messages listlengths 2 2 | conversation_id int64 0 72.4k |
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[
{
"content": "What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction.",
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},
{
"content": "To determine the coefficient of \\(x^2y^6\\) in the expansion of \\(\\left(\\frac{3}{5}x - \\frac{y}{2}\\r... | 0 |
[
{
"content": "Given the quadratic function $f(x)=ax^{2}+bx+c$ with its derivative $f′(x)$, where $f′(0) > 0$, and $f(x)\\geqslant 0$ for any real number $x$, find the minimum value of $\\frac{f(1)}{f′(0)}$.",
"role": "user"
},
{
"content": "To solve the given problem, we need to find the minimum val... | 1 |
[
{
"content": "If $n\\left(n\\neq 0\\right)$ is a root of the equation $x^{2}+mx+3n=0$ with respect to $x$, then the value of $m+n$ is ______.",
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},
{
"content": "To solve the problem, we need to determine the value of \\(m + n\\) given that \\(n \\, (n \\neq 0)\\) is a root of the equa... | 2 |
[
{
"content": "Find the smallest positive $a$ such that $a$ is a multiple of $4$ and $a$ is a multiple of $14.$",
"role": "user"
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"content": "To solve this problem, we need to find the smallest positive integer \\( a \\) that is a multiple of both 4 and 14. This means \\( a \\) has to be the leas... | 3 |
[
{
"content": "The product of 8 and 40 is \\_\\_\\_\\_\\_\\_, and 5 times $\\frac {1}{6}$ is \\_\\_\\_\\_\\_\\_. The sum of 6 instances of 500 ends with \\_\\_\\_\\_\\_\\_ zeros, and the sum is \\_\\_\\_\\_\\_\\_.",
"role": "user"
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{
"content": "To solve this problem, we need to break it down into ... | 4 |
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{
"content": "Let $S$ be a set of real numbers with mean $M$ . If the means of the sets $S\\cup \\{15\\}$ and $S\\cup \\{15,1\\}$ are $M + 2$ and $M + 1$ , respectively, then how many elements does $S$ have?",
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{
"content": "To solve this problem, let's denote:\n- \\( n ... | 5 |
[
{
"content": "Given positive numbers $m$ and $n$ satisfying $2m+n=1$, find the minimum value of $\\frac {1}{m}+ \\frac {2}{n}$.",
"role": "user"
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{
"content": "To solve this problem, we need to find the minimum value of the expression \\(\\frac{1}{m} + \\frac{2}{n}\\) subject to the constraint \\(... | 6 |
[
{
"content": "Petya picked four distinct digits, none of which are zero. Then he formed all possible four-digit numbers without repeating any of the digits. The sum of all these numbers turned out to be 73,326. What four digits did Petya pick?",
"role": "user"
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{
"content": "To solve this problem,... | 7 |
[
{
"content": "Find all real values of $x$ which satisfy\n\\[\\frac{1}{x + 1} + \\frac{6}{x + 5} \\ge 1.\\]",
"role": "user"
},
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"content": "To solve the inequality \\(\\frac{1}{x + 1} + \\frac{6}{x + 5} \\ge 1\\), we will break down the solution into several clear steps with the aid of Python for a... | 8 |
[
{
"content": "On Tuesday, Max's mom gave him $8 dollars for a hot dog at the Grand Boulevard park. On Wednesday, his mom gave him 5 times as much money as she gave him on Tuesday. On Thursday, his mom gave him $9 more in money than she gave him on Wednesday. How much more money did his mom give him on Thursday ... | 9 |
[
{
"content": "Given that the function $y=f\\left(x\\right)$ is an odd function, defined on $R$, and satisfies $f\\left(x+4\\right)=f\\left(x\\right)$. When $x\\in \\left(0,2\\right)$, $f\\left(x\\right)=2^{x}+\\log _{3}(x+2)$, find $f\\left(2023\\right)$.",
"role": "user"
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{
"content": "To solve t... | 10 |
[
{
"content": "The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$ , $P_{0}P_{3}P_{6}$ , $P_{0}P_{6}P_{7}$ , $P_{0}P_{7}P_{8}$ , $P_{1}P_{2}P_{3}$ , $P_{3}P_{4}P_{6}$ , $P_{4}P_{5}P_{6}$ . In how ma... | 11 |
[
{
"content": "If the complex number $z = (x^2 - 3x + 2) + (x - 1)i$ is a pure imaginary number, then the real number $x = \\boxed{2}$.",
"role": "user"
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{
"content": "To determine the value of \\( x \\) for which the given complex number \\( z = (x^2 - 3x + 2) + (x - 1)i \\) is a pure imaginary nu... | 12 |
[
{
"content": "By the first of September, Vlad bought several ballpoint and gel pens. He noticed that if all the purchased pens were gel pens, he would have paid 4 times more than he actually did. Conversely, if all the pens were ballpoint pens, the purchase would have cost him half of the actual amount. How man... | 13 |
[
{
"content": "Find the least integer value of $x$ for which $2|x| + 7 < 17$.",
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{
"content": "To find the least integer value of \\( x \\) for which \\( 2|x| + 7 < 17 \\), let's reason through the inequality step by step.\n\nFirst, isolate the term involving the absolute value \\( |... | 14 |
[
{
"content": "A psychologist believes one in every four people has a disorder he calls ZYX syndrome. The clinic where he works has recently doubled its previous number of 26 patients. How many patients is the psychologist likely to diagnose with ZYX syndrome?",
"role": "user"
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{
"content": "The pr... | 15 |
[
{
"content": "Let $(F_n)_{n\\in{N^*}}$ be the Fibonacci sequence defined by\r $F_1=1$ , $F_2=1$ , $F_{n+1}=F_n+F_{n-1}$ for every $n\\geq{2}$ . Find\r\nthe limit: \\[ \\lim_{n \\to \\infty}(\\sum_{i=1}^n{\\frac{F_i}{2^i}}) \\]",
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{
"content": "To find the limit of the series \... | 16 |
[
{
"content": "A positive integer is called a \"lucky number\" if it satisfies the following conditions:\n(1) It consists of only the digits \"6\" and \"8\";\n(2) It has at least two consecutive 8's.\nFor example, 886 is a three-digit \"lucky number\", 8866 is a four-digit \"lucky number\", and 686888 is a six-d... | 17 |
[
{
"content": "In the expansion of $(2x^{3}+\\frac{1}{\\sqrt{x}})^{6}$, the coefficient of $x^{4}$ is ______.",
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{
"content": "To solve the problem, we need to find the coefficient of \\(x^4\\) in the expansion of \\((2x^3 + \\frac{1}{\\sqrt{x}})^6\\). We can use the Binomial Theorem... | 18 |
[
{
"content": "There are $4$ different gifts. If they are all to be distributed to two people, A and B, with each person receiving at least one gift, the total number of different ways to distribute the gifts is ______.",
"role": "user"
},
{
"content": "To solve this problem using Python and sympy, w... | 19 |
[
{
"content": "Linda owns a store. She sells jeans at 11 dollars each and tees at 8 dollars each. If she sold 7 tees and 4 jeans in a day, how much money, in dollars, did she have at the end of the day?",
"role": "user"
},
{
"content": "To determine how much money Linda made at the end of the day, we... | 20 |
[
{
"content": "The difference between a two-digit number and the number obtained by reversing its digits is $5$ times the sum of the digits of either number. What is the sum of the two digit number and its reverse?\n$\\textbf{(A) }44\\qquad \\textbf{(B) }55\\qquad \\textbf{(C) }77\\qquad \\textbf{(D) }99\\qquad... | 21 |
[
{
"content": "Dave breaks 2 guitar strings per night when playing live. If he performs 6 shows a week for 12 weeks, how many guitar strings will he need to replace?",
"role": "user"
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{
"content": "To determine the total number of guitar strings Dave will need to replace, we need to calculate the ... | 22 |
[
{
"content": "John has to restring 3 basses with 4 strings each, twice as many guitars which have 6 strings each, and 3 fewer 8 string guitars than the normal guitars. How many strings does he need?",
"role": "user"
},
{
"content": "To solve this problem, let's break it down step by step and then v... | 23 |
[
{
"content": "The real-coefficient polynomials \\( f(x) \\) and \\( g(x) \\) satisfy \\( (1+\\mathrm{i} x)^{2001}=f(x)+\\mathrm{i} g(x) \\). Then the sum of the coefficients of \\( f(x) \\) is _______.",
"role": "user"
},
{
"content": "To solve this problem, we need to carefully analyze the given eq... | 24 |
[
{
"content": "Find the number of subsets \\( S \\) of \\(\\{1,2, \\ldots 6\\}\\) satisfying the following conditions:\n- \\( S \\) is non-empty.\n- No subset of \\( S \\) has the property that the sum of its elements is 10.",
"role": "user"
},
{
"content": "To solve this problem, we need to find the... | 25 |
[
{
"content": "( Fixed points and limits) Find all functions from $\\mathbb{R}_{+}^{*}$ to $\\mathbb{R}_{+}^{*}$ such that for all $x, y > 0$, \\( f(x f(y)) = y f(x) \\) and \\( \\lim _{x \\rightarrow+\\infty} f(x) = 0 \\).",
"role": "user"
},
{
"content": "To solve this problem, we'll find functions... | 26 |
[
{
"content": "In how many ways can all ten digits be arranged in a row such that the digits $2, 5, 8$ are placed next to each other?",
"role": "user"
},
{
"content": "To determine the number of ways to arrange all ten digits such that the digits \\(2, 5, 8\\) are placed next to each other, we can us... | 27 |
[
{
"content": "Let the sequence $\\{a_n\\}$ have a sum of the first $n$ terms denoted by $S_n$. It is known that $a_1 + 2a_2 + 3a_3 + \\ldots + na_n = (n-1)S_n + 2n$ ($n \\in \\mathbb{N}^*$).\n1. Find the values of $a_2$ and $a_3$.\n2. Prove that the sequence $\\{S_n + 2\\}$ is a geometric sequence.\n3. Let $b_n... | 28 |
[
{
"content": "A semicircle is inscribed in triangle \\( ABC \\) such that its diameter lies on side \\( BC \\), and the arc touches sides \\( AB \\) and \\( AC \\) at points \\( C_1 \\) and \\( B_1 \\) respectively. Prove that\n\n$$\n\\frac{A C_{1}}{C_{1} B} \\cdot \\frac{B H}{H C} \\cdot \\frac{C B_{1}}{B_{1} ... | 29 |
[
{
"content": "Two farmers agree that pigs are worth $\\$300$ and that goats are worth $\\$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $\\$390$ debt could be paid with two pigs, with one goat re... | 30 |
[
{
"content": "Find the value of \\(10 \\cdot \\operatorname{ctg}(\\operatorname{arcctg} 3 + \\operatorname{arcctg} 7 + \\operatorname{arcctg} 13 + \\operatorname{arcctg} 21)\\).",
"role": "user"
},
{
"content": "To solve the given problem \\(10 \\cdot \\operatorname{ctg}(\\operatorname{arcctg} 3 + \... | 31 |
[
{
"content": "Given two sets $A=\\{1, 2, 3\\}$ and $B=\\{2, 3\\}$. Define the operation \"$*$\" between sets $A$ and $B$ as $A*B=\\{x | x=x_1+x_2, x_1 \\in A, x_2 \\in B\\}$. Determine the largest element in the set $A*B$ and the number of all proper subsets of $A*B$.",
"role": "user"
},
{
"content"... | 32 |
[
{
"content": "The head of a fish weighs as much as the tail and half of the body, the body weighs as much as the head and the tail together. The tail weighs 1 kg. How much does the fish weigh?",
"role": "user"
},
{
"content": "To solve this problem, let's define variables and set up equations based ... | 33 |
[
{
"content": "Prove that for any four points \\( A, B, C, D \\) the following equality holds: \\(\\overline{A B} \\cdot \\overline{C D}+\\overline{A C} \\cdot \\overline{D B}+\\overline{A D} \\cdot \\overline{B C}=0\\).",
"role": "user"
},
{
"content": "To prove the given statement, we'll utilize th... | 34 |
[
{
"content": "If \\(2^{x}=15\\) and \\(15^{y}=32\\), the value of \\(xy\\) is:\n(A) 5\n(B) 8\n(C) 16\n(D) 6\n(E) 4",
"role": "user"
},
{
"content": "To find the value of \\(xy\\), we first need to solve for \\(x\\) and \\(y\\). \n\nGiven:\n\\[2^x = 15\\]\n\\[15^y = 32\\]\n\nWe can take the logarithm... | 35 |
[
{
"content": "In an arithmetic sequence $\\{a_n\\}$, it is known that $a_1 = 3$ and $a_9 = 11$. Find the sum of the first 9 terms, $S_9$.",
"role": "user"
},
{
"content": "To solve this problem, we'll first determine the common difference of the arithmetic sequence, and then use it to find the sum o... | 36 |
[
{
"content": "Find the largest 12-digit number \\( N \\) that satisfies the following two conditions:\n\n- The decimal representation of \\( N \\) contains six digits \"4\" and six digits \"7\";\n- In the decimal representation of \\( N \\), no four consecutive digits form the number \"7444\".",
"role": "us... | 37 |
[
{
"content": "In a lake, there is a patch of water lilies. The area of the water lilies doubles every day. If it takes 48 days for the water lilies to cover the entire lake, how many days does it take for them to cover half of the lake?",
"role": "user"
},
{
"content": "To solve this problem, let's ... | 38 |
[
{
"content": "The function $y=\\sin \\frac{\\pi }{3}x$ takes at least $2$ maximum values in the interval $[0,t]$. The minimum value of the positive integer $t$ is ______.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the minimum value of the positive integer \\( t \\) such ... | 39 |
[
{
"content": "A hospital has 5 internists and 6 surgeons. They need to dispatch 4 doctors to join a disaster relief medical team. If both internists and surgeons are required to be part of the team, how many ways are there to choose the doctors (answer with a number)?",
"role": "user"
},
{
"content"... | 40 |
[
{
"content": "In the arithmetic sequence $\\{a_n\\}$, it is known that $a_1=1$ and $a_5=9$. Find the sum of the first $5$ terms, $S_n=$ ______.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the sum of the first 5 terms of the arithmetic sequence given the initial term \\( a... | 41 |
[
{
"content": "A chessboard of size $8 \\times 8$ is considered.\n\nHow many ways are there to place 6 rooks such that no two rooks are ever on the same row or column?",
"role": "user"
},
{
"content": "To solve the problem of placing 6 rooks on an 8x8 chessboard such that no two rooks are ever on the... | 42 |
[
{
"content": "The base of the pyramid is an isosceles triangle \\( A B C \\) with a base \\( AC = 2 \\) and a lateral side \\( \\sqrt{7} \\). The face \\( A C D \\) is perpendicular to the base plane and is an equilateral triangle. Find the edge \\( B D \\), as well as the areas of all those cross-sections of t... | 43 |
[
{
"content": "If $a,b,c$ are integers from the set of positive integers less than $7$ such that \\begin{align*}\nabc&\\equiv 1\\pmod 7,\\\\\n5c&\\equiv 2\\pmod 7,\\\\\n6b&\\equiv 3+b\\pmod 7,\n\\end{align*}then what is the remainder when $a+b+c$ is divided by $7$?",
"role": "user"
},
{
"content": "L... | 44 |
[
{
"content": "Each turtle lays a clutch of 20 eggs. If 40% of the eggs successfully hatch, how many hatchlings do 6 turtles produce?",
"role": "user"
},
{
"content": "To solve this problem, we need to follow these steps:\n1. Calculate the total number of eggs laid by 6 turtles.\n2. Determine how man... | 45 |
[
{
"content": "Convert the binary number $101_{(2)}$ to a decimal number.",
"role": "user"
},
{
"content": "To solve the problem of converting the binary number \\(101_{(2)}\\) to a decimal number, we can follow these steps:\n\n1. Understand binary to decimal conversion: Each digit in a binary number... | 46 |
[
{
"content": "A regular polygon has an exterior angle that measures $15$ degrees. How many sides does the polygon have?",
"role": "user"
},
{
"content": "To solve the problem, we need to use the following property of regular polygons:\n\n- The sum of the exterior angles of any polygon is always $360... | 47 |
[
{
"content": "If \\(x \\%\\) of 60 is 12, what is \\(15\\%\\) of \\(x\\)? \n(A) \\(\\frac{3}{4}\\) \n(B) \\(\\frac{1}{3}\\) \n(C) 4 \n(D) 3 \n(E) 9",
"role": "user"
},
{
"content": "To solve this problem step-by-step, we will follow these procedures:\n\n1. Convert the word problem into mathemat... | 48 |
[
{
"content": "The Absent-minded Scientist constructed a device consisting of a sensor and a transmitter. The average lifespan (expected value) of the sensor is 3 years, and the average lifespan of the transmitter is 5 years. Knowing the distribution of the lifespan of the sensor and the transmitter, the Absent-... | 49 |
[
{
"content": "I randomly pick an integer $p$ between $1$ and $10$ inclusive. What is the probability that I choose a $p$ such that there exists an integer $q$ so that $p$ and $q$ satisfy the equation $pq - 4p - 2q = 2$? Express your answer as a common fraction.",
"role": "user"
},
{
"content": "To s... | 50 |
[
{
"content": "Kai will celebrate his 25th birthday in March 2020. In what year was Kai born?\n(A) 1975\n(B) 1990\n(C) 1995\n(D) 2000\n(E) 1955",
"role": "user"
},
{
"content": "To determine the year Kai was born, we need to calculate the year by subtracting his age (25 years) from the year he will b... | 51 |
[
{
"content": "Let \\( a = \\frac{\\sqrt{5} + \\sqrt{3}}{\\sqrt{5} - \\sqrt{3}} \\) and \\( b = \\frac{\\sqrt{5} - \\sqrt{3}}{\\sqrt{5} + \\sqrt{3}} \\). Find the value of \\( a^4 + b^4 + (a+b)^4 \\).",
"role": "user"
},
{
"content": "To solve this problem using Python, we'll go through a series of s... | 52 |
[
{
"content": "The numbers \\(1, 2, \\cdots, 9\\) are randomly placed into a \\(3 \\times 3\\) grid such that each cell contains exactly one number and each number is used exactly once. What is the probability that the sum of the numbers in each row and each column is odd?\n\n(A) \\(\\frac{1}{21}\\)\n\n(B) \\(\\... | 53 |
[
{
"content": "Let $S = \\{5^k | k \\in \\mathbb{Z}, 0 \\le k \\le 2004 \\}$. Given that $5^{2004} = 5443 \\cdots 0625$ has $1401$ digits, how many elements of $S$ begin with the digit $1$?\n",
"role": "user"
},
{
"content": "To solve the problem of finding how many elements of the set \\( S = \\{5^k... | 54 |
[
{
"content": "Let \\( B \\) be a point on the circle centered at \\( O \\) with diameter \\( AC \\) and let \\( D \\) and \\( E \\) be the circumcenters of the triangles \\( OAB \\) and \\( OBC \\) respectively. Given that \\( \\sin \\angle BOC = \\frac{4}{5} \\) and \\( AC = 24 \\), find the area of the triang... | 55 |
[
{
"content": "Given quadratic polynomials \\( f \\) and \\( g \\) with the same leading coefficients, it is known that the sum of the four roots of these polynomials is \\( p \\). Find the sum of the roots of the polynomial \\( f+g \\), provided that it has two roots.",
"role": "user"
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{
"content"... | 56 |
[
{
"content": "Kolya and Vitya play the following game on an infinite grid paper. Starting with Kolya, they take turns marking grid nodes, which are the points of intersection of vertical and horizontal lines. Each player must mark a node in such a way that after the move, all marked nodes lie at the vertices of... | 57 |
[
{
"content": "Janet has 10 coins consisting of nickels, dimes, and quarters. Seven of the coins are either dimes or quarters, and eight of the coins are either dimes or nickels. How many dimes does Janet have?",
"role": "user"
},
{
"content": "To solve this problem, we need to set up a system of equ... | 58 |
[
{
"content": "A positive integer will be called \"sparkly\" if its smallest positive divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers \\(2, 3, \\ldots, 2003\\) are sparkly?",
"role": "user"
},
{
"content": "To determine how many numbers between 2 and ... | 59 |
[
{
"content": "Given the parabola $x^{2}=ay$, find the value of $a$ if the equation of its directrix is $y=- \\frac {1}{4}$.",
"role": "user"
},
{
"content": "To solve this problem, we need to understand the relationship between the parabola \\(x^2 = ay\\) and its directrix. \n\nThe standard form of ... | 60 |
[
{
"content": "Given that \\( x \\geq 1 \\), \\( y \\geq 1 \\), and \\(\\lg ^{2} x+\\lg ^{2} y=\\lg 10 x^{2}+\\lg 10 y^{2}\\), find the maximum value of \\( u=\\lg x y \\).",
"role": "user"
},
{
"content": "To solve the given problem, we need to carefully analyze the equation \\(\\lg^2 x + \\lg^2 y =... | 61 |
[
{
"content": "Dongfang Elementary School organized a tree-planting activity. The fifth grade planted 114 trees, which was twice as many as the third grade. The fourth grade planted 32 more trees than the third grade. How many trees did the fourth grade plant?",
"role": "user"
},
{
"content": "To sol... | 62 |
[
{
"content": "What is the domain of the function $f(x)=\\log_2(\\log_3(\\log_4(\\log_5x)))$?",
"role": "user"
},
{
"content": "To solve for the domain of the function \\( f(x) = \\log_2(\\log_3(\\log_4(\\log_5x))) \\), we need to ensure that the arguments in each of the logarithmic functions are pos... | 63 |
[
{
"content": "A factory needs to build a warehouse. According to market research analysis, shipping costs are directly proportional to the distance between the factory and the warehouse, while storage costs are inversely proportional to this distance. When the distance between the factory and the warehouse is 4... | 64 |
[
{
"content": "Given that $(2x-1)^{6}=a\\_0+a\\_1x+a\\_2x^{2}+a\\_3x^{3}+...+a\\_6x^{6}.$, find the value of $a\\_1+2a\\_2+3a\\_3+...+6a\\_6 =$ ___.",
"role": "user"
},
{
"content": "To solve the problem, we need to expand the expression \\((2x - 1)^6\\) and then find the coefficients \\(a_1, a_2, a_... | 65 |
[
{
"content": "A polygon is drawn on graph paper with a perimeter of 36, and its sides follow the grid lines. What is the maximum possible area it can have?",
"role": "user"
},
{
"content": "To find the maximum possible area of a polygon with a given perimeter that follows grid lines, the best approa... | 66 |
[
{
"content": "Find the smallest integer $k$ for which the conditions\n(1) $a_1,a_2,a_3\\cdots$ is a nondecreasing sequence of positive integers\n(2) $a_n=a_{n-1}+a_{n-2}$ for all $n>2$\n(3) $a_9=k$\nare satisfied by more than one sequence.",
"role": "user"
},
{
"content": "To solve this problem, we ... | 67 |
[
{
"content": "The insurance contract provides an insured amount of 500,000 rubles and a deductible of 1%. The actual damage amounted to 4,000 rubles. What will be the amount of insurance compensation?",
"role": "user"
},
{
"content": "To determine the amount of insurance compensation, we need to cal... | 68 |
[
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of\n\\[\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z}.\\]",
"role": "user"
},
{
"content": "Given the constraints \\( x, y, z > 0 \\) and \\( x + y + z = 1 \\), we want to find the minimum val... | 69 |
[
{
"content": "The company \"Horns and Hooves\" split into the company \"Horns\" and the company \"Hooves\" with a different number of employees. The director of the company \"Horns\" receives the same salary as the director of the company \"Hooves\", and the average salary of all other employees in the company ... | 70 |
[
{
"content": "Given that $a$ and $b$ are two consecutive integers, and $a < \\sqrt{13} < b$, find the value of $a + b$.",
"role": "user"
},
{
"content": "To solve this problem, we need to identify the two consecutive integers \\( a \\) and \\( b \\) such that \\( a < \\sqrt{13} < b \\). \n\nHere is ... | 71 |
[
{
"content": "\nCalculate the lengths of the arcs of the curves given by the parametric equations.\n\n$$\n\\begin{aligned}\n& \\left\\{\\begin{array}{l}\nx=\\left(t^{2}-2\\right) \\sin t+2 t \\cos t \\\\\ny=\\left(2-t^{2}\\right) \\cos t+2 t \\sin t\n\\end{array}\\right. \\\\\n& 0 \\leq t \\leq 2 \\pi\n\\end{al... | 72 |
[
{
"content": "John buys 5 notebooks that have 40 pages each. He uses 4 pages per day. How many days do the notebooks last?",
"role": "user"
},
{
"content": "To solve this problem, we need to determine the total number of pages John has and then divide that by the number of pages he uses per day. L... | 73 |
[
{
"content": "Given point $P(t, 4)$ on the parabola $y^{2} = 4x$, and the focus of the parabola is $F$, find $|PF|$ = \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "To solve this problem, we need to follow these steps:\n\n1. **Identify the focus of the parabola:**\n The equation of the para... | 74 |
[
{
"content": "A bag contains 5 red, 6 green, 7 yellow, and 8 blue jelly beans. A jelly bean is selected at random. What is the probability that it is blue?",
"role": "user"
},
{
"content": "To determine the probability of selecting a blue jelly bean from the bag, we'll use the following method:\n... | 75 |
[
{
"content": "Given the quadratic equation $x^{2}-2x-2=0$, after completing the square, we get $\\left(x-1\\right)^{2}=n$. Find the value of $n$.",
"role": "user"
},
{
"content": "To solve for the value of \\( n \\) after completing the square for the given quadratic equation \\( x^2 - 2x - 2 = 0 \\... | 76 |
[
{
"content": "Calculate the limit of the function:\n\n\\[\n\\lim _{x \\rightarrow 0} \\frac{\\cos x+\\ln (1+x) \\sqrt{2+\\cos \\left(\\frac{1}{x}\\right)}}{2+e^{x}}\n\\]",
"role": "user"
},
{
"content": "To solve the problem of finding the limit:\n\n\\[\n\\lim_{x \\rightarrow 0} \\frac{\\cos x + \\l... | 77 |
[
{
"content": "The figure shows the map of an (imaginary) country consisting of five states. The goal is to color this map with green, blue, and yellow in such a way that no two neighboring states share the same color. In how many different ways can the map be painted?\n\n(a) 12\n(b) 6\n(c) 10\n(d) 24\n(e) 120",... | 78 |
[
{
"content": "A Louis Vuitton store in Shanghai had a number of pairs of sunglasses which cost an average of \\ $$ 900 $ per pair. LiAngelo Ball stole a pair which cost \\$ $ 2000$ . Afterwards, the average cost of sunglasses in the store dropped to \\ $$ 890$ per pair. How many pairs of sunglasses were in th... | 79 |
[
{
"content": "\\( A, P, B \\) are three points on a circle with center \\( O \\).\nIf \\( \\angle APB = 146^\\circ \\), find \\( \\angle OAB \\).",
"role": "user"
},
{
"content": "To solve this problem, we need to use some properties of circles and angles.\n\n1. **Understand the geometry of the prob... | 80 |
[
{
"content": "Solve in prime numbers the equation \\( p^2 - 6pq + q^2 + 3q - 1 = 0 \\).",
"role": "user"
},
{
"content": "To solve the problem of finding prime numbers \\( p \\) and \\( q \\) that satisfy the equation \\( p^2 - 6pq + q^2 + 3q - 1 = 0 \\), we can proceed with the following steps:\n\n... | 81 |
[
{
"content": "If the square roots of a positive number are $m-3$ and $m-7$, find:<br/>$(1)$ the positive number;<br/>$(2)$ the cube root of $m^{2}+2$.",
"role": "user"
},
{
"content": "To solve this problem, follow these steps:\n\n1. Carefully read and understand the problem statement.\n2. Break dow... | 82 |
[
{
"content": "Calculate the value of $\\lg 8 + 3\\lg 5$ (Simplify the answer to its most basic form).",
"role": "user"
},
{
"content": "To solve the given problem, we need to simplify the expression \\(\\lg 8 + 3\\lg 5\\) using logarithmic properties. Let's break it down step-by-step:\n\n1. **Recall... | 83 |
[
{
"content": "What is the largest six-digit number that can be obtained by removing nine digits from the number 778157260669103, without changing the order of its digits?\n(a) 778152\n(b) 781569\n(c) 879103\n(d) 986103\n(e) 987776",
"role": "user"
},
{
"content": "To solve this problem, we need to r... | 84 |
[
{
"content": "If angle $A$ lies in the second quadrant and $\\sin A = \\frac{3}{4},$ find $\\cos A.$",
"role": "user"
},
{
"content": "To find the value of \\(\\cos A\\) given that \\(\\sin A = \\frac{3}{4}\\) and that the angle \\(A\\) lies in the second quadrant, we can utilize the Pythagorean ide... | 85 |
[
{
"content": "Is it true that for any number equal to the product of two consecutive natural numbers, you can append some two digits to the end so that you get a square of a natural number?",
"role": "user"
},
{
"content": "To determine whether for any number equal to the product of two consecutive ... | 86 |
[
{
"content": "Given $a=2+\\sqrt{3}$ and $b=2-\\sqrt{3}$, find the value of $a^{2}+2ab-b\\left(3a-b\\right)$.",
"role": "user"
},
{
"content": "Let's solve the problem step by step.\n\nGiven:\n\\[ a = 2 + \\sqrt{3} \\]\n\\[ b = 2 - \\sqrt{3} \\]\n\nWe need to find the value of:\n\\[ a^{2} + 2ab - b(3... | 87 |
[
{
"content": "Given the inverse proportion function $y=\\frac{k}{x}$ passes through the point $\\left(-2,-3\\right)$, find $k$.",
"role": "user"
},
{
"content": "To find the constant \\( k \\) for the inverse proportion function \\( y = \\frac{k}{x} \\) that passes through the point \\((-2, -3)\\), ... | 88 |
[
{
"content": "Can 54 notebooks be divided into three piles so that each pile contains an odd number of notebooks?",
"role": "user"
},
{
"content": "To determine whether 54 notebooks can be divided into three piles with each pile containing an odd number of notebooks, we need to consider the properti... | 89 |
[
{
"content": "To enhance students' mathematical literacy, the school offers three elective courses for 11th-grade students: \"History of Mathematics\", \"Symmetry and Groups\", and \"Geometry on the Sphere\". It is known that there are 600 students in the 11th grade, and each of them participates in only one el... | 90 |
[
{
"content": "Given that point $F$ is the focus of the parabola $C$: $y^{2}=4x$, and point $B$ is on parabola $C$, $A(5,4)$, when the perimeter of $\\triangle ABF$ is minimum, the area of this triangle is $\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
{
"content": "To solve this problem, we need to follo... | 91 |
[
{
"content": "Solve\n\\[\\arcsin x + \\arcsin 2x = \\frac{\\pi}{3}.\\]",
"role": "user"
},
{
"content": "To solve the equation \\(\\arcsin x + \\arcsin 2x = \\frac{\\pi}{3}\\), let's use a numerical approach.\n\nFirst, let's start by defining the range of \\(x\\). The arcsine function is defined for... | 92 |
[
{
"content": "Given the equation $5x^2+kx-10=0$ has one root as $-5$, find the other root and the value of $k$.",
"role": "user"
},
{
"content": "To solve this problem, we need to use the properties of quadratic equations. Given that one of the roots is \\( -5 \\), let's denote the other root by \\(... | 93 |
[
{
"content": "A plane passes through the midpoints of edges $AB$ and $CD$ of pyramid $ABCD$ and divides edge $BD$ in the ratio $1:3$. In what ratio does this plane divide edge $AC$?",
"role": "user"
},
{
"content": "To solve this problem, we can use the coordinate geometry approach and linear algebr... | 94 |
[
{
"content": "Let \\( S = \\{1, 2, \\cdots, n\\} \\). Suppose the 3-element subsets \\( A_1, A_2, \\cdots, A_m \\) (\\( m > 1 \\)) satisfy:\n(1) Every 2-element subset of \\( S \\) is contained in exactly one \\( A_i \\) (\\( 1 \\leq i \\leq m \\));\n(2) Each \\( A_i \\) and \\( A_j \\) (\\( i \\neq j, 1 \\leq ... | 95 |
[
{
"content": "Compute the product of the roots of the equation \\[x^3 - 12x^2 + 48x + 28 = 0.\\]",
"role": "user"
},
{
"content": "To find the product of the roots of the cubic polynomial equation \\(x^3 - 12x^2 + 48x + 28 = 0\\), we can utilize Vieta's formulas. For a cubic equation of the form \\(... | 96 |
[
{
"content": "The perimeter of a square is $20 \\mathrm{~cm}$. By how many square centimeters will the area of the square increase if its perimeter is increased by $12 \\mathrm{~cm}$?",
"role": "user"
},
{
"content": "To solve the problem, we'll follow these steps:\n\n1. **Understand the problem:** ... | 97 |
[
{
"content": "The value of $x$ in the sequence $1$, $1$, $2$, $3$, $x$, $8$, $13$, $21$, $…$ is \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "To solve this problem, we first need to recognize the pattern in the given sequence. The sequence provided is \\(1, 1, 2, 3, x, 8, 13, 21, \\ldots\\).... | 98 |
[
{
"content": "Let \\( S = \\left\\{(x, y) \\mid x^{2} - y^{2} \\text{ is odd}, \\, x, y \\in \\mathbf{R} \\right\\} \\) and \\( T = \\left\\{(x, y) \\mid \\sin \\left(2 \\pi x^{2}\\right) - \\sin \\left(2 \\pi y^{2}\\right) - \\cos \\left(2 \\pi x^{2}\\right) - \\cos \\left(2 \\pi y^{2}\\right), \\, x, y \\in \... | 99 |
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