conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "Solve $\\log_4 x + \\log_2 x^2 = 10$."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\log_4 x + \\log_2 x^2 = 10\\):\n\n1. Convert \\(\\log_4 x\\) to base 2 using the change of base formula:\n \\[\n \\log_4 x = \\frac{\\log_2 x}{\\log_2 4} = \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the number of different partitions of a set with \\( n \\) elements into 2 classes, denoted as \\( {S_2}^n \\)."
},
{
"role": "assistant",
"content": "To find the number of different partitions of a set with \\( n \\) elements into 2 classes, we need to determi... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "Find the points where the function \\( G(u) = \\frac{u+1}{u^2 - u - 6} \\) is not defined and determine whether each discontinuity is removable."
},
{
"role": "assistant",
"content": "To determine where the function \\( G(u) = \\frac{u+1}{u^2 - u - 6} \\) is not def... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Let \\( M_n \\) be the set of all \\( n \\times n \\) matrices with real entries, and let \\( E = \\text{span}(\\{AB - BA : A, B \\in M_n\\}) \\). Find the dimension of \\( E \\)."
},
{
"role": "assistant",
"content": "To determine the dimension of \\( E = \\text{sp... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the expected length of time until a head is tossed with a biased coin?"
},
{
"role": "assistant",
"content": "To find the expected length of time until a head is tossed with a biased coin, we can use the properties of the geometric distribution. Let \\( p \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $\\cos 210^\\circ$."
},
{
"role": "assistant",
"content": "To compute \\(\\cos 210^\\circ\\), we follow these steps:\n\n1. **Determine the Quadrant**: The angle \\(210^\\circ\\) is in the third quadrant because it is between \\(180^\\circ\\) and \\(270^\\cir... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many common points can at least two lines have in a set of 3 plan lines?"
},
{
"role": "assistant",
"content": "To determine the maximum number of common points where at least two lines intersect in a set of three plane lines, we need to consider the possible co... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given a positive integer $n$ and a number $c$, where $-1 < c < 1$, for how many values of $q$ in $[0, 2\\pi)$ is $\\sin(nq) = c$?"
},
{
"role": "assistant",
"content": "To determine the number of values of \\( q \\) in the interval \\([0, 2\\pi)\\) such that \\(\\si... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Eight distinct points are in the plane such that five of them lie on a line \\( \\ell \\), and the other three points lie off the line, in a way such that if some three of the eight points lie on a line, they lie on \\( \\ell \\). How many triangles can be formed using some thr... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$."
},
{
"role": "assistant",
"content": "To find the largest value of \\( c \\) such that \\(-2\\) is in the range of the function \\( f(x) = x^2 + 3x + c \\), we need to determine the m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "There are 8 balls in a box, 4 of which are blue, and 4 of which are red. They are otherwise identical. Daniel removes four of the balls without replacement. What is the probability that Daniel removes more blue balls than red balls?"
},
{
"role": "assistant",
"conte... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the first perfect square? Is it 0 or 1?"
},
{
"role": "assistant",
"content": "To determine the first perfect square, we need to recall the definition of a perfect square. A perfect square is an integer that can be expressed as the square of another integer.... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Let $a\\star b = ab+2b-a$. If $5\\star x = 37$, find the value of $x$."
},
{
"role": "assistant",
"content": "To solve the problem where the operation \\( a \\star b = ab + 2b - a \\) is given and we need to find \\( x \\) such that \\( 5 \\star x = 37 \\):\n\n1. Su... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "In a business school, 75% of students adopted marketing, 62% adopted finance, and 88% adopted HR as electives. What is the minimum number of students specializing in all three subjects?"
},
{
"role": "assistant",
"content": "To determine the minimum number of studen... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many vertices will the simple graph G have if it contains 15 edges, 3 vertices of degree 4, and the other vertices of degree 3?"
},
{
"role": "assistant",
"content": "To determine the number of vertices in the simple graph \\( G \\) with 15 edges, where 3 vertic... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral $$\\int_{0}^{1} \\frac{\\sin(x) + \\sin(\\frac{1}{x})}{x} dx.$$"
},
{
"role": "assistant",
"content": "To evaluate the integral \n\\[\n\\int_{0}^{1} \\frac{\\sin(x) + \\sin\\left(\\frac{1}{x}\\right)}{x} \\, dx,\n\\]\nwe start by splitting the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Jenna spent $175.00 on Christmas gifts for 23 nurses at the hospital. She bought candles for some nurses at $5.00 each and scarves for the others at $10.00 each. How many candles and how many scarves did she buy?"
},
{
"role": "assistant",
"content": "To solve the p... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$?"
},
{
"role": "assistant",
"content": "To find \\( g(f(-2)) \\), we need to evaluate the inner function \\( f(-2) \\) first and then use that result as the input for the function \\( g \\).\n\n1. Evalua... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that their product is an even number? Express your answer as a common fraction."
},
{
"role": "assistant",
"content": "To find the probability ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve for \\( C \\):\n\n\\[ C = \\frac{P \\cdot A}{P \\cdot B + D} \\]"
},
{
"role": "assistant",
"content": "To solve for \\( C \\) in the equation\n\n\\[ C = \\frac{P \\cdot A}{P \\cdot B + D}, \\]\n\nwe observe that \\( C \\) is already isolated on one side of th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Camden invested $22,055 more at 8% than Carter did at 4% (both at simple interest). If the combined total annual return was $2005, how much did Camden invest?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine how much Camden invested ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the congruence equation \\(4x^2 - 3 \\equiv 0 \\pmod{7}\\)."
},
{
"role": "assistant",
"content": "To solve the congruence equation \\(4x^2 - 3 \\equiv 0 \\pmod{7}\\), we follow these steps:\n\n1. **Simplify the equation:**\n \\[\n 4x^2 - 3 \\equiv 0 \\pmo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the probability that each drawn pair is of different colors (one Black and one White) when drawing two balls at a time from a bag containing 4 Black balls and 4 White balls until the bag is empty?"
},
{
"role": "assistant",
"content": "To find the probabilit... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( T = TNYWR \\). Let \\( A \\) and \\( B \\) be distinct digits in base \\( T \\), and let \\( N \\) be the largest number of the form \\( ABA_T \\). Compute the value of \\( N \\) in base 10.\n\nNote: \\( TNYWR \\) typically stands for \"The Number You Will Receive,\" wh... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "Calculate the intersection \\(\\bigcap_{e>0} (x - e, x + e)\\)."
},
{
"role": "assistant",
"content": "To find the intersection \\(\\bigcap_{e>0} (x - e, x + e)\\), we need to determine the set of all points that belong to every interval \\((x - e, x + e)\\) for all... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A retail jeweler sells a pair of gold earrings for $213.90. The jeweler uses a 130% markup based on cost. Based on this information, what was the wholesale cost of the earrings?"
},
{
"role": "assistant",
"content": "To find the wholesale cost of the earrings, we ne... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "What is the minimum number of distinct Set cards such that a set is guaranteed?"
},
{
"role": "assistant",
"content": "To determine the minimum number of distinct Set cards required to guarantee a set, we need to understand the structure of the game and the concept ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $F(a, b, c, d) = a^b + c \\times d$, what is the value of $x$ such that $F(2, x, 4, 11) = 300$?"
},
{
"role": "assistant",
"content": "To find the value of \\( x \\) such that \\( F(2, x, 4, 11) = 300 \\), we start by substituting the given values into the functi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the sequence \\( (u_n) \\) defined by:\n\\[ u_1 = 3 \\]\n\\[ u_n = 2u_{n-1} + 5n \\]\nFind a formula for \\( u_n \\) in terms of \\( n \\)."
},
{
"role": "assistant",
"content": "To find a formula for the sequence \\( (u_n) \\) defined by:\n\\[ u_1 = 3 \\]\n\\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Grady distributed $x$ pieces of candy evenly among nine Halloween bags such that every bag received the greatest possible number of whole pieces of candy, but some candy was left over. What is the greatest possible number of pieces that could have been left over?"
},
{
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the number of combinations of a given length \\( k \\) that use elements from a set of \\( n \\) elements, allowing repetition."
},
{
"role": "assistant",
"content": "To find the number of combinations of length \\( k \\) from a set of \\( n \\) elements, allow... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the value of \\( N \\), where\n\\[ N = 818^3 - 6 \\cdot 818^2 \\cdot 209 + 12 \\cdot 818 \\cdot 209^2 - 8 \\cdot 209^3 \\]"
},
{
"role": "assistant",
"content": "To find the value of \\( N \\) where\n\\[ N = 818^3 - 6 \\cdot 818^2 \\cdot 209 + 12 \\cdot 818 ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\sin x = 3 \\cos x,$ then what is $\\sin x \\cos x$?"
},
{
"role": "assistant",
"content": "Given the equation \\(\\sin x = 3 \\cos x\\), we need to find the value of \\(\\sin x \\cos x\\).\n\nFirst, we use the Pythagorean identity \\(\\sin^2 x + \\cos^2 x = 1\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the expression $(\\omega+1)^{3n}$, where $\\omega$ is a cube root of unity."
},
{
"role": "assistant",
"content": "To evaluate the expression \\((\\omega + 1)^{3n}\\) where \\(\\omega\\) is a cube root of unity, we start by using the properties of cube root... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine all positive integers $n$ such that $\\binom{n}{0}, \\binom{n}{1}, \\dots, \\binom{n}{n}$ are all odd numbers."
},
{
"role": "assistant",
"content": "To determine all positive integers \\( n \\) such that the binomial coefficients \\(\\binom{n}{0}, \\binom... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the greatest three-digit multiple of 19?"
},
{
"role": "assistant",
"content": "To find the greatest three-digit multiple of 19, we start by dividing the largest three-digit number, 999, by 19:\n\n\\[\n999 \\div 19 = 52.578\\ldots\n\\]\n\nThe integer part of... | You are a helpful assistant. |
[
{
"role": "user",
"content": "(ii) Given a set $A = \\{a_1, a_2, a_3, \\ldots, a_n\\}$, find the total number of binary operations on $A$ such that $a_i * a_j \\neq a_i * a_k$ if $j \\neq k$."
},
{
"role": "assistant",
"content": "To find the total number of binary operations on a set \\( A = \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can you choose a pair of cards from a set of 2 cards?"
},
{
"role": "assistant",
"content": "To determine the number of ways to choose a pair of cards from a set of 2 cards, we use the combination formula \\( C(n, k) = \\frac{n!}{k!(n - k)!} \\), where... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways are there for 10 people to be friends, where it is possible that they are all friends with each other or none of them are friends with each other?"
},
{
"role": "assistant",
"content": "To determine the number of ways 10 people can be friends, where ea... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the number of ordered triples of sets $(A, B, C)$ such that $A \\cup B \\cup C = \\{ 1, 2, \\cdots, 2003 \\}$ and $A \\cap B \\cap C = \\varnothing$."
},
{
"role": "assistant",
"content": "To find the number of ordered triples of sets \\((A, B, C)\\) such that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the function \\( f(x) = \\frac{x^2 + 2x}{x^2 - 3x - 10} \\), find the vertical asymptote."
},
{
"role": "assistant",
"content": "To find the vertical asymptote of the function \\( f(x) = \\frac{x^2 + 2x}{x^2 - 3x - 10} \\), we need to determine where the denom... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $f(x)=2\\sqrt{x} + \\frac{12}{\\sqrt{x}}$ and let $g(x)=2x^2-2x-3$. What is $f(g(3))$?"
},
{
"role": "assistant",
"content": "To find \\( f(g(3)) \\), we need to follow these steps:\n\n1. Calculate \\( g(3) \\).\n2. Use the result from step 1 as the input to the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ordered sets of $(a,b,c,d)$ are such that $abcd = -2013^4$?"
},
{
"role": "assistant",
"content": "To determine the number of ordered sets \\((a, b, c, d)\\) such that \\(abcd = -2013^4\\), we start by factorizing 2013 into its prime factors:\n\n\\[ 2013 = ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "By purchasing rupees 50 shares for rupees 80 each, a man gets a 4% profit on his investment. What is his dividend if he buys 60 shares?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the dividend for 60 shares given that the shares... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "How do you evaluate the integral $\\int \\frac{1}{1-x} dx$ and show that it equals $-\\ln(1-x)$?"
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\frac{1}{1 - x} \\, dx\\), we use the substitution method. Let \\(u = 1 - x\\). Then, the diffe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The city has 80 buses. 30% of them are new and cause less pollution. If the city replaces 14 old buses with new ones, what percent of the buses will be new? Round your answer to the nearest tenth."
},
{
"role": "assistant",
"content": "To solve the problem, we need ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $\\cos 135^\\circ$."
},
{
"role": "assistant",
"content": "To compute \\(\\cos 135^\\circ\\), we can use the properties of the unit circle and the reference angle.\n\n1. **Identify the quadrant**: The angle \\(135^\\circ\\) is in the second quadrant because ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Express the volume $W$ obtained by rotating the region $R$ bounded by the curve $y=x^2-x^4$ and the $x$-axis in the first quadrant about the $y$-axis as an integral using the cylindrical shells method (with $dx$)."
},
{
"role": "assistant",
"content": "To find the v... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the number of different four-digit PINs that have a digit sum of 6."
},
{
"role": "assistant",
"content": "To determine the number of different four-digit PINs with a digit sum of 6, we need to find the number of non-negative integer solutions to the equatio... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In triangle $ABC$, $AB = 16$, $AC = 24$, $BC = 19$, and $AD$ is an angle bisector. Find the ratio of the area of triangle $ABD$ to the area of triangle $ACD$. (Express your answer as a fraction in lowest terms.)"
},
{
"role": "assistant",
"content": "To solve the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the largest multiple of $9$ whose negation is greater than $-100$?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the largest multiple of \\(9\\) whose negation is greater than \\(-100\\), we follow these steps:\n\n1. **Interpret the ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the area enclosed by the graph of $|3x|+|4y|=12$?"
},
{
"role": "assistant",
"content": "To find the area enclosed by the graph of \\(|3x| + |4y| = 12\\), we first determine the vertices of the shape formed by this equation. \n\nBy considering the different ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "On a table, there are 2016 coins. Two players take turns, and in each turn, a player can take 1, 2, or 3 coins. The player who takes the last coin wins. Which player has a winning strategy?"
},
{
"role": "assistant",
"content": "To determine which player has a winni... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If the sides of a cyclic quadrilateral are the roots of the polynomial \\(x^4 - 2sx^3 + tx^2 - qx + 2p = 0\\), express the sum of the sides \\(S\\) in terms of \\(p\\), \\(q\\), \\(s\\), and \\(t\\)."
},
{
"role": "assistant",
"content": "To find the sum of the side... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\((1 + x + x^2)^n = a_0 + a_1 x + \\cdots + a_{2n} x^{2n}\\), find \\(a_0 + a_2 + a_4 + \\cdots + a_{2n}\\)."
},
{
"role": "assistant",
"content": "To find the sum of the even-indexed coefficients in the expansion of \\((1 + x + x^2)^n\\), we start by evaluating... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $AAA_4$ can be expressed as $33_b$, where $A$ is a digit in base 4 and $b$ is a base greater than 5, what is the smallest possible sum $A+b$?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to find the smallest possible sum \\( A + b \\) such... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What geometric shape, if any, does the equation $(x-a)^2-(y-b)^2=r^2$ represent?"
},
{
"role": "assistant",
"content": "To determine the geometric shape represented by the equation \\((x - a)^2 - (y - b)^2 = r^2\\), we start by comparing it to the standard forms of ... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7."
},
{
"role": "assistant",
"content": "To find the units digit of \\( n \\) given that \\( mn = 21^6 \\) and \\( m \\) has a units digit of 7, we can proceed as follows:\n\n1. **Deter... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values?"
},
{
"role": "assistant",
"content": "To find the smallest possible perimeter of a triangle with consecutive integer side lengths, we represe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the quadratic equation \\( ax^2 - 2x\\sqrt{2} + c = 0 \\) with real constants \\( a \\) and \\( c \\), and the discriminant is zero, determine the nature of the roots."
},
{
"role": "assistant",
"content": "To determine the nature of the roots of the quadratic... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine the units in the ring \\( R \\) of the form \\( a + b\\sqrt{-5} \\) where \\( a \\) and \\( b \\) are integers."
},
{
"role": "assistant",
"content": "To determine the units in the ring \\( R = \\{ a + b\\sqrt{-5} \\mid a, b \\in \\mathbb{Z} \\} \\), we ne... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "How many ways can Snow White redistribute the presents so that each of the \\( n \\) dwarves gets a different present than what they had before?"
},
{
"role": "assistant",
"content": "To determine the number of ways Snow White can redistribute the presents such that... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "In a group of 50 people, 36 like dogs and 25 like cats. Let \\( x \\) denote the number of people who like both dogs and cats. How many different values can \\( x \\) take?"
},
{
"role": "assistant",
"content": "To determine the number of different values \\( x \\) ... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Let \\( A \\) be a set of \\( kn \\) elements where \\( k \\) and \\( n \\) are natural numbers. A \"k\" grouping of \\( A \\) is a partition of \\( A \\) into subsets, each containing \\( k \\) elements. Find the number of different \"k\" groupings of \\( A \\)."
},
{
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $x \\not= y$ and $\\frac{x}{y} + x = \\frac{y}{x} + y$, find the sum $\\frac{1}{x} + \\frac{1}{y}$."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\frac{x}{y} + x = \\frac{y}{x} + y\\) for \\(x \\neq y\\) and find the sum \\(\\frac{1}{x} + \\fra... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "How many arrangements of the word \"SENIOR\" have the three vowels occurring together?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the number of arrangements of the word \"SENIOR\" where the three vowels (E, I, O) occur together, we can fo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than 1. If Moonbeam gives Sunny a head start of $h$ meters, how many meters must Moonbeam run to overtake Sunny?"
},
{
"role": "assistant",
"content": "To solve the probl... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $1+2x+3x^2 + \\dotsb=9$, find $x$."
},
{
"role": "assistant",
"content": "To solve the problem \\(1 + 2x + 3x^2 + \\dotsb = 9\\), we start by recognizing that the series can be written as \\(\\sum_{n=0}^{\\infty} (n+1)x^n\\). This series converges for \\(|x| < 1\... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "There are $n-1$ red balls, $n$ green balls, and $n+1$ blue balls in a bag. The number of ways of choosing two balls from the bag that have different colors is $299$. Find the value of $n$."
},
{
"role": "assistant",
"content": "To find the value of \\( n \\), we sta... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\(a, b \\in \\mathbb{R}\\) and \\(\\{1, a+b, a\\} = \\{0, \\frac{b}{a}, b\\}\\), then find \\(b - a\\)."
},
{
"role": "assistant",
"content": "Given that \\(a\\) and \\(b\\) are real numbers and the sets \\(\\{1, a+b, a\\}\\) and \\(\\{0, \\frac{b}{a}, b\\}\\) a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many prime numbers are between 30 and 40?"
},
{
"role": "assistant",
"content": "To determine how many prime numbers are between 30 and 40, we first list the numbers in this range: 31, 32, 33, 34, 35, 36, 37, 38, 39.\n\nA prime number is a number greater than 1 ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A city has a system of bus routes with the following properties:\n- Each route has exactly 11 bus stops.\n- It is possible to travel between any two bus stops without changing routes.\n- Any two bus routes have exactly one bus stop in common.\nWhat is the number of bus routes i... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "John recently bought a used car for $\\$5000$ for his pizza delivery job. He gets $\\$10$ for each pizza he delivers, but he has to spend $\\$3$ on gas for each pizza he delivers. What is the minimum whole number of pizzas John must deliver in order to earn back the money he sp... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Let \\( F(x) = x^{\\cos x + \\sin x} \\). Find \\( F'(1) \\)."
},
{
"role": "assistant",
"content": "To find \\( F'(1) \\) where \\( F(x) = x^{\\cos x + \\sin x} \\), we use logarithmic differentiation.\n\n1. Let \\( y = x^{\\cos x + \\sin x} \\). Taking the natural... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a history class, the probability of earning an A is .7 times the probability of earning a B, and the probability of earning a C is 1.4 times the probability of earning a B. Assuming that all grades are A, B, or C, how many B's will there be in a history class of 31 students?... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can the letters in the word \"MIXED\" be rearranged such that no letter maintains its original position?"
},
{
"role": "assistant",
"content": "To determine the number of ways to rearrange the letters in the word \"MIXED\" such that no letter maintains... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Solve the recurrence relation $a_{n} = 9 a_{n-1} - 56 n + 63$ with the initial condition $a_{0} = 2$."
},
{
"role": "assistant",
"content": "To solve the recurrence relation \\( a_n = 9a_{n-1} - 56n + 63 \\) with the initial condition \\( a_0 = 2 \\), we proceed as ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many total days were there in the years 2001 through 2004?"
},
{
"role": "assistant",
"content": "To determine the total number of days from the years 2001 through 2004, we need to identify which of these years are leap years and which are regular years.\n\nA re... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\(f(x) = \\frac{x}{x+1}\\). If \\(f^n(x)\\) is the result of composing \\(f\\) with itself \\(n\\) times, compute \\(f^{2004}(2)\\)."
},
{
"role": "assistant",
"content": "To solve the problem of finding \\( f^{2004}(2) \\) where \\( f(x) = \\frac{x}{x+1} \\), ... | You are a helpful assistant. |
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