conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "When $f(x) = ax^3 - 6x^2 + bx - 5$ is divided by $x - 1,$ the remainder is $-5.$ When $f(x)$ is divided by $x + 2,$ the remainder is $-53.$ Find the ordered pair $(a,b).$"
},
{
"role": "assistant",
"content": "To solve the problem, we use the Remainder Theorem whi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a case with five different men, if nine victims randomly guess which voice belongs to the criminal, what is the probability that all nine victims would select the same person?"
},
{
"role": "assistant",
"content": "To find the probability that all nine victims ra... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "How many prime numbers are between 30 and 40?"
},
{
"role": "assistant",
"content": "To determine how many prime numbers are between 30 and 40, we first list the numbers in this range: 31, 32, 33, 34, 35, 36, 37, 38, 39.\n\nA prime number is a number greater than 1 ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Adam and Simon start on bicycle trips from the same point at the same time. Adam travels east at 8mph and Simon travels south at 6mph. After how many hours are they 60 miles apart?"
},
{
"role": "assistant",
"content": "To solve the problem of finding after how many... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many 5-digit numbers have at least one zero?"
},
{
"role": "assistant",
"content": "To determine the number of 5-digit numbers that have at least one zero, we can use the principle of complementary counting. This involves finding the total number of 5-digit numb... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.)"
},
{
"role": "assistant",
"content": "To determine the numbe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What are the first differences and second differences of the function $y = -2x^2 + 1$?"
},
{
"role": "assistant",
"content": "To find the first and second differences of the function \\( y = -2x^2 + 1 \\), we will follow these steps:\n\n1. **Choose a set of consecut... | 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": "Evaluate: $\\frac{10^{-2}5^0}{10^{-3}}$"
},
{
"role": "assistant",
"content": "To evaluate the expression \\(\\frac{10^{-2} \\cdot 5^0}{10^{-3}}\\), we proceed as follows:\n\n1. Simplify the numerator:\n - Recall that any number to the power of zero is 1, so \\(5^... | 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": "Let $f(x) = 4x + c$ and $g(x) = cx + 2.$ If $f(g(x)) = 12x + d,$ then find $d.$"
},
{
"role": "assistant",
"content": "To find the value of \\( d \\), we start by composing the functions \\( f(x) = 4x + c \\) and \\( g(x) = cx + 2 \\). Substituting \\( g(x) \\) int... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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. |
[
{
"role": "user",
"content": "Given that \\( p^2 + \\frac{1}{p^2} = m \\), find the value of \\( p + \\frac{1}{p} \\)."
},
{
"role": "assistant",
"content": "To find the value of \\( p + \\frac{1}{p} \\) given that \\( p^2 + \\frac{1}{p^2} = m \\), we can proceed as follows:\n\nFirst, we start b... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Carbon-10 is a radioactive isotope with a half-life of 19.255 seconds. Suppose you start with 1 gram of carbon-10. Determine the function \\( f(t) \\) that represents the amount of carbon-10 left after \\( t \\) seconds. Use a formula of the form \\( f(t) = A \\cdot 2^{-t/T} \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$?"
},
{
"role": "assistant",
"content": "To find the measure of ang... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many different 10-letter arrangements of the letters in the word \"GOPALADINS\" are there?"
},
{
"role": "assistant",
"content": "To determine the number of different 10-letter arrangements of the letters in the word \"GOPALADINS,\" we first identify the letters... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the inverse Laplace transform $\\mathcal{L}^{-1} \\left[ \\frac{P-1}{P^2-2P+5} \\right]$."
},
{
"role": "assistant",
"content": "To find the inverse Laplace transform \\(\\mathcal{L}^{-1} \\left[ \\frac{P-1}{P^2-2P+5} \\right]\\), we start by completing the squ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit $\\underset{x\\to \\infty }{\\mathop{\\lim }}\\,\\left( \\sin \\left( x+1 \\right)-\\sin x \\right)$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{x \\to \\infty} \\left( \\sin(x+1) - \\sin(x) \\right)\\), we will use a t... | 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": "During a one-hour race, the velocities of two cars are given by \\( V_1(t) = 52(1 - \\cos(\\pi t)) \\) and \\( V_2(t) = 99t \\), where \\( 0 \\leq t \\leq 1 \\). If both cars start at the mile 0 mark, which car travels the farthest during the hour?"
},
{
"role": "assist... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the sequence \\( (u_n) \\) defined by:\n\\[ u_1 = 3 \\]\n\\[ u_n = 2u_{n-1} + 5n \\]\nFind a formula for \\( u_n \\) in terms of \\( n \\)."
},
{
"role": "assistant",
"content": "To find a formula for the sequence \\( (u_n) \\) defined by:\n\\[ u_1 = 3 \\]\n\\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $10^{-1}\\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive."
},
{
"role": "assistant",
"content": "To compute \\(10^{-1} \\pmod{1001}\\), we need to find an integer \\(x\\) such that \\(10x \\equiv 1 \\pmod{1001}\\). This means we ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the derivative $\\frac{d}{dx} \\sqrt{2x-1}$."
},
{
"role": "assistant",
"content": "To find the derivative of \\(\\sqrt{2x - 1}\\) with respect to \\(x\\), we will use the chain rule. The chain rule states that if \\(y = f(g(x))\\), then the derivative \\(y'\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find \\((1+z)^n\\), where \\(z = \\cos \\frac{2}{3} \\pi + i \\sin \\frac{2}{3} \\pi\\)."
},
{
"role": "assistant",
"content": "To find \\((1+z)^n\\) where \\(z = \\cos \\frac{2}{3} \\pi + i \\sin \\frac{2}{3} \\pi\\), we start by recognizing that \\(z\\) is a compl... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the limit $\\lim _{n \\rightarrow \\infty} \\sqrt[n]{n^{e}+e^{n}}$."
},
{
"role": "assistant",
"content": "To evaluate the limit \\(\\lim_{n \\to \\infty} \\sqrt[n]{n^e + e^n}\\), we start by analyzing the behavior of the terms inside the radical as \\(n\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The median of a set of 22 consecutive even integers is 25. What is the largest number in the set?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the largest number in a set of 22 consecutive even integers with a median of 25, we proceed as fo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In order to compute the area of a particular circle, Juan first measures the length of its diameter. The actual diameter is 20 cm, but Juan's measurement has an error of up to $20\\%$. What is the largest possible percent error, in percent, in Juan's computed area of the circle... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the length of rectangle ABCD if the width is 3 inches and its diagonal is 20 inches."
},
{
"role": "assistant",
"content": "To find the length of the rectangle \\(ABCD\\) where the width is 3 inches and the diagonal is 20 inches, we can use the Pythagorean theo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the integral\n$$\\int\\limits_{0}^{1}dx\\int\\limits_{x}^{1}dy\\int\\limits_{y}^{1}e^{z^3}dz$$"
},
{
"role": "assistant",
"content": "To solve the given triple integral\n\\[\n\\int_{0}^{1} \\int_{x}^{1} \\int_{y}^{1} e^{z^3} \\, dz \\, dy \\, dx,\n\\]\nwe ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the radius of convergence of the power series $\\sum_{n=0}^\\infty n^2 x^n$."
},
{
"role": "assistant",
"content": "To find the radius of convergence of the power series \\(\\sum_{n=0}^\\infty n^2 x^n\\), we can use the ratio test. The ratio test involves compu... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The square root of $x$ is greater than 2 and less than 4. How many integer values of $x$ satisfy this condition?"
},
{
"role": "assistant",
"content": "To solve the problem, we start with the given inequality:\n\n\\[ 2 < \\sqrt{x} < 4 \\]\n\nWe need to find the inte... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Monica tosses a fair 6-sided die. If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars). If the roll is composite, she wins nothing. Otherwise, she loses 3 dollars. What is the expected value of her... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the geometric mean of all real numbers in the interval $(0,1]$?"
},
{
"role": "assistant",
"content": "To find the geometric mean of all real numbers in the interval \\((0,1]\\), we use the formula for the geometric mean of a continuous distribution, which i... | 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": "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": "Evaluate the series \\( \\sum_{k=1}^{\\infty}\\frac{3^{k}}{k!} \\)."
},
{
"role": "assistant",
"content": "To evaluate the series \\( \\sum_{k=1}^{\\infty} \\frac{3^k}{k!} \\), we start by recalling the Taylor series expansion for the exponential function \\( e^x \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An equilateral triangle and a square have equal areas. Each shape has a circle inscribed within it. What is the ratio of the area of the circle inscribed in the triangle to the area of the circle inscribed in the square?"
},
{
"role": "assistant",
"content": "To fin... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Suppose $a$ and $b$ are unknown but neither of them are equal to 2, and there are exactly 65 unattainable scores. Determine $a$ and $b$."
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the values of \\(a\\) and \\(b\\) such that the... | 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 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": "Let $x$ be a real number such that $x^3+4x=8$. Determine the value of $x^7+64x^2$."
},
{
"role": "assistant",
"content": "Given the equation \\( x^3 + 4x = 8 \\), we need to determine the value of \\( x^7 + 64x^2 \\).\n\nFirst, rearrange the given equation to expres... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The area of triangle $ABC$ is $30 \\text{ cm}^2$. Point $D$ on $\\overline{AC}$ divides the side in the ratio $AD:DC = 2:3$. The length of the perpendicular $\\overline{DE}$ dropped to $\\overline{BC}$ is $9 \\text{ cm}$. Find $BC$."
},
{
"role": "assistant",
"conte... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Simplify $(2x - 5)(x + 7) - (x + 5)(2x - 1)$."
},
{
"role": "assistant",
"content": "To simplify the expression \\((2x - 5)(x + 7) - (x + 5)(2x - 1)\\), we will expand both products and then combine like terms.\n\nFirst, we expand \\((2x - 5)(x + 7)\\):\n\\[\n(2x - ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find all relative maxima or minima of the function \\( y = (\\log x)^2 \\) for \\( x > 0 \\)."
},
{
"role": "assistant",
"content": "To find the relative maxima or minima of the function \\( y = (\\ln x)^2 \\) for \\( x > 0 \\), we will follow these steps:\n\n1. **F... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the sum of the reciprocals of the natural-number factors of 6?"
},
{
"role": "assistant",
"content": "To find the sum of the reciprocals of the natural-number factors of 6, we first identify the factors of 6. The factors of 6 are the natural numbers that div... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A department store offers two promotions. Promotion A says, \"Buy one pair of shoes, get the second pair for half the price.\" Promotion B says, \"Buy one pair of shoes, get $\\$10$ off the second pair.\" Jane wants to buy two pairs of shoes that cost $\\$30$ each. She can only... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A giant, circular and perfectly flat pizza must be shared by 22 people. What is the least number of cuts to be made on the pizza so that each person can have a piece of pizza? (The pieces are not necessarily the same shape or with the same area.)"
},
{
"role": "assistan... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?"
},
{
"role": "assistant",
"content": "To determine the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock, we can follow these steps:\n\n1. *... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the limit \\( \\lim_{h \\rightarrow 0} \\frac{(x+h)^{x+h} - x^x}{h} \\)."
},
{
"role": "assistant",
"content": "To calculate the limit \\(\\lim_{h \\rightarrow 0} \\frac{(x+h)^{x+h} - x^x}{h}\\), we recognize that this expression represents the derivative ... | 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": "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": "If $\\displaystyle\\frac{q}{r} = 9$, $\\displaystyle\\frac{s}{r} = 6$, and $\\displaystyle \\frac{s}{t} = \\frac{1}{2}$, then what is $\\displaystyle\\frac{t}{q}$?"
},
{
"role": "assistant",
"content": "To find the value of \\(\\frac{t}{q}\\) given the equations \\(... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Suppose that $\\alpha$ is inversely proportional to $\\beta$. If $\\alpha = -3$ when $\\beta = -6$, find $\\alpha$ when $\\beta = 8$. Express your answer as a fraction."
},
{
"role": "assistant",
"content": "Given that \\(\\alpha\\) is inversely proportional to \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If \\( V \\) is invertible, can we derive a formula for its inverse \\( V^{-1} \\)?"
},
{
"role": "assistant",
"content": "To derive a formula for the inverse of an invertible matrix \\( V \\), we can use the relationship involving the adjugate matrix and the determ... | 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": "If $x \\not= y$ and $\\frac{x}{y} + x = \\frac{y}{x} + y$, find the sum $\\frac{1}{x} + \\frac{1}{y}$."
},
{
"role": "assistant",
"content": "To solve the equation \\(\\frac{x}{y} + x = \\frac{y}{x} + y\\) for \\(x \\neq y\\) and find the sum \\(\\frac{1}{x} + \\fra... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $f(x)=x^3+3$ and $g(x) = 2x^2 + 2x +1$. What is $g(f(-2))$?"
},
{
"role": "assistant",
"content": "To find \\( g(f(-2)) \\), we need to evaluate the inner function \\( f(-2) \\) first and then use that result as the input for the function \\( g \\).\n\n1. Evalua... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Jonah is inside a whale at sea. At the dawn of each day, there is a 25% chance that he will write in his diary, and at noon, there is a 1/3 chance that the whale will swallow him. If he remains stuck indefinitely, what is the probability that Jonah will be swallowed without wri... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve for $x$:\n\\[\n\\log_2 (x-5) + \\log_2 (x-3) = \\log_5 (5^3)\n\\]"
},
{
"role": "assistant",
"content": "To solve the equation \\(\\log_2 (x-5) + \\log_2 (x-3) = \\log_5 (5^3)\\), we will follow these steps:\n\n1. Simplify the right-hand side of the equation.\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, 2 cm, and 3 cm. Assuming the three holes a... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Nikolai is thinking of a positive real number $x$ such that $-5 < x < 1$. What is the probability that if his number is subtracted from $\\frac{1}{2}$, then the result is positive?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the sequence defined by \\( u_0 = \\frac{1}{3} \\) and \\( u_{n+1} = 2u_n^2 - 1 \\), find a closed form for \\( u_n \\)."
},
{
"role": "assistant",
"content": "To find a closed-form expression for the sequence defined by \\( u_0 = \\frac{1}{3} \\) and \\( u_{n... | You are a helpful assistant. |
[
{
"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": "Solve the following equation for real numbers \\(x\\) and \\(y\\):\n\\[\n(x^2+1)(y^2+1) - 2(x+1)(y+1) + 4 = 0\n\\]"
},
{
"role": "assistant",
"content": "To solve the equation \\((x^2 + 1)(y^2 + 1) - 2(x + 1)(y + 1) + 4 = 0\\) for real numbers \\(x\\) and \\(y\\), w... | 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": "Let $f(x)=-3x^2+x-4$, $g(x)=-5x^2+3x-8$, and $h(x)=5x^2+5x+1$. Express $f(x)+g(x)+h(x)$ as a single polynomial, with the terms in order by decreasing degree."
},
{
"role": "assistant",
"content": "To solve the problem of expressing \\( f(x) + g(x) + h(x) \\) as a si... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to determine the tot... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A merchant buys goods at $25\\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\\%$ on the marked price and still clear a profit of $25\\%$ on the selling price. What percent of the list price must he mark the goods?"
},
{
"role":... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $f(x)=2\\sqrt{x} + \\frac{12}{\\sqrt{x}}$ and let $g(x)=2x^2-2x-3$. What is $f(g(3))$?"
},
{
"role": "assistant",
"content": "To find \\( f(g(3)) \\), we need to follow these steps:\n\n1. Calculate \\( g(3) \\).\n2. Use the result from step 1 as the input to the... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine a vector equation of the plane through the point \\((-4,5,1)\\) and parallel to the vectors \\((-3,-5,3)\\) and \\((2,-1,-5)\\)."
},
{
"role": "assistant",
"content": "To determine the vector equation of the plane through the point \\((-4,5,1)\\) and paral... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An octopus told me that his underwater cave is $567_{8}$ years old. How many years is this in base ten?"
},
{
"role": "assistant",
"content": "To convert the octal number \\(567_8\\) to its decimal (base 10) equivalent, we need to express it as a sum of powers of 8.... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Among all the solutions, real and complex, of the system of equations\n\n\\[ x^2 + y^2 + x + y = 8 \\]\n\\[ xy + x + y = 5 \\]\n\nwhat is the least possible real part of \\( x \\)?"
},
{
"role": "assistant",
"content": "To solve the system of equations:\n\n\\[ x^2 +... | 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": "How many vertical and horizontal asymptotes does the graph of the function $f(x)=\\frac{e^x+e^{-x}}{e^x-e^{-x}}$ have?"
},
{
"role": "assistant",
"content": "To determine the number of vertical and horizontal asymptotes of the function \\( f(x) = \\frac{e^x + e^{-x}... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the value of $ \\frac{1}{2+\\frac{1}{2+\\frac{1}{2+\\frac{1}{2}}}}$."
},
{
"role": "assistant",
"content": "To compute the value of \\( \\frac{1}{2+\\frac{1}{2+\\frac{1}{2+\\frac{1}{2}}}} \\), we will simplify the expression step by step from the innermost f... | You are a helpful assistant. |
[
{
"role": "user",
"content": "An infinite geometric series has common ratio $\\frac{-1}{3}$ and sum $25.$ What is the second term of the sequence?"
},
{
"role": "assistant",
"content": "Given an infinite geometric series with a common ratio \\( r = \\frac{-1}{3} \\) and a sum \\( S = 25 \\), we... | 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": "Write the expression $\\frac{4+3c}{7}+2$ as a single fraction."
},
{
"role": "assistant",
"content": "To combine the expression \\(\\frac{4+3c}{7} + 2\\) into a single fraction, follow these steps:\n\n1. **Identify the common denominator**: The first term \\(\\frac{... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The quadratic $x^2-20x+36$ can be written in the form $(x+b)^2+c$, where $b$ and $c$ are constants. What is $b+c$?"
},
{
"role": "assistant",
"content": "To rewrite the quadratic \\( x^2 - 20x + 36 \\) in the form \\( (x + b)^2 + c \\), we start by completing the sq... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the equation of a hyperbola that has vertices at $(-2,-1)$ and $(10,-1)$ and has one asymptote with slope $\\frac{4}{3}$."
},
{
"role": "assistant",
"content": "To find the equation of the hyperbola with vertices at \\((-2, -1)\\) and \\((10, -1)\\) and one asy... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How can synthetic division be used to evaluate the polynomial \\( f(x) = 12x^3 - 107x^2 - 15x + 54 \\) at a given value of \\( x \\)?"
},
{
"role": "assistant",
"content": "To evaluate the polynomial \\( f(x) = 12x^3 - 107x^2 - 15x + 54 \\) at a specific value \\( x... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet?"
},
{
"role": "assistant",
"content": "To determine the fewest number of 3-inch by 4-inch ti... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the equations:\n\\[\n(x+y)(x+z)=6,\n\\]\n\\[\n(x+y)(y+z)=30,\n\\]\n\\[\n(y+z)(x+z)=45,\n\\]\nfind the value of \\( |x+y+z| \\)."
},
{
"role": "assistant",
"content": "Given the equations:\n\\[\n(x+y)(x+z)=6,\n\\]\n\\[\n(x+y)(y+z)=30,\n\\]\n\\[\n(y+z)(x+z)=45,\... | 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 is never found in the units place of an even number, we first identify the possible digits for the units ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( A = \\{1, 2, 3, 4, 5\\} \\) and \\( B = \\{0, 1, 2\\} \\). Find the number of pairs of functions \\((f, g)\\) where both \\( f \\) and \\( g \\) map the set \\( A \\) into the set \\( B \\) and there are exactly two elements \\( x \\in A \\) such that \\( f(x) = g(x) \\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In right triangle $ABC$, $AB=10$, $AC=6$ and $BC=8$ units. What is the distance from $C$ to the midpoint of segment $AB$?"
},
{
"role": "assistant",
"content": "To find the distance from point \\(C\\) to the midpoint of segment \\(AB\\) in right triangle \\(ABC\\) ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( f : \\mathbb{Q} \\rightarrow \\mathbb{Z} \\) be a homomorphism of additive groups. Determine \\( f(r) \\) for all \\( r \\in \\mathbb{Q} \\)."
},
{
"role": "assistant",
"content": "To determine \\( f(r) \\) for all \\( r \\in \\mathbb{Q} \\) where \\( f : \\... | 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": "The base $9$ representation of a positive integer is $AB$ and its base $7$ representation is $BA$. What is the integer expressed in base $10$?"
},
{
"role": "assistant",
"content": "To solve the problem, we need to find a positive integer that has the base 9 represe... | 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": "The perimeter of a rectangle is 56 meters. The ratio of its length to its width is 4:3. What is the length in meters of a diagonal of the rectangle?"
},
{
"role": "assistant",
"content": "To solve the problem, we start by using the given information about the perime... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let $ABCDEF$ be a cyclic hexagon such that $AB \\parallel DE$. What result is obtained when applying Pascal's Theorem to the hexagon $ABCDEF$?"
},
{
"role": "assistant",
"content": "To solve the problem of applying Pascal's Theorem to the cyclic hexagon \\(ABCDEF\\)... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\( f_n \\) denote the number of ways to toss a fair coin \\( n \\) times such that no two consecutive heads occur. Find the recursion for \\( f_n \\)."
},
{
"role": "assistant",
"content": "To find the recursion for \\( f_n \\), which denotes the number of ways... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If a leaky faucet collects 1/2 cup of water from 8am to 4pm, how many gallons of water would this amount to in 1 day, given that there are 16 cups in a gallon?"
},
{
"role": "assistant",
"content": "To determine how many gallons of water a leaky faucet collects in o... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\[f(x) =\n\\begin{cases}\n3x^2 + 2&\\text{if } x\\le 3, \\\\\nax - 1 &\\text{if } x>3.\n\\end{cases}\n\\]Find $a$ if the graph of $y=f(x)$ is continuous (which means the graph can be drawn without lifting your pencil from the paper)."
},
{
"role": "assistant",
... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the remainder of $8^6 + 7^7 + 6^8$ divided by 5?"
},
{
"role": "assistant",
"content": "To find the remainder when \\(8^6 + 7^7 + 6^8\\) is divided by 5, we will use modular arithmetic properties to simplify each term individually.\n\n1. Calculate \\(8^6 \\m... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the greatest three-digit multiple of 19?"
},
{
"role": "assistant",
"content": "To find the greatest three-digit multiple of 19, we start by dividing the largest three-digit number, 999, by 19:\n\n\\[\n999 \\div 19 = 52.578\\ldots\n\\]\n\nThe integer part of... | You are a helpful assistant. |
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