful assistant. | 0.294481 |
[
{
"role": "user",
"content": "Let \\( n \\in \\mathbb{N}, n \\ge 2 \\). Determine which of the following statements are true:\na) Any finite group \\( G \\) of order \\( n \\) is isomorphic to a subgroup of \\( GL_n(\\mathbb{R}) \\)."
},
{
"role": "assistant",
"content": "To determine whether an... | You are a helpful assistant. | 0.291399 |
[
{
"role": "user",
"content": "Two fair dice were rolled and the sum of the obtained numbers was even. What is the probability that both numbers were equal?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the probability that both numbers on two fair dice are equal given th... | You are a helpful assistant. | 0.290064 |
[
{
"role": "user",
"content": "If \\(a\\) is real, what is the only real number that could be a multiple root of the equation \\(x^3 + ax + 1 = 0\\)?"
},
{
"role": "assistant",
"content": "To determine the only real number that could be a multiple root of the equation \\(x^3 + ax + 1 = 0\\), we u... | You are a helpful assistant. | 0.288201 |
[
{
"role": "user",
"content": "Find the number of quadruples $(x, y, z, w)$ of non-negative integers that satisfy the inequality $w + x + y + z \\le 1992$."
},
{
"role": "assistant",
"content": "To find the number of quadruples \\((w, x, y, z)\\) of non-negative integers that satisfy the inequali... | You are a helpful assistant. | 0.287496 |
[
{
"role": "user",
"content": "Evaluate the integral \\(\\int e^{1/z} \\, dz\\) along the boundary of the rectangle centered at \\(1 + 2i\\) with width 5 and height 6. The boundary is divided into four contours."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int e^{1/z} \\, ... | You are a helpful assistant. | 0.285994 |
[
{
"role": "user",
"content": "Given the matrix of the quadratic form \\( f \\) in the basis \\( W \\) and the matrix \\( L_{W}^{V} \\) of the linear transformation \\( L \\) from basis \\( V \\) to basis \\( W \\), how can the matrix of the quadratic form \\( f \\circ L \\) be presented?"
},
{
"role... | You are a helpful assistant. | 0.281598 |
[
{
"role": "user",
"content": "Let $f(x) = 4x + c$ and $g(x) = cx + 2.$ If $f(g(x)) = 12x + d,$ then find $d.$"
},
{
"role": "assistant",
"content": "To find the value of \\( d \\), we start by composing the functions \\( f(x) = 4x + c \\) and \\( g(x) = cx + 2 \\). Substituting \\( g(x) \\) int... | You are a helpful assistant. | 0.278262 |
[
{
"role": "user",
"content": "Trickster Rabbit agrees to double Foolish Fox's money every time Fox crosses the bridge by Rabbit's house, but Fox must pay 40 coins in toll to Rabbit after each crossing. After crossing the bridge three times, Fox discovers that all his money is gone. How many coins did Fox ha... | You are a helpful assistant. | 0.275304 |
[
{
"role": "user",
"content": "Evaluate the limit $\\lim_{h\\to0}\\left[\\frac{1}{h}\\int^{x^2+h}_{x^2}\\log[3+\\arcsin(t)]\\,dt\\right]$ in terms of $x$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{h\\to0}\\left[\\frac{1}{h}\\int^{x^2+h}_{x^2}\\log[3+\\arcsin(t)]\\,dt\... | You are a helpful assistant. | 0.271311 |
[
{
"role": "user",
"content": "A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?"
},
{
"role": "assis... | You are a helpful assistant. | 0.270874 |
[
{
"role": "user",
"content": "Given the linear transformation \\( f: \\mathbb{R} \\rightarrow \\mathbb{R}^2 \\) defined by \\( f(u) = v \\), where \\( u = \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\) and \\( v = \\begin{pmatrix} x + y \\\\ x - y \\end{pmatrix} \\), if the points \\( P', Q', R' \\) are the i... | You are a helpful assistant. | 0.269108 |
[
{
"role": "user",
"content": "In a case with five different men, if nine victims randomly guess which voice belongs to the criminal, what is the probability that all nine victims would select the same person?"
},
{
"role": "assistant",
"content": "To find the probability that all nine victims ra... | You are a helpful assistant. | 0.267594 |
[
{
"role": "user",
"content": "How many positive two-digit integers are there in which each of the two digits is prime?"
},
{
"role": "assistant",
"content": "To determine how many positive two-digit integers have each digit as a prime number, we first identify the single-digit prime numbers. The... | You are a helpful assistant. | 0.259766 |
[
{
"role": "user",
"content": "Determine the type of graph represented by the equation $x^2 = (2y + 3)^2$. Is it a hyperbola or intersecting lines?"
},
{
"role": "assistant",
"content": "To determine the type of graph represented by the equation \\(x^2 = (2y + 3)^2\\), we start by rewriting it in... | You are a helpful assistant. | 0.238305 |
[
{
"role": "user",
"content": "Evaluate the integral \\[ \\int_0^\\infty \\frac{x^3}{e^x - 1} \\, dx \\]"
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int_0^\\infty \\frac{x^3}{e^x - 1} \\, dx\\), we can use the relationship between the integral of \\(x^{s-1}/(e^x - 1)\\) a... | You are a helpful assistant. | 0.235207 |
[
{
"role": "user",
"content": "Integrate $\\sec{x}$."
},
{
"role": "assistant",
"content": "To integrate \\(\\sec x\\), we use a specific technique involving multiplying by a form of 1 to simplify the integrand. Here is the clean, final solution:\n\n1. Start with the integral:\n \\[\n \\int \... | You are a helpful assistant. | 0.231312 |
[
{
"role": "user",
"content": "Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation."
},
{
"role": "assistant",
"content": "To solve the problem of finding all positive values of \\( c \\) such that the inequality... | You are a helpful assistant. | 0.230584 |
[
{
"role": "user",
"content": "Solve for \\( x \\) in the equation \\( \\sqrt[3]{x^2 + 26} + 3\\sqrt{x} + \\sqrt{x + 3} = 8 \\)."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\sqrt[3]{x^2 + 26} + 3\\sqrt{x} + \\sqrt{x + 3} = 8\\), we proceed as follows:\n\n1. **Determine the do... | You are a helpful assistant. | 0.229058 |
[
{
"role": "user",
"content": "Solve the equation \\(x^3 - [x] = 3\\)."
},
{
"role": "assistant",
"content": "To solve the equation \\(x^3 - [x] = 3\\), where \\([x]\\) denotes the floor function of \\(x\\), we start by letting \\([x] = n\\), where \\(n\\) is an integer. This implies \\(x \\in [n... | You are a helpful assistant. | 0.229004 |
[
{
"role": "user",
"content": "Calculate the indefinite integral $\\int \\left(\\tan x + \\frac{1}{\\tan x}\\right)^2 \\, dx$."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. | 0.226232 |
[
{
"role": "user",
"content": "How many different 10-letter arrangements of the letters in the word \"GOPALADINS\" are there?"
},
{
"role": "assistant",
"content": "To determine the number of different 10-letter arrangements of the letters in the word \"GOPALADINS,\" we first identify the letters... | You are a helpful assistant. | 0.225577 |
[
{
"role": "user",
"content": "Let\n\\[z = \\frac{(-11 + 13i)^3 \\cdot (24 - 7i)^4}{3 + 4i},\\]and let $w = \\frac{\\overline{z}}{z}.$ Compute $|w|.$"
},
{
"role": "assistant",
"content": "To compute \\( |w| \\) where \\( w = \\frac{\\overline{z}}{z} \\) and \\( z \\) is given by the expression\... | You are a helpful assistant. | 0.224641 |
[
{
"role": "user",
"content": "Let \\( f: \\mathbb{N} \\rightarrow \\mathbb{N} \\) be a function such that \\( f(f(n)) + 2f(n) = 3n + 4 \\) for all \\( n \\in \\mathbb{N} \\). Find \\( f(2006) \\)."
},
{
"role": "assistant",
"content": "To solve for \\( f(2006) \\) given the functional equation \... | You are a helpful assistant. | 0.217497 |
[
{
"role": "user",
"content": "How do you factor $x^n - 1$, where $n \\geq 3$, into irreducibles in $\\mathbb{Z}[x]$?"
},
{
"role": "assistant",
"content": "To factor the polynomial \\( x^n - 1 \\) into irreducibles in \\( \\mathbb{Z}[x] \\) for \\( n \\geq 3 \\), we use the concept of cyclotomic... | You are a helpful assistant. | 0.216452 |
[
{
"role": "user",
"content": "Suppose that the number of people entering a department store in a given day is a random variable with mean 50. Suppose further that the amount of money spent by these customers are random variables with a common mean of 8. Assume also that the amount of money spent by a custom... | You are a helpful assistant. | 0.216281 |
[
{
"role": "user",
"content": "How many other points with integer coordinates lie on the line passing through $P(2,2)$ and $Q(50,998)$ that are strictly between $P$ and $Q$?"
},
{
"role": "assistant",
"content": "To determine the number of integer coordinate points strictly between \\( P(2, 2) \\... | You are a helpful assistant. | 0.21322 |
[
{
"role": "user",
"content": "Evaluate $\\log_{\\sqrt{6}} (216\\sqrt{6})$."
},
{
"role": "assistant",
"content": "To evaluate \\(\\log_{\\sqrt{6}} (216\\sqrt{6})\\), we start by expressing the base and the argument in terms of base 6.\n\nFirst, note that \\(\\sqrt{6} = 6^{1/2}\\). Therefore, the... | You are a helpful assistant. | 0.213065 |
[
{
"role": "user",
"content": "How many vertical asymptotes does the function \\( f(x) = \\frac{x^2 - 6x - 16}{x^3 - 5x^2 - 2x + 24} \\) have?"
},
{
"role": "assistant",
"content": "To determine the number of vertical asymptotes of the function \\( f(x) = \\frac{x^2 - 6x - 16}{x^3 - 5x^2 - 2x + 2... | You are a helpful assistant. | 0.207522 |
[
{
"role": "user",
"content": "For some positive integer $k$, when 60 is divided by $k^2$, the remainder is 6. What is the remainder when 100 is divided by $k$?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by noting that when 60 is divided by \\( k^2 \\), the remainder is... | You are a helpful assistant. | 0.205349 |
[
{
"role": "user",
"content": "If \\( \\log_6 x = 2.5 \\), what is the value of \\( x \\)?"
},
{
"role": "assistant",
"content": "To solve the equation \\( \\log_6 x = 2.5 \\), we start by converting it to its exponential form. The equation \\( \\log_6 x = 2.5 \\) implies that \\( x = 6^{2.5} \\)... | You are a helpful assistant. | 0.199796 |
[
{
"role": "user",
"content": "Solve the inequality: $\\max(\\log_{3}{x}, \\log_{\\frac{1}{2}}{x}) \\leq 3$."
},
{
"role": "assistant",
"content": "To solve the inequality \\(\\max(\\log_{3}{x}, \\log_{\\frac{1}{2}}{x}) \\leq 3\\), we need to ensure that both \\(\\log_{3}{x} \\leq 3\\) and \\(\\l... | You are a helpful assistant. | 0.199233 |
[
{
"role": "user",
"content": "Compute the limit of the sequence $\\{z_n\\}$ where $z_n = i^{n!}$."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. | 0.194379 |
[
{
"role": "user",
"content": "Given the general term of a sequence \\( t_n = 3 \\cdot 5^{n-1} \\), derive the recursive formula for the sequence."
},
{
"role": "assistant",
"content": "Given the general term of the sequence \\( t_n = 3 \\cdot 5^{n-1} \\), we need to derive the recursive formula ... | You are a helpful assistant. | 0.192257 |
[
{
"role": "user",
"content": "How many ways are there to distribute 28 identical apples to 4 people such that each person receives at least one apple?"
},
{
"role": "assistant",
"content": "To determine the number of ways to distribute 28 identical apples to 4 people such that each person receiv... | You are a helpful assistant. | 0.189578 |
[
{
"role": "user",
"content": "Calculate the sum $1 + 3 + 5 + \\cdots + 15 + 17$."
},
{
"role": "assistant",
"content": "To calculate the sum of the sequence \\(1 + 3 + 5 + \\cdots + 15 + 17\\), we can use the formula for the sum of an arithmetic sequence. The sequence is an arithmetic sequence w... | You are a helpful assistant. | 0.187924 |
[
{
"role": "user",
"content": "Points A and B are on different sides of a line \\( l \\). The distance from point A to line \\( l \\) is 10 inches, and the distance from point B to line \\( l \\) is 4 inches. What is the distance between the midpoint O of segment AB and the line \\( l \\)?"
},
{
"rol... | You are a helpful assistant. | 0.182705 |
[
{
"role": "user",
"content": "Two years ago there were 20 trailer homes on Elm Street with an average age of 18 years. At that time, a group of brand new trailer homes was then added to Elm Street. Today, the average age of all the trailer homes on Elm Street is 14 years. How many new trailer homes were add... | You are a helpful assistant. | 0.18006 |
[
{
"role": "user",
"content": "Jessica works at a bakery, and every day she makes 30 pie crusts that each use up $\\frac16$ cup of flour. One day, she decides that she instead wants to make 20 larger pie crusts using the same total amount of flour. How many cups of flour will go into each of the new crusts? ... | You are a helpful assistant. | 0.175539 |
[
{
"role": "user",
"content": "Morgan walks \\( x \\) yards north, then walks \\( (2x-1) \\) yards west. She is now \\( (2x+1) \\) yards from her starting point. What is the distance, in yards, she travels?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by noting that Morga... | You are a helpful assistant. | 0.167519 |
[
{
"role": "user",
"content": "Let $a$ and $b$ be real numbers such that $a + b = 8$. Let $c$ be the minimum possible value of $x^2 + ax + b$ over all real numbers $x$. Find the maximum possible value of $c$ over all such $a$ and $b$."
},
{
"role": "assistant",
"content": "To find the maximum pos... | You are a helpful assistant. | 0.160677 |
[
{
"role": "user",
"content": "Given a number with last digit \\( a \\) in base \\( b \\), consider the sequence \\( 1, a, a^2, a^3, \\ldots \\). What is the period of the last digits of this sequence for \\( a = 17 \\) and \\( b = 10 \\)?"
},
{
"role": "assistant",
"content": "To find the period... | You are a helpful assistant. | 0.15867 |
[
{
"role": "user",
"content": "Suppose \\(\\gcd(p, q) = 1\\), \\(\\gcd(p, 10) = 1\\), and \\(\\gcd(q, 10) = 1\\). If the decimal expansion of \\(1/p\\) has period \\(r\\) and the decimal expansion of \\(1/q\\) has period \\(s\\), find the period of the decimal expansion of \\(1/(pq)\\)."
},
{
"role":... | You are a helpful assistant. | 0.157501 |
[
{
"role": "user",
"content": "In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle... | You are a helpful assistant. | 0.155401 |
[
{
"role": "user",
"content": "What is the remainder when $6x^3-15x^2+21x-23$ is divided by $3x-6$?"
},
{
"role": "assistant",
"content": "To find the remainder when \\(6x^3 - 15x^2 + 21x - 23\\) is divided by \\(3x - 6\\), we can use the Remainder Theorem. The Remainder Theorem states that the r... | You are a helpful assistant. | 0.154336 |
[
{
"role": "user",
"content": "How many additive abelian groups of order 16 have the property that $x + x + x + x = 0$?"
},
{
"role": "assistant",
"content": "To determine how many additive abelian groups of order 16 satisfy the condition that \\(x + x + x + x = 0\\) for all elements \\(x\\), we ... | You are a helpful assistant. | 0.152212 |
[
{
"role": "user",
"content": "The base three number $12012_3$ is equal to which base ten number?"
},
{
"role": "assistant",
"content": "To convert the base three number \\(12012_3\\) to a base ten number, we need to evaluate each digit's positional value and sum them up. Each digit in the base t... | You are a helpful assistant. | 0.151036 |
[
{
"role": "user",
"content": "Given $\\log 1.23 = 0.07$, find $10^{1.07}$."
},
{
"role": "assistant",
"content": "To find \\(10^{1.07}\\) given that \\(\\log 1.23 = 0.07\\), we can proceed as follows:\n\n1. Recognize that \\(\\log 1.23 = 0.07\\) implies \\(\\log_{10} 1.23 = 0.07\\). By the defin... | You are a helpful assistant. | 0.148633 |
[
{
"role": "user",
"content": "Evaluate the integral \\(\\int \\frac{3x-6}{\\sqrt{x^2-4x+11}} \\, dx\\)."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\frac{3x-6}{\\sqrt{x^2-4x+11}} \\, dx\\), we start by simplifying the expression under the square root in the denomina... | You are a helpful assistant. | 0.148516 |
[
{
"role": "user",
"content": "How many five-digit numbers are there in which the sum of the first and the last digits is six?"
},
{
"role": "assistant",
"content": "To determine the number of five-digit numbers where the sum of the first and last digits is six, we can follow these steps:\n\n1. *... | You are a helpful assistant. | 0.145536 |
[
{
"role": "user",
"content": "The three roots of the equation \\[ax^3+bx^2+cx+d=0\\]are $1,$ $2,$ and $3.$ Compute $\\frac{c}{d}.$"
},
{
"role": "assistant",
"content": "To find \\(\\frac{c}{d}\\) for the cubic equation \\(ax^3 + bx^2 + cx + d = 0\\) with roots 1, 2, and 3, we use Vieta's formul... | You are a helpful assistant. | 0.139336 |
[
{
"role": "user",
"content": "How many pages are there in a book if it takes 852 digits to number the pages consecutively?"
},
{
"role": "assistant",
"content": "To determine the number of pages in a book given that it takes 852 digits to number the pages consecutively, we can break down the pro... | You are a helpful assistant. | 0.13171 |
[
{
"role": "user",
"content": "What is $3^{-1} + 3^{-2} \\pmod{25}$? Express your answer as an integer from $0$ to $24$, inclusive."
},
{
"role": "assistant",
"content": "To solve the problem \\(3^{-1} + 3^{-2} \\pmod{25}\\), we need to find the multiplicative inverses of \\(3\\) and \\(3^2\\) mo... | You are a helpful assistant. | 0.120666 |
[
{
"role": "user",
"content": "A new US flag has seven rows of stars, with the middle row having the most stars and the number of stars decreasing by one each time you go up or down from the middle. If the entire flag has 51 stars, how many stars does the middle row have?"
},
{
"role": "assistant",
... | You are a helpful assistant. | 0.110402 |
[
{
"role": "user",
"content": "What is the expected value of the roll of a standard 6-sided die?"
},
{
"role": "assistant",
"content": "To find the expected value of rolling a standard 6-sided die, we need to consider each possible outcome and its associated probability. A standard 6-sided die ha... | You are a helpful assistant. | 0.078639 |
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