conversations listlengths 2 2 | system stringclasses 1
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[
{
"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": "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. |
[
{
"role": "user",
"content": "Given a rectangular prism $ABCDEFGH$ with $A=(0,0,0)$ and $G=(4,3,2)$, and sides parallel to the coordinate axes, find the coordinates of the midpoint $M$ of segment $FG$."
},
{
"role": "assistant",
"content": "To find the coordinates of the midpoint \\( M \\) of se... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a right triangle, the square of the hypotenuse is equal to twice the product of the legs. What is one of the acute angles of the triangle?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by defining the sides of the right triangle. Let the l... | 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": "Find the number of possible solutions to the Diophantine equation \\( w + x + y + z = n \\)."
},
{
"role": "assistant",
"content": "To find the number of possible solutions to the Diophantine equation \\( w + x + y + z = n \\) where \\( w, x, y, z \\) are non-negati... | 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": "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": "For \\( x, y, z, \\) and \\( w \\ge 0 \\), compute the smallest value of \\( x \\) satisfying the following system:\n\\[ y = x - 2001, \\]\n\\[ z = 2y - 2001, \\]\n\\[ w = 3z - 2001. \\]"
},
{
"role": "assistant",
"content": "To find the smallest value of \\( x \\) ... | You are a helpful assistant. |
[
{
"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. |
[
{
"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": "How many ways can 8 queens be arranged on an 8x8 chessboard so that no two queens can attack each other?"
},
{
"role": "assistant",
"content": "To solve the problem of determining how many ways 8 queens can be arranged on an 8x8 chessboard so that no two queens can ... | 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": "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. |
[
{
"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": "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": "In how many ways can the word $MISSISSIPPI$ be arranged such that the vowels are in alphabetical order?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the number of ways to arrange the word \"MISSISSIPPI\" such that the vowels are in alphabet... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Let \\( n, r \\in \\mathbb{N} \\) such that \\( r \\leq n \\). Find the value of \\[ C(n-1,0) + C(n,1) + C(n+1,2) + \\cdots + C(n+r-1,r) \\] where \\( C(n,r) = \\frac{n!}{(n-r)!r!} \\)."
},
{
"role": "assistant",
"content": "To find the value of the sum \\( C(n-1,0)... | You are a helpful assistant. |
[
{
"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. |
[
{
"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. |
[
{
"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": "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": "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": "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": "Given that $\\alpha$ and $\\beta$ are the real roots of the equation $x^2 - 1154x + 1 = 0$, find the value of $\\sqrt[4]{\\alpha} + \\sqrt[4]{\\beta}$."
},
{
"role": "assistant",
"content": "Given the quadratic equation \\(x^2 - 1154x + 1 = 0\\) with real roots \\(\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "It can be shown that for any positive integer $n,$\n\\[\\begin{pmatrix} 1 & 1 \\\\ 1 & 0 \\end{pmatrix}^n = \\begin{pmatrix} F_{n + 1} & F_n \\\\ F_n & F_{n - 1} \\end{pmatrix},\\]where $F_n$ denotes the $n$th Fibonacci number.\n\nCompute $F_{784} F_{786} - F_{785}^2.$"
},
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ordered triples of positive integers $(x, y, z)$ satisfy the equation $x^{2018} + y^{2018} = z^{2018}$?"
},
{
"role": "assistant",
"content": "To determine the number of ordered triples of positive integers \\((x, y, z)\\) that satisfy the equation \\(x^{20... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the equation \\(x^2 + y^2 + z^2 = 1\\), find the maximum value of \\(x + 2y + 3z\\)."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | 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": "Write down a definite integral that will give the value of the area under the curve \\( y = x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) between \\( x = \\frac{1}{2}\\pi \\) and \\( x = \\frac{5}{2}\\pi \\). (The expression \\( x^2 \\cos\\left(\\frac{1}{5}x\\right) \\) takes no ne... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find all points \\( z \\) such that the series \\( \\sum_{n=0}^{\\infty} z^n \\) converges."
},
{
"role": "assistant",
"content": "To find all points \\( z \\) such that the series \\( \\sum_{n=0}^{\\infty} z^n \\) converges, we start by recognizing that this is a g... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Using the balance-scale shown, determine how many ♦ are necessary to replace the ? in the last picture. The given balance equations are:\n1. ▼♦ = ○\n2. ▼ = ♦◘\n3. ○○ = ◘◘◘\n4. ▼ = ?\n\nExplain your reasoning."
},
{
"role": "assistant",
"content": "To determine how m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Integrate \\(\\frac{\\sin x}{x}\\) with respect to \\(x\\)."
},
{
"role": "assistant",
"content": "The integral of \\(\\frac{\\sin x}{x}\\) with respect to \\(x\\) cannot be expressed in terms of elementary functions. Instead, it is represented by a special function... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Determine whether the function \\( g(x) = \\frac{2}{x - \\sqrt{2}} \\) is continuous at other points in its domain."
},
{
"role": "assistant",
"content": "To determine whether the function \\( g(x) = \\frac{2}{x - \\sqrt{2}} \\) is continuous at other points in its ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find all normal subgroups of \\( S_n \\) for all \\( n \\geq 5 \\)."
},
{
"role": "assistant",
"content": "To find all normal subgroups of \\( S_n \\) for \\( n \\geq 5 \\), we start by recalling the definitions and properties of normal subgroups and the symmetric g... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\sqrt{5 + x} + \\sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$?"
},
{
"role": "assistant",
"content": "To solve the problem, we start with the given equation:\n\n\\[\n\\sqrt{5 + x} + \\sqrt{20 - x} = 7\n\\]\n\nLet \\( A = \\sqrt{5 + x} \\) and \\( B... | 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": "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": "Solve for $\\log(i)$ given the expression $Bz \\cdot \\log(i) - Bh \\cdot \\log(i)$ and the result $\\log(i) = \\frac{1}{Bz - Bh}$."
},
{
"role": "assistant",
"content": "To solve for \\(\\log(i)\\) given the expression \\(Bz \\cdot \\log(i) - Bh \\cdot \\log(i)\\) ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In the expansion of $(x^2 + y - z^3)^8$, find the term that involves $x^{10} z^6$ using the binomial theorem approach."
},
{
"role": "assistant",
"content": "To find the term in the expansion of \\((x^2 + y - z^3)^8\\) that involves \\(x^{10} z^6\\), we use the mult... | You are a helpful assistant. |
[
{
"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. |
[
{
"role": "user",
"content": "Abigail, Beatrice, and Carson combine their eggs to sell them at the market. If Abigail has 37 eggs, Beatrice has 49 eggs, and Carson has 14 eggs, and if eggs can only be sold in cartons of 12, how many eggs will be left over if all cartons are sold?"
},
{
"role": "assi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Carol has a fair die with faces labeled $1$ to $6$. She rolls the die once, and if she rolls a $1$, $2$, or $3$, she rolls the die a second time. What is the expected value of her last roll?"
},
{
"role": "assistant",
"content": "To find the expected value of Carol'... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many spheres are there that are tangent to all lines containing the edges of a triangular pyramid?"
},
{
"role": "assistant",
"content": "To determine the number of spheres that are tangent to all lines containing the edges of a triangular pyramid (tetrahedron),... | 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": "How many points of intersection are there when all the diagonals of a decagon are drawn?"
},
{
"role": "assistant",
"content": "To determine the number of intersection points formed by all the diagonals of a decagon, we need to consider the following:\n\n1. **Unders... | You are a helpful assistant. |
[
{
"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. |
[
{
"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": "Find the range and domain of the function \\( f(x) = a \\sin x + b \\cos x \\)."
},
{
"role": "assistant",
"content": "To find the domain and range of the function \\( f(x) = a \\sin x + b \\cos x \\), we proceed as follows:\n\n### Domain:\nThe function \\( f(x) = a... | 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": "Calculate \\( \\frac{d}{d \\epsilon } \\left( \\| \\dot{ \\mathbf{x} } ^{ \\epsilon } \\| ^{2} \\right) \\) where \\( \\dot{ \\mathbf{x} } ^{ \\epsilon } = \\frac{ \\mathbf{x}(t) + \\epsilon \\mathbf{z} (t) }{ \\| \\mathbf{x} (t) + \\epsilon \\mathbf{z} (t) \\| } \\) at the poi... | 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": "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": "What value of $k$ will make $x^2 - 16x + k$ the square of a binomial?"
},
{
"role": "assistant",
"content": "To determine the value of \\( k \\) that makes the quadratic expression \\( x^2 - 16x + k \\) the square of a binomial, we need to recognize the form of a pe... | 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": "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": "Given the production rates of 8 First Babies or 20 Real Babies per hour, and a maximum of 48 hours per week, derive the inequality \\(5x + 2y \\leq 1920\\) where \\(x\\) is the number of First Babies and \\(y\\) is the number of Real Babies. Explain the origin of the constant 1... | You are a helpful assistant. |
[
{
"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. |
[
{
"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": "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": "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": "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": "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": "If 25,197,624 hot dogs are packaged in sets of 4, how many will be left over?"
},
{
"role": "assistant",
"content": "To determine how many hot dogs will be left over if 25,197,624 hot dogs are packaged in sets of 4, we need to find the remainder when 25,197,624 is d... | 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": "Does the infinite geometric series $1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + \\cdots$ converge? If it does, what does it converge to?"
},
{
"role": "assistant",
"content": "To determine if the infinite geometric series \\(1 - 1 + 1 - 1 + 1 - 1 + \\cdots\\) converges, we star... | 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 possible values of $\\sec \\beta$ given that $2 \\tan^2 \\beta = 3 + 9 \\sec \\beta$."
},
{
"role": "assistant",
"content": "To find the possible values of \\(\\sec \\beta\\) given the equation \\(2 \\tan^2 \\beta = 3 + 9 \\sec \\beta\\), we start by using ... | 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": "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 the value of $2^x + 2^{-x}$ is $\\sqrt[3]{3}$, find the exact value of $8^x + 8^{-x}$. Give the final result in its simplest form."
},
{
"role": "assistant",
"content": "Given that \\(2^x + 2^{-x} = \\sqrt[3]{3}\\), we need to find the value of \\(8^x + 8^{-x}\\)... | 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": "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": "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": "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": "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": "In how many ways can three judges vote on three performers A, B, and C, listing their order of preference, such that two judges agree in their order of preference, while the third differs?"
},
{
"role": "assistant",
"content": "To solve the problem of determining th... | 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 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": "Given \\( x = y^2 \\) and \\( y = x^2 \\), where \\( x \\) and \\( y \\) are complex numbers, find the least possible real value of \\( x^2 + x + 1 \\)."
},
{
"role": "assistant",
"content": "Given the equations \\( x = y^2 \\) and \\( y = x^2 \\), we start by subst... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine if the function \\( g: \\mathbb{N} \\to \\mathbb{N} \\) defined by \\( g(x) = \\left\\lfloor \\frac{3x+1}{3} \\right\\rfloor \\) is bijective and find its inverse if it exists."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | 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": "If there is a 20% chance that Jane will bump into Taylor in the hallway between period 2 and 3, and if she bumps into Taylor, there is a 25% chance that she will be late for class, what is the probability that Jane will be late for class on any given day?"
},
{
"role": ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify the following: $\\frac{4+3+2+1}{6+5+3+2+1}$."
},
{
"role": "assistant",
"content": "To simplify the fraction \\(\\frac{4+3+2+1}{6+5+3+2+1}\\), we first need to add the numbers in the numerator and the denominator separately.\n\nFirst, we calculate the numer... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many different AoPS avatars are possible using 3 colors to color the differently oriented rhombuses, given that there are 3 different orientations possible?"
},
{
"role": "assistant",
"content": "To determine the number of different AoPS avatars possible using 3... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the mean value of the measures of the three interior angles of any triangle?"
},
{
"role": "assistant",
"content": "To find the mean value of the measures of the three interior angles of any triangle, we start by recalling a fundamental geometric fact: the s... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of $(2x + 5)^2$ when $x = 3$?"
},
{
"role": "assistant",
"content": "To find the value of \\((2x + 5)^2\\) when \\(x = 3\\):\n\n1. Substitute \\(x = 3\\) into the expression:\n \\[\n (2 \\cdot 3 + 5)\n \\]\n\n2. Calculate the expression insid... | 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": "The 5-digit number $52\\,28\\square$ is a multiple of 6. Which digit is represented by $\\square$?"
},
{
"role": "assistant",
"content": "To determine the digit represented by \\(\\square\\) in the 5-digit number \\(5228\\square\\) such that the number is a multipl... | You are a helpful assistant. |
[
{
"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. |
[
{
"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": "How many different six-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 7?"
},
{
"role": "assistant",
"content": "To determine how many different six-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 7, we follow these steps:\n\n1. **Ide... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Jimmy notices $7$ oranges weigh the same as $5$ apples. If Jimmy has $28$ oranges, how many apples would Jimmy need to equal the weight of his $28$ oranges?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by noting the given relationship: 7 or... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the fourth power of $\\sqrt{1+\\sqrt{1+\\sqrt{1}}}$?"
},
{
"role": "assistant",
"content": "To find the fourth power of \\(\\sqrt{1+\\sqrt{1+\\sqrt{1}}}\\), we start by simplifying the expression step by step.\n\n1. Compute the innermost expression:\n \\[\... | 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": "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. |
[
{
"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": "While shopping at the local mall, Ann walked slowly (at a constant speed) down a descending escalator, taking 10 steps to reach the last step. She then ran up the same (descending) escalator with a speed five times greater than the speed with which she had walked down, taking 2... | You are a helpful assistant. |
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