conversations listlengths 2 2 | system stringclasses 1
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[
{
"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": "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": "A box contains 4032 answer scripts, exactly half of which have an odd number of marks. We randomly choose 2 scripts and follow these rules:\n- If both scripts have odd scores, we add one mark to one of them, put it back in the box, and keep the other script outside.\n- If both ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Graph the region defined by \\(0 \\leq \\theta \\leq \\pi\\) and \\(r < -1\\)."
},
{
"role": "assistant",
"content": "To graph the region defined by \\(0 \\leq \\theta \\leq \\pi\\) and \\(r < -1\\), we need to understand the transformation of negative \\(r\\) value... | 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": "What is the shape of those points in a plane whose distances to three non-collinear points add up to a constant value?"
},
{
"role": "assistant",
"content": "To determine the shape of the points in a plane whose distances to three non-collinear points add up to a co... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "In an \\( m \\times n \\) rectangular grid, how many unit squares does the diagonal connecting two opposite vertices pass through? Express your answer in terms of \\( m \\) and \\( n \\)."
},
{
"role": "assistant",
"content": "To determine the number of unit squares... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can 4 nuts be placed in a $4 \\times 4$ table such that there is at least one nut in every row and every column?"
},
{
"role": "assistant",
"content": "To solve the problem of placing 4 nuts in a \\(4 \\times 4\\) table such that there is at least o... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "There are 20 houses along the shore of a lake, each painted one of four colors at random. Starting from one of these houses, you walk around the lake and laugh each time you pass a house that is the same color as the previous house you passed. What is the expected number of tim... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In the \"Let's Make a Deal\" game, what is the probability of winning the good prize if you switch doors after being shown one of the junky prizes?"
},
{
"role": "assistant",
"content": "To determine the probability of winning the good prize by switching doors in th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that \\(a + b = 17\\) and \\(a, b \\in \\mathbb{Z^+}\\), find the minimum value of \\(a\\) and \\(b\\)."
},
{
"role": "assistant",
"content": "Given that \\(a + b = 17\\) and \\(a, b \\in \\mathbb{Z^+}\\), we need to find the minimum value of \\(a\\) and \\(b\... | 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": "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": "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": "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. |
[
{
"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. |
[
{
"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. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "Bill is sent to a donut shop to purchase exactly six donuts. If the shop has four kinds of donuts and Bill is to get at least one of each kind, how many combinations will satisfy Bill's order requirements?"
},
{
"role": "assistant",
"content": "To determine the numb... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Write the barycentric coordinates of the midpoint $M$ of segment $AB$."
},
{
"role": "assistant",
"content": "To find the barycentric coordinates of the midpoint \\( M \\) of segment \\( AB \\) relative to triangle \\( ABC \\):\n\n1. **Understanding Barycentric Coor... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many arrangements of all eleven letters of the word M I S S I S S I P P I are there in which all the four letters I are consecutive?"
},
{
"role": "assistant",
"content": "To determine the number of arrangements of the letters in \"MISSISSIPPI\" where all four I... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An entity A has a probability of 3/8 to disappear and a probability of 5/8 to produce two new entities A. What is the probability that A produces an infinite descent of entities?\n\nExplanation: The problem statement is extracted directly from the forum post and rephrased for c... | 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": "Determine if the set \\( S = \\{A \\in \\mathbb{M}_n(\\mathbb{R}) \\mid \\text{tr}(A) = 0\\} \\) is nowhere dense in \\( \\mathbb{M}_n(\\mathbb{R}) \\), where \\( n \\ge 2 \\)."
},
{
"role": "assistant",
"content": "To determine if the set \\( S = \\{A \\in \\mathbb... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Does there exist a continuous function $f$ from $(0,1)$ to $(0,1)$?"
},
{
"role": "assistant",
"content": "To determine whether there exists a continuous function \\( f \\) from the open interval \\((0,1)\\) to itself, we need to find a function \\( f \\) such that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In base \\( b \\), how many integers with \\( (2n+1) \\) digits are palindromes?"
},
{
"role": "assistant",
"content": "To determine the number of \\((2n+1)\\)-digit palindromes in base \\( b \\), we need to consider the structure of a palindrome. A \\((2n+1)\\)-dig... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The numbers $1$ through $8$ are written on the vertices of a cube such that each vertex has a distinct number written on it. On the center of each face of this cube, the sum of the numbers written on the four vertices of this face is written. Find the sum of the numbers written... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"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. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yel... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"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": "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. |
[
{
"role": "user",
"content": "Let \\( \\sum a_n \\) be a convergent series with \\( a_n \\geq 0 \\). Determine the nature of the series \\( \\sum \\frac{1}{1 + a_n} \\)."
},
{
"role": "assistant",
"content": "To determine the nature of the series \\( \\sum \\frac{1}{1 + a_n} \\) given that \\( \... | 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": "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. |
[
{
"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. |
[
{
"role": "user",
"content": "Forest Ho-Chen notices that the stars on the US flag are arranged as a $6$ by $5$ rectangle with a $5$ by $4$ rectangle on the inside, allowing for beautiful cross patterns. He wonders what the previous number of states the US could have had where the stars could be arranged in... | 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": "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. |
[
{
"role": "user",
"content": "Find a formula or rule for the sequence: $1, 2, 6, 30, 210, 2310, \\ldots$"
},
{
"role": "assistant",
"content": "To find a formula or rule for the sequence \\(1, 2, 6, 30, 210, 2310, \\ldots\\), we observe the pattern in the sequence and derive the rule as follows:... | 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": "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. |
[
{
"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. |
[
{
"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": "Let \\( A \\) be a set with \\( n(A) = 2 \\). Denote \\( P^1(A) = P(A) \\) and \\( P^k(A) = P(P^{k-1}(A)) \\) for every natural number \\( k \\geq 1 \\). Determine the smallest \\( k \\) such that \\( n(P^k(A)) \\geq 65000 \\)."
},
{
"role": "assistant",
"content": ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum of all numbers formed from the digits 1, 3, 5, 7, 9 that are greater than 10000, with no digits repeated."
},
{
"role": "assistant",
"content": "To find the sum of all numbers formed from the digits 1, 3, 5, 7, 9 that are greater than 10,000 with no dig... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games and every team in the other division $M$ games, with $N > 2M$ and $M > 4$. Each team plays a total of 76 games. How many games does a team play within its own divis... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( A, M, \\) and \\( C \\) be digits such that \\((100A + 10M + C)(A + M + C) = 2005\\). What is the value of \\( A \\)?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by noting that \\( A, M, \\) and \\( C \\) are digits, and the equatio... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation \\( x \\cdot 2^x = 1 \\) algebraically."
},
{
"role": "assistant",
"content": "To solve the equation \\( x \\cdot 2^x = 1 \\) algebraically, we start by rewriting the equation using the natural exponential function:\n\n\\[\nx \\cdot 2^x = 1\n\\]\n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute:\n\\[\n\\frac{1 \\cdot 1! + 2 \\cdot 2! + 3 \\cdot 3! + \\ldots + 7 \\cdot 7! + 1}{6!}\n\\]"
},
{
"role": "assistant",
"content": "To compute the expression\n\\[\n\\frac{1 \\cdot 1! + 2 \\cdot 2! + 3 \\cdot 3! + \\ldots + 7 \\cdot 7! + 1}{6!},\n\\]\nwe start... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the sum ${n\\choose0}^2+{n\\choose1}^2+{n\\choose2}^2+\\dots+{n\\choose{n}}^2$ using a counting argument."
},
{
"role": "assistant",
"content": "To evaluate the sum \\(\\binom{n}{0}^2 + \\binom{n}{1}^2 + \\binom{n}{2}^2 + \\dots + \\binom{n}{n}^2\\) using a... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"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": "Calculate the indefinite integral $\\int \\left(\\tan x + \\frac{1}{\\tan x}\\right)^2 \\, dx$."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is $0.1 \\div 0.004$?"
},
{
"role": "assistant",
"content": "To solve the problem \\(0.1 \\div 0.004\\), we can use the method of eliminating the decimals by multiplying both the dividend and the divisor by the same power of 10. Here are the steps:\n\n1. Identi... | 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": "A square has an area of $25x^2 + 20x + 4$. Find the restrictions on $x$ such that the side length of the square is positive."
},
{
"role": "assistant",
"content": "To find the restrictions on \\( x \\) such that the side length of the square is positive, given that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If real numbers $x$ and $y$ satisfy $(x+2y)^2 + 4(x+2y + 2 - xy) = 0,$ what is $x+2y?$"
},
{
"role": "assistant",
"content": "To solve the equation \\((x + 2y)^2 + 4(x + 2y + 2 - xy) = 0\\) and find the value of \\(x + 2y\\), we proceed as follows:\n\nFirst, let \\(... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "Suppose that \\( X \\) is a random variable with mean 7 and variance 4, and let \\( Y \\) be a random variable independent of \\( X \\), with mean 0 and variance 1. Let \\( Z = X - Y \\). Evaluate \\( \\text{Cov}(X, Z) \\) and hence \\( \\rho(X, Z) \\)."
},
{
"role": "a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the polynomial \\( P(x) \\) with the conditions \\( P(1) = 0 \\), \\( P(2) = 1 \\), and \\( P(x^2) = x^4 + mx^2 + n \\), find \\( P(0) \\)."
},
{
"role": "assistant",
"content": "Given the polynomial \\( P(x) \\) with the conditions \\( P(1) = 0 \\), \\( P(2) ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given matrices $\\boldsymbol{A}=\\begin{bmatrix} 1 & 3 & -4 \\\\ -1 & -3 & 4 \\end{bmatrix}$ and $\\boldsymbol{B}=\\begin{bmatrix} 1 & 3 \\\\ -1 & 8 \\end{bmatrix}$, find the product $AB$."
},
{
"role": "assistant",
"content": "To find the product \\( \\boldsymbol{A... | 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": "What is the value of $T_1 + T_2 + \\ldots + T_{15}$, where $T_n$ denotes the $n$th triangular number?"
},
{
"role": "assistant",
"content": "To find the value of \\( T_1 + T_2 + \\ldots + T_{15} \\), where \\( T_n \\) denotes the \\( n \\)-th triangular number, we u... | 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": "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. |
[
{
"role": "user",
"content": "If the perimeter of rectangle ABCD is 20 inches, what is the least value of diagonal AC in inches?"
},
{
"role": "assistant",
"content": "To find the least value of the diagonal \\( AC \\) in a rectangle \\( ABCD \\) with a perimeter of 20 inches, we start by noting... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Find a binomial coefficient equal to the expression $\\binom{n}{k} + 3\\binom{n}{k-1} + 3\\binom{n}{k-2} + \\binom{n}{k-3}$."
},
{
"role": "assistant",
"content": "To solve the problem of finding a binomial coefficient equal to the expression \\(\\binom{n}{k} + 3\\b... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If Jose has 6 favorite DVDs, each 20 minutes long, and he wants to watch all 6 DVDs in every conceivable order, how long (in hours) will it take him to accomplish this?"
},
{
"role": "assistant",
"content": "To determine how long it will take Jose to watch all 6 of ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers?"
},
{
"role": "assistant",
"content": "To solve the problem where the sum of two numbers is 45 and their difference is 3, we can use a system of equations. Let the two nu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the last digit of $2005^{2005}$?"
},
{
"role": "assistant",
"content": "To determine the last digit of \\(2005^{2005}\\), we need to find \\(2005^{2005} \\mod 10\\).\n\nFirst, observe that the last digit of a number is equivalent to the number modulo 10. The... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are 50%, 25%, and 20%, respectively. All three bags of candy are dumped into one bowl. What is the closest ratio of yellow jelly beans to all beans in the bowl?... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Nathan will roll two six-sided dice. What is the probability that he will roll a number less than three on the first die and a number greater than three on the second die? Express your answer as a common fraction."
},
{
"role": "assistant",
"content": "To find the p... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find all integers $N$ such that $N^3 = 9k + 2$, where $k$ is an integer."
},
{
"role": "assistant",
"content": "To find all integers \\( N \\) such that \\( N^3 = 9k + 2 \\) where \\( k \\) is an integer, we need to determine if there exists an integer \\( N \\) for... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the greatest possible positive integer value of $x$ if $\\displaystyle\\frac{x^4}{x^2} < 10$?"
},
{
"role": "assistant",
"content": "To solve the inequality \\(\\frac{x^4}{x^2} < 10\\) for the greatest possible positive integer value of \\(x\\), we start by ... | 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": "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. |
[
{
"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. |
[
{
"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. |
[
{
"role": "user",
"content": "Given that \\( m \\) and \\( n \\) are positive integers and \\( m^2 + 5n^2 = 36 \\), find the value of \\( m + 5n \\)."
},
{
"role": "assistant",
"content": "To solve the equation \\( m^2 + 5n^2 = 36 \\) for positive integers \\( m \\) and \\( n \\), we will system... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How can you solve for \\( x \\) in the congruence \\( 2x \\equiv 4 \\pmod{5} \\)?"
},
{
"role": "assistant",
"content": "To solve the congruence \\( 2x \\equiv 4 \\pmod{5} \\), we need to isolate \\( x \\). This involves finding the modular inverse of 2 modulo 5, wh... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_{5}$ dollars worth of gold coins, and $124_{5}$ dollars worth of rubbing alcohol. What is the total dolla... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Solve for \\( x \\) in the congruence equation \\( 100 + 7x \\equiv 9 \\pmod{11} \\)."
},
{
"role": "assistant",
"content": "To solve the congruence equation \\( 100 + 7x \\equiv 9 \\pmod{11} \\), follow these steps:\n\n1. **Simplify the equation**:\n \\[\n 100 ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can one choose 6 candies from 8 available brands?"
},
{
"role": "assistant",
"content": "To solve the problem of determining the number of ways to choose 6 candies from 8 available brands, we use the concept of combinations with repetition. The form... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the range of a set of three numbers given... | You are a helpful assistant. |
[
{
"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. |
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