messages listlengths 2 2 | conversation_id int64 0 72.4k |
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"content": "Answer the following questions as required.\n1. Given $10^{m}=12$ and $10^{n}=3$, find the value of $10^{m-n}$.\n2. If $a+3b=3$, find the value of $3^{a}\\times 27^{b}$.\n3. Given $8\\times 2^{m}\\div 16^{m}=2^{6}$, find the value of $m$.",
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{
"content": "Sure, let's s... | 200 |
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"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) be 2006 positive integers (they can be the same) such that \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{2006}} \\) are all different from each other. What is the minimum number of distinct numbers in \\( a_{1}, a_{2}, \\cd... | 201 |
[
{
"content": "If the equation $x^{m+1}-2=1$ with respect to $x$ is a linear equation, then the value of $m$ is ______.",
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"content": "To determine the value of \\( m \\) such that the equation \\( x^{m+1} - 2 = 1 \\) is a linear equation with respect to \\( x \\), we need to analy... | 202 |
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"content": "For what smallest natural number \\( a \\) are there exactly 50 perfect squares in the numerical interval \\( (a, 3a) \\)?",
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"content": "To find the smallest natural number \\( a \\) for which there are exactly 50 perfect squares in the interval \\( (a, 3a) \\), we ... | 203 |
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"content": "The motto \"Green waters and green mountains are golden mountains and silver mountains\" encourages Hope High School to organize tree planting activities for students every year. This year, the school bought a batch of tree seedlings and formed tree planting teams. If each team plants 5 trees, the... | 204 |
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"content": "How many 4-digit positive multiples of 4 can be formed from the digits 0, 1, 2, 3, 4, 5, 6 such that each digit appears without repetition?",
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"content": "To solve the problem, let's break it down into smaller steps and then use Python code to verify the solution.\n\... | 205 |
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"content": "Simplify first, then evaluate: $-7x+2(x^{2}-1)-(2x^{2}-x+3)$, where $x=1$.",
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"content": "Certainly! Let's break down the problem step-by-step and utilize Python with SymPy to simplify and evaluate the expression.\n\n1. **Expression Simplification:**\n We start wit... | 206 |
[
{
"content": "How many three-digit numbers are there in which the digits 1, 2, and 3 each appear exactly once?",
"role": "user"
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"content": "To solve the problem of finding how many three-digit numbers can be formed where the digits 1, 2, and 3 each appear exactly once, we can use the concept of... | 207 |
[
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"content": "Given $x^{2}+y^{2}=4$ and $\\left(x-y\\right)^{2}=5$, find $\\left(x+y\\right)^{2}$.",
"role": "user"
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"content": "Let's solve the problem step-by-step using Python and SymPy. We'll start by setting up the equations given and solve for \\(x\\) and \\(y\\).\n\nThe problem gives us t... | 208 |
[
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"content": "The least common multiple of two numbers that do not divide each other is 90, and their greatest common divisor is 6. Find these numbers.",
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"content": "To solve the problem, we will utilize the relationship between the Least Common Multiple (LCM) and the Greatest Co... | 209 |
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{
"content": "Let $f(x)$ be a function defined on the set of real numbers $R$ and satisfying the following conditions:\n$$\nf(10 + x) = f(10 - x), \\quad f(20 - x) = -f(20 + x) .\n$$\n\nThen $f(x)$ is:\n(A) An even function and a periodic function.\n(B) An even function but not a periodic function.\n(C) An odd ... | 210 |
[
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"content": "Calculate: $-{3}^{2}+|-5|-18\\times{(-\\frac{1}{3})}^{2}$.",
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"content": "To solve the problem \\(-3^2 + |-5| - 18 \\times \\left(-\\frac{1}{3}\\right)^2\\), we can follow these steps:\n\n1. **Simplify the individual components:**\n - \\( -3^2 \\) : Calculate the s... | 211 |
[
{
"content": "Eva the magician has a deck of 100 cards, numbered from 1 to 100, with each card displaying the same number on both of its sides. Initially, she has shuffled her cards arbitrarily and stacked them. Then she performs the following operations: if the card with number $k$ is on the top of the stack, ... | 212 |
[
{
"content": "If the sum of the first three terms of an arithmetic sequence $\\{a_n\\}$ is $S_3 = 9$ and $a_1 = 1$, then $a_2$ equals ____.",
"role": "user"
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{
"content": "To solve this problem, we need to find the second term \\(a_2\\) of an arithmetic sequence given that the sum of the first thr... | 213 |
[
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"content": "From the numbers 1, 2, 3, and 4, form a four-digit number where the first digit is 1 and only two digits are the same. There are $\\boxed{\\_\\_\\_\\_}$ such four-digit numbers.",
"role": "user"
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{
"content": "To solve this problem, we need to count how many four-digit numbers can be... | 214 |
[
{
"content": "Three cans of juice fill $\\frac{2}{3}$ of a one-litre jug. How many cans of juice are needed to completely fill 8 one-litre jugs?",
"role": "user"
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{
"content": "First, we need to determine the volume of juice in one can by using the information that three cans of juice fill \\(\\fr... | 215 |
[
{
"content": "The number $101$ is the smallest three-digit palindromic prime. What is the second-smallest one?",
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"content": "To find the second-smallest three-digit palindromic prime number, let's first understand the definition:\n\n1. A three-digit palindromic number reads the s... | 216 |
[
{
"content": "Jeff bought 6 pairs of shoes and 4 jerseys for $560. Jerseys cost 1/4 price of one pair of shoes. Find the shoe's price total price.",
"role": "user"
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{
"content": "Let's break down and solve the problem step by step and then use a Python code to verify our calculations.\n\n1. Let's ... | 217 |
[
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"content": "Coordinate System and Parametric Equations. \nGiven the curve $C$: \n$$\n\\begin{cases}\nx=\\cos \\theta \\\\\ny=\\sin \\theta\n\\end{cases}\n(\\theta \\text{ is the parameter}).\n$$\n\n$(1)$ Convert the parametric equations of $C$ into a standard equation; \n$(2)$ If the coordinates of each po... | 218 |
[
{
"content": "Given $f(x)=|x-1|-|x|$, find the value of $f[f(\\frac{1}{2})]$.",
"role": "user"
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"content": "To solve the problem, we need to find the value of \\( f(f(\\frac{1}{2})) \\) where \\( f(x) = |x-1| - |x| \\).\n\nLet's break down the steps:\n\n1. **Define the function \\( f(x) \\):**\n... | 219 |
[
{
"content": "In the binomial expansion of $$(2x^{2}- \\frac {1}{x})^{6}$$, the constant term is \\_\\_\\_\\_\\_.",
"role": "user"
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"content": "To find the constant term in the binomial expansion of \\((2x^2 - \\frac {1}{x})^6\\), we need to identify the term in the expansion where the power of ... | 220 |
[
{
"content": "Oleg has 550 rubles and wants to give his mother tulips for March 8th. The number of tulips must be odd, and each tulip must be of a different shade. In the store, each tulip costs 49 rubles, and there are eleven different shades available. How many ways are there for Oleg to give the flowers to h... | 221 |
[
{
"content": "Determine the least possible value of the natural number $n$ such that $n!$ ends in exactly $1987$ zeros.\n\n<details><summary>Note</summary>Note. Here (and generally in MathLinks) natural numbers supposed to be positive.</details>",
"role": "user"
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{
"content": "To determine t... | 222 |
[
{
"content": "Given an arithmetic sequence $\\{a_n\\}$, where $a_3+a_7=8$, find the sum of the first $9$ terms, $S_9=$ ______.",
"role": "user"
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{
"content": "To solve this problem, we need to use the properties of an arithmetic sequence.\n\n1. **Understand the Arithmetic Sequence:**\n The gener... | 223 |
[
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"content": "The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.\n\n*2018 CCA Math Bonanza Lightning Round #3.1*",
"role": "user"
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"content": "To determine the sum of the four distinct prime numbers that divide \\(16^4 + 16^2 + 1\\), we ... | 224 |
[
{
"content": "Let $x$ be a value such that $8x^2 + 7x - 1 = 0$ and $24x^2+53x-7 = 0.$ What is the value of $x$? Express your answer as a simplified common fraction.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the value of \\( x \\) that satisfies both quadratic equations:... | 225 |
[
{
"content": "For any \\( x \\in \\left(0, \\frac{\\pi}{2}\\right) \\), which of the following statements is true?\n(A) \\( \\sin (\\sin x) < \\cos x < \\cos (\\cos x) \\)\n(B) \\( \\sin (\\sin x) > \\cos x > \\cos (\\cos x) \\)\n(C) \\( \\sin (\\cos x) > \\cos x > \\cos (\\sin x) \\)\n(D) \\( \\sin (\\cos x) <... | 226 |
[
{
"content": "Find the expected value of a random variable \\( X \\) that is uniformly distributed in the interval \\((a, b)\\).",
"role": "user"
},
{
"content": "To find the expected value of a random variable \\(X\\) that is uniformly distributed in the interval \\((a, b)\\), we can use the follow... | 227 |
[
{
"content": "Let the sequence \\\\(\\{a_n\\}\\) be defined by \\\\(a_n = 2n - 7 (n \\in \\mathbb{N}^*)\\), then the value of \\\\(|a_1| + |a_2| + \\ldots + |a_{15}| = \\) ______.",
"role": "user"
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{
"content": "To solve the problem, we need to calculate the sum of the absolute values of the first... | 228 |
[
{
"content": "If $a=\\sqrt{5}+1$ and $b=\\sqrt{5}-1$, find the value of the following algebraic expressions:\n1. $a^{2}b+ab^{2}$\n2. $a^{2}-ab+b^{2}$.",
"role": "user"
},
{
"content": "Let's solve each algebraic expression step-by-step using Python and SymPy.\n\n1. **Expression \\(a^{2}b + ab^{2}\\)... | 229 |
[
{
"content": "What is the smallest positive integer $n$ such that\n\n$$\n\\sqrt{n}-\\sqrt{n-1}<0.01 ?\n$$",
"role": "user"
},
{
"content": "To solve the problem of finding the smallest positive integer \\( n \\) such that:\n\n$$\n\\sqrt{n} - \\sqrt{n-1} < 0.01\n$$\n\nwe can follow these steps:\n\n1.... | 230 |
[
{
"content": "Convert the binary number $10011_{(2)}$ to a decimal number equals \\_\\_\\_\\_\\_.",
"role": "user"
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{
"content": "To convert the binary number \\(10011_{(2)}\\) to a decimal number, you can follow these steps:\n\n1. Understand that each digit in a binary number represents a power o... | 231 |
[
{
"content": "Let's assume that Olga Pavlovna has \\( x \\) liters, and Maria Petrovna has \\( y \\) liters of jam left. The numbers \\( x \\) and \\( y \\) are randomly and independently chosen from the interval from 0 to 1. Consider a random point with coordinates \\((x, y)\\) from the unit square \\( F \\). ... | 232 |
[
{
"content": "Vasya and Tolya exchanged badges. Before the exchange, Vasya had 5 more badges than Tolya. After Vasya exchanged 24% of his badges for 20% of Tolya's badges, Vasya had one badge fewer than Tolya. How many badges did each boy have before the exchange?",
"role": "user"
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{
"content": "L... | 233 |
[
{
"content": "Mrs. Oaklyn buys handmade rugs at $40 each and sells them at $60 each. If she bought 20 rugs, calculate the profit she will make from selling the rugs.",
"role": "user"
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{
"content": "To solve this problem, we need to follow these steps:\n\n1. Calculate the total cost of buying 20 ru... | 234 |
[
{
"content": "On June 1, a group of students is standing in rows, with 15 students in each row. On June 2, the same group is standing with all of the students in one long row. On June 3, the same group is standing with just one student in each row. On June 4, the same group is standing with 6 students in each r... | 235 |
[
{
"content": "A book that is to be recorded onto compact discs takes $412$ minutes to read aloud. Each disc can hold up to $56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain... | 236 |
[
{
"content": "Yura left home for school 5 minutes later than Lena, but he walked twice as fast as her. How long after Yura started walking will he catch up to Lena?",
"role": "user"
},
{
"content": "To solve this problem, we can set up the following equations and solve them step by step.\n\n1. **Def... | 237 |
[
{
"content": "If the system of equations in terms of $x$ and $y$ $\\left\\{{\\begin{array}{l}{x+my=2}\\\\{mx+16y=8}\\end{array}}\\right.$ has infinitely many solutions, then the value of the real number $m$ is ____.",
"role": "user"
},
{
"content": "To find the value of \\( m \\) such that the given... | 238 |
[
{
"content": "Find the greatest common divisor of $5616$ and $11609$.",
"role": "user"
},
{
"content": "The problem requires finding the greatest common divisor (GCD) of \\(5616\\) and \\(11609\\). The Euclidean algorithm is a well-known and efficient method for computing the GCD of two numbers and ... | 239 |
[
{
"content": "From 22 kg of fresh mushrooms, 2.5 kg of dried mushrooms are obtained, which still contain 12% water. What percent of water is in the fresh mushrooms?",
"role": "user"
},
{
"content": "To solve this problem, let's break it down into manageable steps:\n\n1. **Understand the Problem:**\n... | 240 |
[
{
"content": "In a track and field sports event, there are 216 athletes participating in the 100-meter sprint. The track has 6 lanes, with 1 athlete in each lane. Each race has 6 athletes participating, and only the first-place athlete in each group can advance to the next round. How many races are needed to de... | 241 |
[
{
"content": "What is the decimal representation of the binary number \"1111\"?",
"role": "user"
},
{
"content": "To convert a binary number to its decimal representation, we can use the method of positional weights. Each digit (bit) in a binary number represents an increasing power of 2, starting f... | 242 |
[
{
"content": "On rectangular coordinates, point $A = (1,2)$ , $B = (3,4)$ . $P = (a, 0)$ is on $x$ -axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$ .",
"role": "user"
},
{
"content": "To solve this problem, we need to minimize the sum of distances \\(AP + PB\... | 243 |
[
{
"content": "It is known that among the members of the government of Limonia (a total of 20 members), there is at least one honest member, and among any two members, at least one is a bribe-taker. How many bribe-takers are in the government?",
"role": "user"
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{
"content": "Given the two condition... | 244 |
[
{
"content": "The base of isosceles $\\triangle ABC$ is $24$ and its area is $60$. What is the length of one\nof the congruent sides?\n$\\mathrm{(A)}\\ 5 \\qquad \\mathrm{(B)}\\ 8 \\qquad \\mathrm{(C)}\\ 13 \\qquad \\mathrm{(D)}\\ 14 \\qquad \\mathrm{(E)}\\ 18$",
"role": "user"
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{
"content": "To s... | 245 |
[
{
"content": "Triangle \\(ABC\\) is given with \\(AB = 13\\), \\(BC = 14\\), and \\(CA = 15\\). Let \\(E\\) and \\(F\\) be the feet of the altitudes from \\(B\\) and \\(C\\), respectively. Let \\(G\\) be the foot of the altitude from \\(A\\) in triangle \\(AFE\\). Find \\(AG\\).",
"role": "user"
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{
... | 246 |
[
{
"content": "Tony multiplies together at least two consecutive positive integers. He obtains the six-digit number \\( N \\). The left-hand digits of \\( N \\) are '47', and the right-hand digits of \\( N \\) are '74'. What integers does Tony multiply together?",
"role": "user"
},
{
"content": "To s... | 247 |
[
{
"content": "In a drama club, the number of boys is $80 \\%$ of the number of girls. What percentage is the number of girls in this drama club relative to the number of boys?",
"role": "user"
},
{
"content": "To solve the problem, we need to determine the percentage of the number of girls relative ... | 248 |
[
{
"content": "Given that \\( x_{0} \\) satisfies the equation \\( x^{2} + x + 2 = 0 \\). If \\( b = x_{0}^{4} + 2x_{0}^{3} + 3x_{0}^{2} + 2x_{0} + 1 \\), find the value of \\( b \\).",
"role": "user"
},
{
"content": "First, let's address the given quadratic equation \\( x^2 + x + 2 = 0 \\). We'll fi... | 249 |
[
{
"content": "Is it true that by changing one digit in the decimal representation of any natural number, one can obtain a prime number?",
"role": "user"
},
{
"content": "To solve this problem, we need to take a careful and systematic approach:\n\n1. **Understand the Problem:**\n We need to determi... | 250 |
[
{
"content": "A teacher asks her class what the value of $\\left(5^2-4^2\\right)^3$ is. What is the value of the expression?",
"role": "user"
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{
"content": "To solve the expression \\(\\left(5^2 - 4^2\\right)^3\\), we will work step-by-step:\n\n1. Evaluate \\(5^2\\).\n2. Evaluate \\(4^2\\).\n3. S... | 251 |
[
{
"content": "A particle moves according to the law $s=2t^3$. The average speed of the particle in the time interval $[1, 1.1]$ is \\_\\_\\_\\_\\_\\_ m/s, and the instantaneous speed at $t=1$ is \\_\\_\\_\\_\\_\\_ m/s.",
"role": "user"
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{
"content": "To solve this problem, we need to calculate the... | 252 |
[
{
"content": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$.",
"role": "user"
},
{
"content": "To solve this problem, we will follow logical reasoning combined with some inequalities. Here's the step-by-step reasoning:\n\n1. We need both \\(2x\\) and \\(3x\\) ... | 253 |
[
{
"content": "A boat's speed in still water is $15 \\, \\mathrm{mph}$, and a creek's current speed is $5 \\, \\mathrm{mph}$. If the boat travels back and forth over a certain distance, what is the ratio of the boat’s average speed for the round trip to its speed in still water?\n\n(A) $\\frac{5}{4}$\n(B) $\\fra... | 254 |
[
{
"content": "If $a=\\log_8 225$ and $b=\\log_2 15$, then\n$\\text{(A) } a=b/2\\quad \\text{(B) } a=2b/3\\quad \\text{(C) } a=b\\quad \\text{(D) } b=a/2\\quad \\text{(E) } a=3b/2$",
"role": "user"
},
{
"content": "To solve the problem, we need to compare the expressions \\(a = \\log_8 225\\) and \\(... | 255 |
[
{
"content": "Teresa scored 70 marks in science, 80 in music, 85 in social studies, and the physics exam, which was the hardest test that Teresa had ever taken in her entire life, she scored half as many marks as she scored in music. Calculate the total marks Teresa scored in all the subjects.",
"role": "us... | 256 |
[
{
"content": "In economics, the marginal function $Mf(x)$ of a function $f(x)$ is defined as $Mf(x) = f(x+1) - f(x)$. A certain company can produce up to $10$ lithography machines of a certain type per month. The revenue function for producing $x$ machines ($x \\geqslant 1$, $x \\in \\mathbb{N}^{*}$) is $R(x) =... | 257 |
[
{
"content": "In rectangle $ABCD$, $DC = 2 \\cdot CB$ and points $E$ and $F$ lie on $\\overline{AB}$ so that $\\overline{ED}$ and $\\overline{FD}$ trisect $\\angle ADC$ as shown. What is the ratio of the area of $\\triangle DEF$ to the area of rectangle $ABCD$?\n\n$\\textbf{(A)}\\ \\ \\frac{\\sqrt{3}}{6}\\qquad... | 258 |
[
{
"content": "Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$.",
"role": "user"
},
{
"content": "To find the least positive integer \\( n \\) for which \\( 2^n + 5^n - n \\) is a multiple of \\( 1000 \\), we need to solve for \\( n \\) such that:\n\n\\[ 2^n + 5^... | 259 |
[
{
"content": "Let the coefficient of the $x^{5}$ term in the expansion of $\\left(x-\\frac{2}{\\sqrt{x}}\\right)^{8}$ be $A$, and the binomial coefficient be $B$. Then, $\\frac{A}{B}=$\\_\\_\\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the coefficient of... | 260 |
[
{
"content": "Evaluate $\\log_\\frac{1}{3}9$.",
"role": "user"
},
{
"content": "To find the value of \\(\\log_\\frac{1}{3}9\\), we need to determine the exponent \\(x\\) such that:\n\n\\[\n\\left(\\frac{1}{3}\\right)^x = 9\n\\]\n\nWe can use the properties of logarithms, conversion, and exponents to... | 261 |
[
{
"content": "There are 27 ones written on the board. Every minute, Karlson erases two arbitrary numbers and writes their sum on the board, then eats the number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could have eaten in 27 minutes?",
"role": "user... | 262 |
[
{
"content": "The graph of the power function $f(x)=x^{a}$ passes through the point $(4, \\frac{1}{2})$. Determine the value of $f(\\frac{1}{4})$.",
"role": "user"
},
{
"content": "To solve this problem, we need to find the value of \\(a\\) such that the function \\(f(x) = x^a\\) passes through the ... | 263 |
[
{
"content": "In a certain city, residents are charged for water usage in a tiered manner. If the monthly water usage of a household does not exceed $5$ tons, the charge is $2$ yuan per ton; if it exceeds $5$ tons, the excess part is charged at $3.5$ yuan per ton. Let $x$ be the monthly water usage in tons of a... | 264 |
[
{
"content": "Given point $P(a+1,2a-3)$, $Q(2,3)$, when point $P$ lies on the angle bisector in the first and third quadrants, the coordinates of point $P$ are ______; when $PQ$ is parallel to the $x$-axis, $PQ=______$.",
"role": "user"
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{
"content": "Alright, let's break down the problem into man... | 265 |
[
{
"content": "For which non-zero real numbers $x$ is $\\frac{|x-|x|\\-|}{x}$ a positive integer? \n$\\textbf{(A)}\\ \\text{for negative } x \\text{ only} \\qquad \\\\ \\textbf{(B)}\\ \\text{for positive } x \\text{ only} \\qquad \\\\ \\textbf{(C)}\\ \\text{only for } x \\text{ an even integer} \\qquad \\\\ \\te... | 266 |
[
{
"content": "A busy restaurant is counting how many customers they had during that Friday to try to predict how many they might get on Saturday. During breakfast, they had 73 customers. During lunch, they had 127 customers. During dinner, they had 87 customers. If they predict they'll get twice the amount of c... | 267 |
[
{
"content": "Let be a natural number $ n\\ge 3. $ Find $$ \\inf_{\\stackrel{ x_1,x_2,\\ldots ,x_n\\in\\mathbb{R}_{>0}}{1=P\\left( x_1,x_2,\\ldots ,x_n\\right)}}\\sum_{i=1}^n\\left( \\frac{1}{x_i} -x_i \\right) , $$ where $ P\\left( x_1,x_2,\\ldots ,x_n\\right) :=\\sum_{i=1}^n \\frac{1}{x_i+n-1} , $ and... | 268 |
[
{
"content": "Find the triplets of real numbers \\(x, y, z\\) that satisfy the following system of equations:\n\n\\[\n\\begin{aligned}\nx + y + z & = 2 \\\\\nx^2 + y^2 + z^2 & = 26 \\\\\nx^3 + y^3 + z^3 & = 38\n\\end{aligned}\n\\]",
"role": "user"
},
{
"content": "To solve the system of equations us... | 269 |
[
{
"content": "If the sequence \\(\\{a_n\\}\\) is defined by \\(a_1=1\\) and \\(4a_n a_{n+1} = (a_n + a_{n+1} - 1)^2\\) with \\(a_n > a_{n-1}\\), then \\(a_n\\) equals\nA. \\(n^2\\)\nB. \\(n\\)\nC. \\(n(n-1)\\)\nD. \\(n^3\\)",
"role": "user"
},
{
"content": "To solve the problem, we start by understa... | 270 |
[
{
"content": "If \\( m \\) and \\( n \\) are positive integers with \\( m^2 - n^2 = 43 \\), find the value of \\( m^3 - n^3 \\).",
"role": "user"
},
{
"content": "To solve the given problem, we start with the equation \\( m^2 - n^2 = 43 \\). This can be factored as:\n\\[ (m - n)(m + n) = 43 \\]\n\nS... | 271 |
[
{
"content": "Carlos has some gold quarters. If he spends them in a store, they are worth the regular amount. If he melts them down, he can get $100 per ounce. Each quarter weighs 1/5 of an ounce. How many times more money would he get from melting them down instead of spending them in a store?",
"role": "u... | 272 |
[
{
"content": "There are $20n$ members in the Trumpington marching band, and when they line up in rows of 26, there are 4 band members left over. If $n$ is an integer and there are fewer than 1000 band members, what is the maximum number of people that could be in the Trumpington marching band?",
"role": "us... | 273 |
[
{
"content": "Find all positive integers \\( n \\) such that for any integer \\( k \\), there exists an integer \\( a \\) for which \\( a^3 + a - k \\) is divisible by \\( n \\).",
"role": "user"
},
{
"content": "To solve this problem, we need to find all positive integers \\( n \\) such that for an... | 274 |
[
{
"content": "A class needs to select 4 students from 8 to participate in the 4×100 meters relay race at the school sports meeting. Among them, students A and B must be selected, and they must run either the first or the last leg. The total number of different arrangements is (answer in numbers).",
"rol... | 275 |
[
{
"content": "How many ordered triples of [integers](https://artofproblemsolving.com/wiki/index.php/Integer) $(a,b,c)$, with $a \\ge 2$, $b\\ge 1$, and $c \\ge 0$, satisfy both $\\log_a b = c^{2005}$ and $a + b + c = 2005$?\n$\\mathrm{(A)} \\ 0 \\qquad \\mathrm{(B)} \\ 1 \\qquad \\mathrm{(C)} \\ 2 \\qquad \\mat... | 276 |
[
{
"content": "Given $x + 3y = 5$ and $2x - y = 2$, find the value of $2x^2 + 5xy - 3y^2$.",
"role": "user"
},
{
"content": "To solve the problem, we need to follow these steps:\n\n1. Solve the system of linear equations to find the values of \\(x\\) and \\(y\\):\n \\[\n \\begin{cases}\n x +... | 277 |
[
{
"content": "A group composed of doctors and lawyers has an average (arithmetic mean) age of 40 years. If the average age of the doctors is 35 years and the average age of the lawyers is 50 years, what is the ratio of the number of doctors to the number of lawyers?\n(A) $3: 2$\n(B) $3: 1$\n(C) $2: 3$\n(D) $2: ... | 278 |
[
{
"content": "At the Sterling Auto Store, they have 500 cars for sale. If they have 10 sales professionals, and each salesperson sells 10 cars per month, how many months will it take for them to sell all of the cars?",
"role": "user"
},
{
"content": "Let's break down the problem step-by-step:\n\n1.... | 279 |
[
{
"content": "If $64$ is divided into three parts proportional to $2$, $4$, and $6$, the smallest part is:\n$\\textbf{(A)}\\ 5\\frac{1}{3}\\qquad\\textbf{(B)}\\ 11\\qquad\\textbf{(C)}\\ 10\\frac{2}{3}\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ \\text{None of these answers}$",
"role": "user"
},
{
... | 280 |
[
{
"content": "Find the greatest common factor of 5! and 6!. (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.)",
"role": "user"
},
{
"content": "To find the greatest common factor (GCF) of \\(5!\\) and \\(6!\\), we will... | 281 |
[
{
"content": "Given a positive integer \\( a \\), define a sequence of integers \\( x_1, x_2, x_3, \\ldots \\) such that \\( x_1 = a \\) and \\( x_{n+1} = 2x_n + 1 \\). Let \\( y_n = 2^{x_n} - 1 \\). Determine the largest possible integer \\( k \\) such that for some positive integer \\( a \\), the numbers \\( ... | 282 |
[
{
"content": "Last Sunday at noon the date on the calendar was 15 (April 15, 2007). What will be the date on the calendar one million minutes after that time?",
"role": "user"
},
{
"content": "To determine the date one million minutes after 12:00 PM on April 15, 2007, we need to follow these steps:... | 283 |
[
{
"content": "If the function $f(x) = \\frac{2}{3^x + 1} - a$ is an odd function, then the value of the real number $a$ is _______.",
"role": "user"
},
{
"content": "To determine the value of \\(a\\) such that the function \\( f(x) = \\frac{2}{3^x + 1} - a \\) is an odd function, let's proceed with ... | 284 |
[
{
"content": "Calculate the arc length of the cardioid $\\rho = 2(1+\\cos \\varphi)$ for $0 \\leq \\varphi \\leq 2\\pi$.",
"role": "user"
},
{
"content": "To solve the problem of calculating the arc length of the cardioid $\\rho = 2(1+\\cos \\varphi)$ for \\(0 \\leq \\varphi \\leq 2\\pi\\), we need ... | 285 |
[
{
"content": "Find all integer values of \\(a\\), not exceeding 15 in absolute value, for which the inequality\n\n$$\n\\frac{4x - a - 4}{6x + a - 12} \\leq 0\n$$\n\nholds for all \\(x\\) in the interval \\([2, 3]\\). In the answer, indicate the sum of all such \\(a\\).",
"role": "user"
},
{
"content... | 286 |
[
{
"content": "The slope of the tangent line to the curve $y=x^2+x$ at point A $(2,6)$ is.",
"role": "user"
},
{
"content": "To find the slope of the tangent line to the curve \\( y = x^2 + x \\) at the point \\( A(2, 6) \\), we need to take the derivative of the function and evaluate it at \\( x = 2... | 287 |
[
{
"content": "João, Jorge, José, and Jânio are good friends. One time, João was out of money, but his friends had some. So Jorge gave João a fifth of his money, José gave João a fourth of his money, and Jânio gave João a third of his money. If all of them gave the same amount of money to João, what fraction of ... | 288 |
[
{
"content": "Given a four-digit number where the digits are distinct non-negative integers and their sum is 12, find the number of such four-digit numbers.",
"role": "user"
},
{
"content": "To solve this problem, we need to count the number of distinct four-digit numbers where the sum of the digits... | 289 |
[
{
"content": "Let \\( S_{n} = 1 + 2 + \\cdots + n \\). How many of \\( S_{1}, S_{2}, \\cdots, S_{2015} \\) are multiples of 2015?\n",
"role": "user"
},
{
"content": "To solve the problem, we need to calculate how many of the partial sums \\( S_{1}, S_{2}, \\cdots, S_{2015} \\) are multiples of 2015.... | 290 |
[
{
"content": "Calculate the number of multiplication and addition operations required in the process of evaluating the polynomial \\( f(x) = 5x^5 + 4x^4 + 3x^3 + 2x^2 + x + 1 \\) at \\( x = 2 \\) using the Horner's method (秦九韶算法).",
"role": "user"
},
{
"content": "To solve this problem, we utilize H... | 291 |
[
{
"content": "Given vectors $\\overrightarrow{a}=(1,0)$ and $\\overrightarrow{b}=(m,n)$, if $\\overrightarrow{b}- \\overrightarrow{a}$ is parallel to $\\overrightarrow{a}$, then the value of $n$ is ______.",
"role": "user"
},
{
"content": "To solve this problem, we need to determine the value of \\(... | 292 |
[
{
"content": "Solve the equation \\(2021 \\cdot \\sqrt[202]{x^{2020}} - 1 = 2020 x\\) for \\(x \\geq 0\\).",
"role": "user"
},
{
"content": "Let's solve the equation \\(2021 \\cdot \\sqrt[202]{x^{2020}} - 1 = 2020 x\\) for \\(x \\geq 0\\) step-by-step using Python and sympy.\n\n1. **Rewrite the Equa... | 293 |
[
{
"content": "Christine is subbing aquafaba for egg whites in baking. Every 2 tablespoons of aquafaba is equivalent to 1 egg white. She's making 2 angel food cakes that call for 8 egg whites each. How many tablespoons of aquafaba will she need in order not to use egg whites?",
"role": "user"
},
{
... | 294 |
[
{
"content": "Given the general term formula of the sequence $\\{a_{n}\\}$ as $a_{n}=(-1)^{n-1}(n-1)$, let $S_{n}$ denote the sum of its first $n$ terms. Calculate the value of $S_{15}$.",
"role": "user"
},
{
"content": "To solve this problem, we need to find the sum of the first 15 terms of the seq... | 295 |
[
{
"content": "Calculate the value of $$\\frac {\\cos10^\\circ(1+ \\sqrt {3}\\tan10^\\circ)}{\\cos50 ^\\circ }.$$",
"role": "user"
},
{
"content": "To calculate the value of the given expression \\(\\frac {\\cos10^\\circ(1+ \\sqrt {3}\\tan10^\\circ)}{\\cos50 ^\\circ }\\), we can use SymPy, a Python l... | 296 |
[
{
"content": "In the three Chinese phrases \"尽心尽力\", \"力可拔山\", and \"山穷水尽\", each Chinese character represents a unique digit between 1 and 8. The same character represents the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each phrase ... | 297 |
[
{
"content": "If the equation with respect to \\(x\\), \\( k \\cos x + \\arccos \\frac{\\pi}{4} = 0 \\), has a solution that belongs to the third quadrant, then \n(A) \\( k \\geq -\\arccos \\frac{\\pi}{4} \\) \n(B) \\( k < \\frac{\\sqrt{2}}{2} \\) \n(C) \\( k > \\arccos \\frac{\\pi}{4} \\) \n(D) \\( \\fr... | 298 |
[
{
"content": "In the geometric sequence $\\left\\{ a_n \\right\\}$, $a_2=5$ and $a_6=33$. Find the value of $a_3 \\cdot a_5$.",
"role": "user"
},
{
"content": "To find the value of \\(a_3 \\cdot a_5\\) in the given geometric sequence, we need to use the properties of geometric sequences and the info... | 299 |
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