conversations listlengths 2 2 | system stringclasses 1
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[
{
"role": "user",
"content": "How many zeros are at the end of the base-3 expression of \\( 27! \\)?"
},
{
"role": "assistant",
"content": "To determine the number of trailing zeros in the base-3 expression of \\(27!\\), we need to find the highest power of 3 that divides \\(27!\\). This is equi... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the remainder when $x^3$ is divided by $x^2 + 5x + 1.$"
},
{
"role": "assistant",
"content": "To find the remainder when \\( x^3 \\) is divided by \\( x^2 + 5x + 1 \\), we can use polynomial long division or express \\( x^3 \\) in terms of the divisor.\n\nFirst... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Adam is playing a game: he rolls a regular die until he rolls a $6$. The probability that he rolls the die at most twice can be expressed in the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime. Find $m+n$."
},
{
"role": "assistant",
"content": "To find t... | 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": "If \\(0 < p \\le |a_n| \\le q\\) for all \\(n \\in \\mathbb{N}\\), find the radius of convergence of \\(\\sum a_n x^n\\)."
},
{
"role": "assistant",
"content": "To determine the radius of convergence \\( R \\) of the power series \\(\\sum a_n x^n\\), we use the root... | 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": "Find the limit \\(\\lim_{x \\to 0^+} \\left\\lfloor \\frac{\\tan x}{x} \\right\\rfloor\\), where \\(\\left\\lfloor \\cdot \\right\\rfloor\\) denotes the greatest integer function."
},
{
"role": "assistant",
"content": "To find the limit \\(\\lim_{x \\to 0^+} \\left\... | 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": "Find an explicit formula for $a_n$, where $a_n$ is the number of ways (where the order is significant) the natural number $n$ can be written as a sum of $1$s and $2$s."
},
{
"role": "assistant",
"content": "To find an explicit formula for \\( a_n \\), where \\( a_n ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the perimeter of the right triangle whose exradius of the hypotenuse is 30?"
},
{
"role": "assistant",
"content": "To find the perimeter of the right triangle whose exradius of the hypotenuse is 30, we start by recalling the formula for the exradius correspo... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral $\\int \\csc^4 x \\, dx$."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\csc^4 x \\, dx\\), we can use the trigonometric identity \\(1 + \\cot^2 x = \\csc^2 x\\). Here is the clean and detailed solution:\n\n1. **Rewr... | You are a helpful assistant. |
[
{
"role": "user",
"content": "If $\\log_b 2 = a$ and $\\log_b 3 = c$, then compute $\\log_b 3456b$."
},
{
"role": "assistant",
"content": "To compute \\(\\log_b 3456b\\) given \\(\\log_b 2 = a\\) and \\(\\log_b 3 = c\\), we proceed as follows:\n\n1. Use the logarithm property \\(\\log_b (xy) = \... | You are a helpful assistant. |
[
{
"role": "user",
"content": "From the first 15 natural numbers, 5 are selected so that no two of them are consecutive. If the number of such selections is 33 × k, find k."
},
{
"role": "assistant",
"content": "To solve the problem of selecting 5 numbers from the first 15 natural numbers such th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute the two rightmost digits in the sum: $4! + 8! + 12! + 16! + 20!$."
},
{
"role": "assistant",
"content": "To find the two rightmost digits of the sum \\(4! + 8! + 12! + 16! + 20!\\), we need to compute the sum modulo 100.\n\nFirst, we calculate each factorial... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The sum of three numbers $x$ ,$y$, $z$ is 165. When the smallest number $x$ is multiplied by 7, the result is $n$. The value $n$ is obtained by subtracting 9 from the largest number $y$. This number $n$ also results by adding 9 to the third number $z$. What is the product of th... | 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": "The solutions to $4x^2 + 3 = 3x - 9$ can be written in the form $x = a \\pm b i,$ where $a$ and $b$ are real numbers. What is $a + b^2$? Express your answer as a fraction."
},
{
"role": "assistant",
"content": "To solve the equation \\(4x^2 + 3 = 3x - 9\\) and find ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Calculate the indefinite integral $\\int \\left(\\tan x + \\frac{1}{\\tan x}\\right)^2 \\, dx$."
},
{
"role": "assistant",
"content": "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Convert $10101_3$ to a base 10 integer."
},
{
"role": "assistant",
"content": "To convert the base 3 number \\(10101_3\\) to a base 10 integer, we need to evaluate the value of each digit based on its position. The positions of the digits are as follows, starting fr... | You are a helpful assistant. |
[
{
"role": "user",
"content": "The size of the increase from each term to the next in the list $a, 225\\frac{1}{2}, c, d, 284$ is always the same. What is the value of $a$?"
},
{
"role": "assistant",
"content": "To solve for the value of \\( a \\) in the arithmetic sequence \\( a, 225.5, c, d, 28... | 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": "Evaluate $\\log_2\\frac{1}{16}$."
},
{
"role": "assistant",
"content": "To evaluate \\(\\log_2 \\frac{1}{16}\\), we need to determine the exponent \\(x\\) such that \\(2^x = \\frac{1}{16}\\).\n\nFirst, recognize that \\(16\\) can be written as a power of \\(2\\):\n\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the degree measure of an angle whose measure is double the measure of its complement?"
},
{
"role": "assistant",
"content": "To solve the problem of finding the degree measure of an angle whose measure is double the measure of its complement, we proceed as f... | 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": "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": "What is the 100th digit after the decimal point of the decimal representation of 1/7?"
},
{
"role": "assistant",
"content": "To determine the 100th digit after the decimal point in the decimal representation of \\( \\frac{1}{7} \\), we start by noting that \\( \\fra... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the series\n\\[ A = \\sum_{n=1}^{\\infty} \\frac{1}{(2n-1)^4} \\]\nand\n\\[ B = \\sum_{n=1}^{\\infty} \\frac{1}{(2n)^4}, \\]\nfind the value of \\(\\frac{A}{B}\\)."
},
{
"role": "assistant",
"content": "To find the value of \\(\\frac{A}{B}\\) where\n\\[ A = \\... | 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": "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": "Let \\( P(x) = a_0 + a_1 x + a_2 x^2 + \\cdots + a_n x^n \\) be a polynomial where each \\( a_i \\) is a non-negative integer. Given that \\( P(1) = 4 \\) and \\( P(5) = 136 \\), find the value of \\( P(3) \\)."
},
{
"role": "assistant",
"content": "Given the polyno... | 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 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": "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": "Find the greedy Egyptian fraction representations for $2/13$."
},
{
"role": "assistant",
"content": "To find the greedy Egyptian fraction representation for \\(\\frac{2}{13}\\), we follow these steps:\n\n1. **Identify the largest unit fraction less than \\(\\frac{2}... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\(\\int \\frac{1}{x^2 + 6x + 13} \\, dx\\)."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\frac{1}{x^2 + 6x + 13} \\, dx\\), we start by completing the square in the denominator.\n\n1. **Complete the square:**\n T... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of $(x - y)(x + y)$ if $x = 10$ and $y = 15$?"
},
{
"role": "assistant",
"content": "To find the value of \\((x - y)(x + y)\\) when \\(x = 10\\) and \\(y = 15\\), we can use the difference of squares formula, which states that \\((x - y)(x + y) = x... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Khali has to shovel snow off the sidewalk in front of his house. The sidewalk is 20 feet long and 2 feet wide. If the snow is $\\frac{1}{2}$ foot deep, how many cubic feet of snow does Khali have to shovel off the sidewalk?"
},
{
"role": "assistant",
"content": "To ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Evaluate the integral \\(\\int \\log(x + \\sqrt{x^2 + a^2}) \\, dx\\)."
},
{
"role": "assistant",
"content": "To evaluate the integral \\(\\int \\log(x + \\sqrt{x^2 + a^2}) \\, dx\\), we use integration by parts. Let:\n\n\\[\nu = \\log(x + \\sqrt{x^2 + a^2}) \\quad ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A square has an area of $25x^2 + 20x + 4$. Find the restrictions on $x$ such that the side length of the square is positive."
},
{
"role": "assistant",
"content": "To find the restrictions on \\( x \\) such that the side length of the square is positive, given that ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "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": "Integrate $\\sec{x}$."
},
{
"role": "assistant",
"content": "To integrate \\(\\sec x\\), we use a specific technique involving multiplying by a form of 1 to simplify the integrand. Here is the clean, final solution:\n\n1. Start with the integral:\n \\[\n \\int \... | You are a helpful assistant. |
[
{
"role": "user",
"content": "You and nine friends initially spend $4000$ dollars on tickets to attend a concert. Six friends cancel last minute due to the flu. You and the remaining friends split the original cost of $4000$ dollars equally. What percent of the total cost does each remaining individual have... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Goldilocks has a $1/2$ chance of choosing the correct path at each fork. If she chooses the wrong path more than two times, she gets lost. What is the probability she gets lost if she encounters five forks?"
},
{
"role": "assistant",
"content": "To determine the pro... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many different 7-digit numbers can be formed using each of the digits 2, 5, 2, 5, 2, 5, 0 exactly once?"
},
{
"role": "assistant",
"content": "To determine the number of different 7-digit numbers that can be formed using the digits 2, 5, 2, 5, 2, 5, 0 exactly on... | 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": "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": "My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team to have 3 boys and 3 girls?"
},
{
"role": "assistant",
"content": "To determine the n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Determine the limit $$\\lim_{n\\to\\infty}\\left(\\dfrac{\\sqrt[n]{a}+\\sqrt[n]{b}+\\sqrt[n]{c}}{3}\\right)^n$$ where \\(a, b, c > 0\\)."
},
{
"role": "assistant",
"content": "To determine the limit \n\n\\[\n\\lim_{n\\to\\infty}\\left(\\dfrac{\\sqrt[n]{a}+\\sqrt[n]{... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many ways can the letters in the word \"MIXED\" be rearranged such that no letter maintains its original position?"
},
{
"role": "assistant",
"content": "To determine the number of ways to rearrange the letters in the word \"MIXED\" such that no letter maintains... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Characterize all graphs with $\\Delta(G) \\leq 2$."
},
{
"role": "assistant",
"content": "To characterize all graphs with \\(\\Delta(G) \\leq 2\\), we need to understand the structure of such graphs. Recall that \\(\\Delta(G)\\) denotes the maximum degree of a graph... | You are a helpful assistant. |
[
{
"role": "user",
"content": "A box contains 4032 answer scripts, exactly half of which have an odd number of marks. We randomly choose 2 scripts and follow these rules:\n- If both scripts have odd scores, we add one mark to one of them, put it back in the box, and keep the other script outside.\n- If both ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given parallelogram ABCD, with AB = 10, AD = 12, and BD = $2\\sqrt{13}$, find AC."
},
{
"role": "assistant",
"content": "To find the length of \\( AC \\) in parallelogram \\( ABCD \\) where \\( AB = 10 \\), \\( AD = 12 \\), and \\( BD = 2\\sqrt{13} \\), we can use t... | 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": "Figures $0$, $1$, $2$, and $3$ consist of $1$, $5$, $13$, and $25$ squares, respectively. If this pattern continues, how many squares does figure $100$ have?"
},
{
"role": "assistant",
"content": "To determine the number of squares in figure 100, we first need to id... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the minimum possible value for $y$ in the equation $y = x^2 + 12x + 5$?"
},
{
"role": "assistant",
"content": "To find the minimum value of the quadratic function \\( y = x^2 + 12x + 5 \\), we first determine the vertex of the parabola, since the parabola op... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the value of the 25th term of the arithmetic sequence $2,\n5, 8, \\ldots$?"
},
{
"role": "assistant",
"content": "To find the 25th term of the arithmetic sequence \\(2, 5, 8, \\ldots\\), we start by identifying the first term \\(a_1\\) and the common differe... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Khali has to shovel snow off the sidewalk in front of his house. The sidewalk is 20 feet long and 2 feet wide. If the snow is $\\frac{1}{2}$ foot deep, how many cubic feet of snow does Khali have to shovel off the sidewalk?"
},
{
"role": "assistant",
"content": "To ... | 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": "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": "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": "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": "Brian’s lawn is in the shape of a circle with radius 10 meters. Brian can throw a frisbee up to 50 meters from where he stands. What is the area of the region (in square meters) in which the frisbee can land, if Brian can stand anywhere on his lawn?"
},
{
"role": "assis... | 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": "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": "Find $\\ln{-1}$."
},
{
"role": "assistant",
"content": "To find \\(\\ln(-1)\\), we need to consider the complex logarithm. The natural logarithm of a complex number \\(z\\) is given by the formula:\n\n\\[\n\\ln(z) = \\ln|z| + i \\arg(z)\n\\]\n\nwhere \\(|z|\\) is th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum of the geometric series \\(3 + 6 + 12 + \\ldots + 3072\\)."
},
{
"role": "assistant",
"content": "To find the sum of the geometric series \\(3 + 6 + 12 + \\ldots + 3072\\), we start by identifying the first term \\(a\\) and the common ratio \\(r\\).\n\n... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Integrate the function \\( 4x(x^2 + 1)e^{x^2 + 1} \\)."
},
{
"role": "assistant",
"content": "To integrate the function \\( 4x(x^2 + 1)e^{x^2 + 1} \\), we will use the method of substitution followed by integration by parts.\n\n1. **Substitution:**\n Let \\( u = x... | 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": "Solve for \\( A \\) and \\( B \\) given the equations:\n\\[ A10 - B100 = 0 \\]\n\\[ \\int_{0}^{10} (Ax - Bx^2) \\, dx = 1 \\]"
},
{
"role": "assistant",
"content": "To solve for \\( A \\) and \\( B \\) given the equations:\n\\[ 10A - 100B = 0 \\]\n\\[ \\int_{0}^{10}... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the sum \\(\\frac{1}{1+2} + \\frac{1}{1+2+3} + \\cdots + \\frac{1}{1+2+\\cdots+71}\\)."
},
{
"role": "assistant",
"content": "To find the sum \\(\\frac{1}{1+2} + \\frac{1}{1+2+3} + \\cdots + \\frac{1}{1+2+\\cdots+71}\\), we start by recognizing that each term i... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In the \"Let's Make a Deal\" game, what is the probability of winning the good prize if you switch doors after being shown one of the junky prizes?"
},
{
"role": "assistant",
"content": "To determine the probability of winning the good prize by switching doors in th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find a simple expression for the sum \\(\\sum_{k=1}^{\\infty} \\frac{1}{k^2 + 2}\\) in terms of elementary functions."
},
{
"role": "assistant",
"content": "To find a simple expression for the sum \\(\\sum_{k=1}^{\\infty} \\frac{1}{k^2 + 2}\\), we can use the known ... | 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": "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": "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": "A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit?"
},
{
"role": "assis... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given \\( 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": "Given that $\\gcd{(a,b)} = 2$ and $\\text{lcm}(a,b) = 174$, compute the value of $a \\cdot b$."
},
{
"role": "assistant",
"content": "To solve the problem, we need to find the value of \\(a \\cdot b\\) given that \\(\\gcd(a, b) = 2\\) and \\(\\text{lcm}(a, b) = 174\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Solve the equation \\(x^4 - 7x^2 - 18 = 0\\) for both real and imaginary solutions."
},
{
"role": "assistant",
"content": "To solve the equation \\(x^4 - 7x^2 - 18 = 0\\) for both real and imaginary solutions, we start by making a substitution. Let \\(y = x^2\\). Th... | You are a helpful assistant. |
[
{
"role": "user",
"content": "What is the reflection of the vector $\\begin{pmatrix}2 \\\\ 0 \\\\ 2 \\end{pmatrix}$ across a line with direction $\\begin{pmatrix}1 \\\\ -1 \\\\ 2 \\end{pmatrix}$?"
},
{
"role": "assistant",
"content": "To find the reflection of the vector \\(\\begin{pmatrix}2 \\\... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Decompose the rational function $\\frac{5x^3+2x}{x^2-4}$ into partial fractions."
},
{
"role": "assistant",
"content": "To decompose the rational function \\(\\frac{5x^3 + 2x}{x^2 - 4}\\) into partial fractions, we follow these steps:\n\n1. **Perform Polynomial Long... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Compute $(4+8)^2+(4^2+8^2)$."
},
{
"role": "assistant",
"content": "To compute \\((4+8)^2 + (4^2 + 8^2)\\), we will break it down into manageable parts and solve each part step-by-step.\n\nFirst, we calculate \\((4+8)^2\\):\n\\[\n4 + 8 = 12\n\\]\n\\[\n12^2 = 12 \\ti... | 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": "Solve the following differential equation by separation of variables (or otherwise):\n\\[ \\frac{dy}{dx} - 1 = e^{x-y} \\]"
},
{
"role": "assistant",
"content": "To solve the differential equation \\(\\frac{dy}{dx} - 1 = e^{x-y}\\), we start by making a substitution... | 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": "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": "What is $4\\cdot 6+8\\cdot 3-28\\div 2$?"
},
{
"role": "assistant",
"content": "To solve the expression \\(4 \\cdot 6 + 8 \\cdot 3 - 28 \\div 2\\), we need to follow the order of operations (PEMDAS/BODMAS):\n\n1. **Parentheses/Brackets**: There are no parentheses or... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given that $\\sum_{r=1}^{\\infty} \\frac{1}{r^{2}} = \\frac{\\pi^{2}}{6}$, find the value of $1 - \\frac{1}{2^{2}} + \\frac{1}{3^{2}} - \\frac{1}{4^{2}} + \\cdots$."
},
{
"role": "assistant",
"content": "To find the value of the series \\(1 - \\frac{1}{2^2} + \\frac... | 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": "A very large number $x$ is equal to $2^23^34^45^56^67^78^89^9$. What is the smallest positive integer that, when multiplied with $x$, produces a product that is a perfect square?"
},
{
"role": "assistant",
"content": "To find the smallest positive integer that, when... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Let \\(f(x) = \\frac{x}{x+1}\\). If \\(f^n(x)\\) is the result of composing \\(f\\) with itself \\(n\\) times, compute \\(f^{2004}(2)\\)."
},
{
"role": "assistant",
"content": "To solve the problem of finding \\( f^{2004}(2) \\) where \\( f(x) = \\frac{x}{x+1} \\), ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "In a cube-shaped box with an edge length of 5, there are two balls. The radius of one ball is 2. This ball touches the base and two side faces of the cube. The other ball touches the first ball, the base, and the other two side faces of the cube. Find the radius of the second b... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Factor the expression \\(\\frac{a^{2n} + (a^n)(b^n)}{3a^{3n} + (3a^{2n})(b^n)}\\)."
},
{
"role": "assistant",
"content": "To factor and simplify the expression \\(\\frac{a^{2n} + a^n b^n}{3a^{3n} + 3a^{2n} b^n}\\), we proceed as follows:\n\n1. **Factor the numerator... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Given the function \\( F(x, y) = x^3 + 3x^2y - 2xy^5 \\), find the first partial derivative with respect to \\( y \\) when \\( x \\) is held constant."
},
{
"role": "assistant",
"content": "To find the first partial derivative of the function \\( F(x, y) = x^3 + 3x^... | 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": "I flip a fair coin once and roll a regular six-sided die. What is the probability the coin will show heads and the die will show a 2?"
},
{
"role": "assistant",
"content": "To find the probability that a fair coin shows heads and a regular six-sided die shows a 2, ... | You are a helpful assistant. |
[
{
"role": "user",
"content": "Find the polynomial whose roots are three more than those of the polynomial \\(x^3 - 6x^2 + 3x - 10\\)."
},
{
"role": "assistant",
"content": "To find the polynomial whose roots are three more than those of the polynomial \\(x^3 - 6x^2 + 3x - 10\\), we need to subst... | You are a helpful assistant. |
[
{
"role": "user",
"content": "How many points on the Cartesian plane \\( (x, y) \\) satisfy \\( ((x - 20)^{20} - 20)^{20} + ((y - 20)^{20} - 20)^{20} = 0 \\)?"
},
{
"role": "assistant",
"content": "To solve the equation \\(((x - 20)^{20} - 20)^{20} + ((y - 20)^{20} - 20)^{20} = 0\\), we need to ... | 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": "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. |
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