conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "Given the equation \\((x-7)^2 + (y + 4)^2 = 9\\), how do we determine whether to leave it as a circle equation or to divide by the constant to fit the ellipse equation?"
},
{
"role": "assistant",
"content": "Given the equation \\((x-7)^2 + (y+4)^2 = 9\\), we need to... | You are a helpful assistant. |
[
{
"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": "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": "Let \\( f \\) be an entire function on \\( \\mathbb{C} \\). Let \\( g(z) = \\overline{f(\\overline{z})} \\). Find when \\( f = g \\)."
},
{
"role": "assistant",
"content": "To determine when \\( f = g \\) for an entire function \\( f \\) and \\( g(z) = \\overline{f(... | 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 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": "A circle passes through vertices A and D and touches side BC of a square with side length 2. What is the radius of the circle?"
},
{
"role": "assistant",
"content": "Given a square with side length 2, vertices \\( A(0,0) \\), \\( B(2,0) \\), \\( C(2,2) \\), and \\( ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum:\n\\[\n\\frac{\\sqrt{1+1} - \\sqrt{1}}{\\sqrt{1^2 + 1}} + \\frac{\\sqrt{2+1} - \\sqrt{2}}{\\sqrt{2^2 + 2}} + \\frac{\\sqrt{3+1} - \\sqrt{3}}{\\sqrt{3^2 + 3}} + \\ldots + \\frac{\\sqrt{15+1} - \\sqrt{15}}{\\sqrt{15^2 + 15}}\n\\]"
},
{
"role": "assistant",
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( S \\) be the set of permutations of \\( n \\geq 100 \\) objects. Consider the experiment of choosing an element of \\( S \\) randomly with equal probabilities. Let \\( f \\) be the random variable defined by \\( f(\\sigma) = |\\{j : \\sigma(j) = j\\}| \\), so \\( f(\\si... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( f(x) = \\sqrt{x-1} \\). Is \\( f \\) differentiable at all \\( x \\) in its domain? If not, where and why does it fail to be differentiable?"
},
{
"role": "assistant",
"content": "To determine if the function \\( f(x) = \\sqrt{x-1} \\) is differentiable at a... | 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": "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": "What is the largest prime factor of $$(25!)^3 - (24!)^3$$?"
},
{
"role": "assistant",
"content": "To find the largest prime factor of \\((25!)^3 - (24!)^3\\), we start by factoring the expression.\n\nFirst, note that \\(25! = 25 \\times 24!\\). Therefore, we can rew... | 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": "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": "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": "Find all limit points of the sequence \\( x_n = \\cos(n) \\)."
},
{
"role": "assistant",
"content": "To find all limit points of the sequence \\( x_n = \\cos(n) \\), we need to analyze the behavior of the sequence \\( \\cos(n) \\) where \\( n \\) is an integer.\n\n#... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of $\\dfrac{3 \\times 4}{6}?$"
},
{
"role": "assistant",
"content": "To solve the expression \\(\\frac{3 \\times 4}{6}\\), we follow the order of operations, which dictates that we perform multiplication before division. Here are the steps:\n\n1. F... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many multiples of 10 are between 9 and 101?"
},
{
"role": "assistant",
"content": "To determine how many multiples of 10 are between 9 and 101, we first identify the smallest and largest multiples of 10 within this range.\n\nThe smallest multiple of 10 greater t... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given a polynomial \\( P(x) \\) such that \\( P(2) = 5 \\) and \\( P(3) = 7 \\), find the polynomial \\( u(x) \\) in the expression \\( P(x) = (x-2)(x-3)p(x) + u(x) \\)."
},
{
"role": "assistant",
"content": "To find the polynomial \\( u(x) \\) in the expression \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be non-zero real numbers such that $a + b + c = 0.$ Find all possible values of\n\\[\\frac{a^3 + b^3 + c^3}{abc}.\\]Enter all the possible values, separated by commas."
},
{
"role": "assistant",
"content": "Given non-zero real numbers \\(a\\), \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many triangles can be formed by connecting 5 points on one line and 12 points on a parallel line?"
},
{
"role": "assistant",
"content": "To determine the number of triangles that can be formed by connecting 5 points on one line and 12 points on a parallel line, ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is $3^{-1} + 3^{-2} \\pmod{25}$? Express your answer as an integer from $0$ to $24$, inclusive."
},
{
"role": "assistant",
"content": "To solve the problem \\(3^{-1} + 3^{-2} \\pmod{25}\\), we need to find the multiplicative inverses of \\(3\\) and \\(3^2\\) mo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given \\( f(t) = t + \\ln(t) \\), find \\( \\frac{d(f^{-1})}{dt} \\) at \\( t = 1 + e \\). Provide exact values."
},
{
"role": "assistant",
"content": "To find \\( \\frac{d(f^{-1})}{dt} \\) at \\( t = 1 + e \\) for the function \\( f(t) = t + \\ln(t) \\), we follow ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways are there to color the vertices of a square with six colors if rotations are considered the same?"
},
{
"role": "assistant",
"content": "To determine the number of distinct ways to color the vertices of a square using six colors, considering that rotat... | 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": "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": "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": "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": "Evaluate the limit $\\lim_{x\\rightarrow 6} f(x)$ for the piecewise function:\n\\[ f(x) = \\begin{cases} \nx^2 - 4, & 0 < x < 6 \\\\ \n4, & x = 6 \\\\ \n8x + 2, & x > 6 \n\\end{cases} \\]"
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{x \\to ... | 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": "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": "In triangle PQR, the sides PQ and QR have lengths 6 inches and 8 inches, respectively. The median PM is 4 inches. What is the length of PR in inches?"
},
{
"role": "assistant",
"content": "To find the length of \\( PR \\) in triangle \\( PQR \\) where \\( PQ = 6 \\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Consider the relation \\((x^2 + x)^N = 1\\), where \\(N\\) is a positive integer and \\(x \\in \\mathbb{C}\\). As \\(N\\) gets larger, the complex solutions trace out a graph of an equation. What is this equation?"
},
{
"role": "assistant",
"content": "To determine ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In how many ways can a postman deliver seven letters to seven neighbors such that exactly two of the neighbors receive the wrong letters?"
},
{
"role": "assistant",
"content": "To determine the number of ways a postman can deliver seven letters to seven neighbors su... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $K$ be a compact convex set with non-zero interior in $\\mathbb{R}^n$. Let $E_1$ and $E_2$ be the ellipsoids of minimal and maximal volume that contain and are contained in $K$, respectively. What is the best upper bound for the ratio of their volumes?"
},
{
"role":... | 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": "What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?"
},
{
"role": "assistant",
"content": "To determine the value of \\( c \\) such that \\( x - 5 \\) is a factor of the polynomial \\( P(x) = x^3 + 2x^2 +... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the product \\(\\sin\\left(\\frac{\\pi}{199}\\right) \\cdot \\sin\\left(\\frac{2\\pi}{199}\\right) \\cdot \\sin\\left(\\frac{3\\pi}{199}\\right) \\cdots \\sin\\left(\\frac{198\\pi}{199}\\right)\\)."
},
{
"role": "assistant",
"content": "To evaluate the prod... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Consider the system of equations:\n\\[w + x + y + z = 0\\]\n\\[3w + 2x + y = 3\\]\nIf the four variables \\(w, x, y, z\\) are integers, determine whether the solution of the system is finite or infinite."
},
{
"role": "assistant",
"content": "To determine whether th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( K \\) be a commutative field. Find all prime ideals of the polynomial ring \\( K[X] \\)."
},
{
"role": "assistant",
"content": "To find all prime ideals of the polynomial ring \\( K[X] \\) where \\( K \\) is a commutative field, we proceed as follows:\n\n1. ... | You are a helpful assistant. |
[
{
"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": "Compute $\\sin 270^\\circ$."
},
{
"role": "assistant",
"content": "To compute \\(\\sin 270^\\circ\\), we use the unit circle, which is a circle with a radius of 1 centered at the origin of the coordinate system. The angle \\(270^\\circ\\) is measured counterclockwis... | 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": "Suppose that $x^4$ is a prime number. Find all $a, b, c, d \\in \\mathbb{Q}$ such that $ax^3 + bx^2 + cx + d = 0$."
},
{
"role": "assistant",
"content": "Given that \\( x^4 \\) is a prime number \\( p \\), we have \\( x^4 = p \\). This implies that \\( x \\) is a ro... | 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": "Andrew's school has 1024 lockers, all initially closed. He starts from locker number one and toggles every locker (opens if closed, closes if open). He then toggles every 2nd locker, every 3rd locker, and so on, up to every 1024th locker. How many lockers are closed after he fi... | 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": "The population of Central Java is $25\\%$ of the population of the island of Java and $15\\%$ of the population of Indonesia. What percentage of Indonesia's population lives outside Java?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A suitcase lock has 3 dials with the digits $0, 1, 2,..., 9$ on each. How many different settings are possible if all three digits have to be different?"
},
{
"role": "assistant",
"content": "To determine the number of different settings for a suitcase lock with 3 d... | 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": "How many positive two-digit integers are there in which each of the two digits is prime?"
},
{
"role": "assistant",
"content": "To determine how many positive two-digit integers have each digit as a prime number, we first identify the single-digit prime numbers. The... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\(a, b, c, d \\in \\mathbb{N}^+\\), \\( \\log_a b = \\frac{3}{2} \\), \\( \\log_c d = \\frac{5}{4} \\), and \\( a - c = 9 \\). Find the value of \\( b - d \\)."
},
{
"role": "assistant",
"content": "Given \\(a, b, c, d \\in \\mathbb{N}^+\\), with \\(\\log_a b =... | 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": "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": "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": "The ''roundness'' of an integer greater than 1 is the sum of the exponents of the prime factorization of the number. For example, $20 = 2^2 \\times 5^1$, so 20 has a roundness of 3. What is the roundness of 1,000,000?"
},
{
"role": "assistant",
"content": "To find t... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit \\(\\lim_{x \\to 0} \\left( \\frac{1^x + 2^x + \\cdots + n^x}{n} \\right)^{1/x}\\)."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{x \\to 0} \\left( \\frac{1^x + 2^x + \\cdots + n^x}{n} \\right)^{1/x}\\), we proceed as foll... | 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": "What is the probability of having 4 boys and 4 girls in a family of 8 children?"
},
{
"role": "assistant",
"content": "To find the probability of having exactly 4 boys and 4 girls in a family of 8 children, we can use the binomial probability formula. Here are the d... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Ten distinct points are identified on the circumference of a circle. How many different convex quadrilaterals can be formed if each vertex must be one of these 10 points?"
},
{
"role": "assistant",
"content": "To determine the number of different convex quadrilatera... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can the word HALL be arranged so that the l's are together?"
},
{
"role": "assistant",
"content": "To solve the problem of finding how many ways the word \"HALL\" can be arranged such that the two L's are together, we can treat the two L's as a single ... | 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": "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": "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": "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": "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": "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": "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": "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": "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": "Let\n\\[z = \\frac{(-11 + 13i)^3 \\cdot (24 - 7i)^4}{3 + 4i},\\]and let $w = \\frac{\\overline{z}}{z}.$ Compute $|w|.$"
},
{
"role": "assistant",
"content": "To compute \\( |w| \\) where \\( w = \\frac{\\overline{z}}{z} \\) and \\( z \\) is given by the expression\... | You are a helpful assistant. |
[
{
"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 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": "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": "Find all pairs of positive integers \\( (x, y) \\) such that \\( x^6 = y^2 + 127 \\)."
},
{
"role": "assistant",
"content": "To find all pairs of positive integers \\((x, y)\\) that satisfy the equation \\(x^6 = y^2 + 127\\), we start by rewriting the equation as a ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the limit of the sequence $\\{z_n\\}$ where $z_n = i^{n!}$."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The roots of the polynomial \\( P(x) = x^3 - ax^2 + bx - c \\) are the real numbers \\( a \\), \\( b \\), and \\( c \\). Given that \\( a, b, c \\ne 0 \\), what is \\( P(10) \\)?"
},
{
"role": "assistant",
"content": "Given the polynomial \\( P(x) = x^3 - ax^2 + bx ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the number of subsets with 20 elements of a set with 52 elements?"
},
{
"role": "assistant",
"content": "To find the number of subsets with 20 elements from a set of 52 elements, we use the concept of combinations. The number of ways to choose 20 elements fr... | 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": "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": "Within a group of 2009 people, every two people have exactly one common friend. Find the least value of the difference between the person with the maximum number of friends and the person with the minimum number of friends."
},
{
"role": "assistant",
"content": "To ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\( \\int \\frac{dx}{x(1+(\\ln x)^2)} \\) without using trigonometric substitution."
},
{
"role": "assistant",
"content": "To evaluate the integral \\( \\int \\frac{dx}{x(1+(\\ln x)^2)} \\), we use the substitution \\( u = \\ln x \\). This subs... | 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": "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": "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": "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": "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": "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": "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": "Find the sum of the digits of the largest even three-digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged."
},
{
"role": "assistant",
"content": "To find the sum of the digits of the largest even three-di... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Andi, Beni, Coki, Doni, and Edo are playing a game where each player is either a mousedeer (who always tells the truth) or a wolf (who always lies). Andi says that Beni is a mousedeer. Coki says Doni is a wolf, while Edo says Andi is not a wolf. Beni says Coki is not a mousedee... | 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": "On an old-fashioned bicycle the front wheel has a radius of $2.5$ feet and the back wheel has a radius of $4$ inches. If there is no slippage, how many revolutions will the back wheel make while the front wheel makes $100$ revolutions?"
},
{
"role": "assistant",
"c... | 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": "Find the number of quadruples $(x, y, z, w)$ of non-negative integers that satisfy the inequality $w + x + y + z \\le 1992$."
},
{
"role": "assistant",
"content": "To find the number of quadruples \\((w, x, y, z)\\) of non-negative integers that satisfy the inequali... | You are a helpful assistant. |
[
{
"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": "From a 50-foot-tall lookout tower, a park ranger sees a fire at an angle of depression of 1.6°. How far is the fire from the tower?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the distance from the 50-foot-tall lookout tower to ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral $\\int_{0}^{t} sgn(x) \\ dx$, where $sgn(x) = -1$ for $x < 0$, $sgn(0) = 0$, and $sgn(x) = 1$ for $x > 0$."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int_{0}^{t} \\text{sgn}(x) \\, dx\\), we need to consider the behavi... | 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. |
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