messages listlengths 2 2 | conversation_id int64 0 72.4k |
|---|---|
[
{
"content": "The volume of the tetrahedron \\( A B C D \\) is \\( V \\). Points \\( K \\), \\( L \\), and \\( M \\) are taken on the edges \\( C D \\), \\( D B \\), and \\( A B \\) respectively, such that \\( 2 C K = C D \\), \\( 3 D L = D B \\), \\( 5 B M = 2 A B \\). Find the volume of the tetrahedron \\( K ... | 300 |
[
{
"content": "Let \\( x_1, x_2, \\ldots, x_{100} \\) be natural numbers greater than 1 (not necessarily distinct). In an \\(80 \\times 80\\) table, numbers are arranged as follows: at the intersection of the \\(i\\)-th row and the \\(k\\)-th column, the number \\(\\log _{x_{k}} \\frac{x_{i}}{16}\\) is written. ... | 301 |
[
{
"content": "Given the mapping $A \\rightarrow B$ with the rule $f: x \\rightarrow 2x+1$, find the element in $A$ that corresponds to the element $3$ in $B$.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the element in set \\(A\\) that maps to the element \\(3\\) in set \\... | 302 |
[
{
"content": "Mary just held tryouts for the high school band. 80% of the 20 flutes got in, half the 30 clarinets got in, 1/3 of the 60 trumpets got in, and 1/10th of the 20 pianists got in. How many people are in the band total?",
"role": "user"
},
{
"content": "To solve this problem, let's break i... | 303 |
[
{
"content": "Four red candies and three green candies can be combined to make many different flavors. Flavors are different if the percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green; and likewise 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is to be made us... | 304 |
[
{
"content": "Sarah decided to pull weeds from her garden. On Tuesday she pulled 25 weeds. The next day she pulled three times the number of weeds she did the day before. On Thursday her allergies bothered her and she could only pull up one-fifth of the weeds she pulled on the day before. Finally, on Friday it ... | 305 |
[
{
"content": "How many of the base-ten numerals for the positive integers less than or equal to $2017$ contain the digit $0$?\n$\\textbf{(A)}\\ 469\\qquad\\textbf{(B)}\\ 471\\qquad\\textbf{(C)}\\ 475\\qquad\\textbf{(D)}\\ 478\\qquad\\textbf{(E)}\\ 481$",
"role": "user"
},
{
"content": "To determine ... | 306 |
[
{
"content": "If the point $(5, b)$ is between two parallel lines $6x - 8y + 1 = 0$ and $3x - 4y + 5 = 0$, \nthen the value of the integer $b$ is",
"role": "user"
},
{
"content": "To solve the problem, we need to determine the value of \\(b\\) such that the point \\((5, b)\\) lies between the two p... | 307 |
[
{
"content": "We will call a natural number \\( n \\) squareable if the numbers from 1 to \\( n \\) can be arranged in such a way that each member of the sequence, when added to its position number, results in a perfect square. For example, the number 5 is squareable because the numbers can be arranged like thi... | 308 |
[
{
"content": "Find constants $A$, $B$, and $C$, such that\n$$\\frac{-x^2+3x-4}{x^3+x}= \\frac{A}{x} +\\frac{Bx+C}{x^2+1} $$Enter your answer as the ordered triplet $(A,B,C)$.",
"role": "user"
},
{
"content": "To find the constants \\( A \\), \\( B \\), and \\( C \\), we need to decompose the given r... | 309 |
[
{
"content": "Given an ellipse $C$: $\\frac{x^2}{25} + \\frac{y^2}{9} = 1$ and a point $P$ on it, the distance from $P$ to the right directrix is $5$. Find the distance from point $P$ to the left focus of the ellipse $C$.",
"role": "user"
},
{
"content": "To solve the problem, we need to use the geo... | 310 |
[
{
"content": "David recently bought a large supply of letter tiles. One day he arrives back to his dorm to find that some of the tiles have been arranged to read $\\textsc{Central Michigan University}$ . What is the smallest number of tiles David must remove and/or replace so that he can rearrange them to re... | 311 |
[
{
"content": "For a positive integer $n$ , let $\\omega(n)$ denote the number of positive prime divisors of $n$ . Find the smallest positive tinteger $k$ such that $2^{\\omega(n)}\\leq k\\sqrt[4]{n}\\forall n\\in\\mathbb{N}$ .",
"role": "user"
},
{
"content": "Let's start by understanding the... | 312 |
[
{
"content": "Let $A$ be the number of 2019-digit numbers, that is made of 2 different digits (For example $10\\underbrace{1...1}_{2016}0$ is such number). Determine the highest power of 3 that divides $A$ .",
"role": "user"
},
{
"content": "To solve this problem, we will first determine how ma... | 313 |
[
{
"content": "In the sequence $\\left\\{a_{n}\\right\\}$, $a_{1}$ and $a_{2}$ are given non-zero integers, and $a_{n+2}=\\left|a_{n+1}-a_{n}\\right|$:\n(1) Let $a_{1}=2, a_{2}=-1$, find $a_{2008}$;\n(2) Prove that it is always possible to choose infinitely many terms from $\\left\\{a_{n}\\right\\}$ to form two ... | 314 |
[
{
"content": "(5 points) The function $y=f(x)$ is the inverse function of $y=ax$, and the graph of $f(x)$ passes through the point $(4, 2)$. Find the value of $a$.",
"role": "user"
},
{
"content": "To solve this problem, we need to determine the value of \\(a\\) such that the function \\(y = f(x)\\)... | 315 |
[
{
"content": "Given a cube with a side length of 4, is it possible to entirely cover three of its faces that share a common vertex with 16 rectangular paper strips of size $1 \\times 3$?",
"role": "user"
},
{
"content": "To determine whether it is possible to entirely cover three faces of a cube wit... | 316 |
[
{
"content": "What is $\\frac{2+4+6}{1+3+5} - \\frac{1+3+5}{2+4+6}$?\n$\\textbf{(A)}\\ -1 \\qquad\\textbf{(B)}\\ \\frac{5}{36} \\qquad\\textbf{(C)}\\ \\frac{7}{12} \\qquad\\textbf{(D)}\\ \\frac{49}{20} \\qquad\\textbf{(E)}\\ \\frac{43}{3}$",
"role": "user"
},
{
"content": "To solve the problem, we ... | 317 |
[
{
"content": "Consider a geometric sequence $\\{a_n\\}$ where $a_1 = 1$ and $a_4 = 8$, find the sum of the first 10 terms, $S_{10}$.",
"role": "user"
},
{
"content": "Let's break down the problem into smaller, manageable parts and solve it step-by-step using Python and SymPy.\n\n1. **Understand the ... | 318 |
[
{
"content": "A plane divides space into ______ parts.",
"role": "user"
},
{
"content": "To determine how many parts a plane divides space into, we can use geometric reasoning rather than SymPy, since this is a combinatorial geometry problem.\n\nHere’s an outline of the reasoning process:\n\n1. In 3... | 319 |
[
{
"content": "Find the greatest natural number that cannot be expressed as the sum of two composite numbers.",
"role": "user"
},
{
"content": "To solve the problem of finding the greatest natural number that cannot be expressed as the sum of two composite numbers, let's start by understanding the es... | 320 |
[
{
"content": "Given that $x_{1}$ and $x_{2}$ are the two roots of the quadratic equation $x^{2}-4x+3=0$, find $x_{1}+x_{2}-x_{1}x_{2}=\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the roots of the quadratic equation \\(x^2 - 4x + 3 = 0\\) and then compu... | 321 |
[
{
"content": "In the Cartesian coordinate system, the distance from point $P(2,-3)$ to the $x$-axis is ______.",
"role": "user"
},
{
"content": "To solve the problem of finding the distance from the point \\( P(2, -3) \\) to the \\( x \\)-axis, we can follow these steps:\n\n1. **Understand the Probl... | 322 |
[
{
"content": "Alice has six magical pies in her pocket - two that increase in size (eat - grow), and the remaining are decreasing (eat - shrink). When Alice met Mary Ann, she blindly took three pies from her pocket and gave them to Mary. Find the probability that one of the girls does not have a single size-inc... | 323 |
[
{
"content": "Four people, Jia, Yi, Bing, and Ding, are each required to formulate a quadratic equation with solutions \\( x_1, x_2 \\) of the form \\( x^2 + px + q = 0 \\) under the following conditions:\n- Jia: \\( p, q, x_1, x_2 \\) are all integers that leave a remainder of 1 when divided by 3.\n- Yi: \\( p... | 324 |
[
{
"content": "How many numbers are in the list $$ 1.5, 5.5, 9.5, 13.5, \\ldots, 41.5, 45.5? $$",
"role": "user"
},
{
"content": "To determine how many numbers are in the list \\(1.5, 5.5, 9.5, 13.5, \\ldots, 41.5, 45.5\\), we can treat it as an arithmetic sequence where the first term \\(a_1\\) is 1... | 325 |
[
{
"content": "Alvita is planning a garden patio to be made from identical square paving stones laid out in a rectangle measuring $x$ stones by $y$ stones. She finds that when she adds a border of width one stone around the patio, the area of the border is equal to the original area of the patio. How many possib... | 326 |
[
{
"content": "What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?",
"role": "user"
},
{
"content": "To solve the problem of finding the smallest possible number of whole 2-by-3 non-overlap... | 327 |
[
{
"content": "To enhance students' physical fitness, our school has set up sports interest classes for seventh graders. Among them, the basketball interest class has $x$ students, the number of students in the soccer interest class is $2$ less than twice the number of students in the basketball interest class, ... | 328 |
[
{
"content": "Consider the function $f(x) = x^2 +2\\sqrt{x}$. Evaluate $2f(2) - f(8)$.",
"role": "user"
},
{
"content": "To solve the problem, let's take the following steps:\n\n1. Define the function \\( f(x) = x^2 + 2\\sqrt{x} \\).\n2. Evaluate the function at \\( x = 2 \\) and at \\( x = 8 \\).\... | 329 |
[
{
"content": "By expanding the expression \\((1 + \\sqrt{13})^{210}\\) using the binomial theorem, we obtain terms of the form \\(C_{210}^{k}(\\sqrt{13})^{k}\\). Find the value of \\(k\\) for which this term has the greatest value.",
"role": "user"
},
{
"content": "To find the value of \\(k\\) such ... | 330 |
[
{
"content": "Let \\((a_{1}, a_{2}, \\ldots, a_{8})\\) be a permutation of \\((1,2, \\ldots, 8)\\). Find, with proof, the maximum possible number of elements of the set\n\\[ \\left\\{a_{1}, a_{1}+a_{2}, \\ldots, a_{1}+a_{2}+\\cdots+a_{8}\\right\\} \\]\nthat can be perfect squares.",
"role": "user"
},
{
... | 331 |
[
{
"content": "Xiao Ming is 9 years old this year. His father's age is 5 times that of Xiao Ling. This year, his father is $\\boxed{\\text{answer}}$ years old. A pen costs 2 yuan, and buying 60 such pens costs $\\boxed{\\text{answer}}$ yuan.",
"role": "user"
},
{
"content": "To solve the problem, we'... | 332 |
[
{
"content": "In a certain month, there are 5 Fridays, 5 Saturdays, and 5 Sundays. On which day of the week does the first day of that month fall?",
"role": "user"
},
{
"content": "To solve the problem of determining the day of the week on which the first day of the month falls, given that the month... | 333 |
[
{
"content": "Given $x>0$, $y>0$, and $4x+y=1$, find the minimum value of $\\frac{1}{x} + \\frac{4}{y}$.",
"role": "user"
},
{
"content": "To solve this optimization problem, we need to minimize the function \\(\\frac{1}{x} + \\frac{4}{y}\\) subject to the constraint \\(4x + y = 1\\), with the condi... | 334 |
[
{
"content": "Given that the terminal side of angle $\\alpha$ passes through point P(1, 2), then $\\tan\\alpha=$ \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "To find \\(\\tan \\alpha\\) given that the terminal side of angle \\(\\alpha\\) passes through point \\(P(1, 2)\\), we can use the de... | 335 |
[
{
"content": "Petya has 25 coins, each with a denomination of 1, 2, 5, or 10 rubles. Among these coins, 19 are not two-ruble coins, 20 are not ten-ruble coins, and 16 are not one-ruble coins. How many five-ruble coins does Petya have?",
"role": "user"
},
{
"content": "To determine the number of five... | 336 |
[
{
"content": "When $a > 0$, $b > 0$ and $a + b = 2$, the maximum value of the determinant \n$$\\begin{vmatrix} \na & 1 \\\\ \n1 & b \n\\end{vmatrix}$$ \nis __________.",
"role": "user"
},
{
"content": "To find the maximum value of the determinant given the conditions \\(a > 0\\), \\(b > 0\\), and \\... | 337 |
[
{
"content": "Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$.\n",
"role": "user"
},
{
"content": "We have \\(a_n = 6^n + 8^n\\). We need to determine the remainder when \\(a_{83} \\equiv 6^{83} + 8^{83} \\pmod{49}\\).\n\nTo simplify this, we'll use properties of modul... | 338 |
[
{
"content": "Given a rectangle with dimensions \\(100 \\times 101\\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal.",
"role": "user"
},
{
"content": "To determine the number of segments into which the grid lines divide the diagon... | 339 |
[
{
"content": "Given the function $f(x)= \\begin{cases} 1,x\\in \\mathbb{Q} \\\\ 0,x\\in \\mathbb{R}\\setminus\\mathbb{Q} \\end{cases}$, find $f(f(2\\pi))$ = \\_\\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "To solve the problem, we need to understand and analyze the function \\( f(x) \\) defined... | 340 |
[
{
"content": "Math teacher Lei listed the following materials on multimedia: Calculate: $-5\\frac{5}{6}+(-9\\frac{2}{3})+17\\frac{3}{4}+(-3\\frac{1}{2})$. Solution: The original expression $=[(-5)+(-\\frac{5}{6})]+[(-9)+(-\\frac{2}{3})]+[17+\\frac{3}{4}]+[(-3)+(-\\frac{1}{2})]$; $=[(-5)+(-9)+(-3)+17]+[(-\\frac{... | 341 |
[
{
"content": "Assume that all roots of the equation \\( a^{x} + 2x - 4 = 0 \\) (where \\( a > 0 \\) and \\( a \\neq 1 \\)) are \\( u_{1}, u_{2}, \\cdots, u_{k} \\), where \\( k \\) is a positive integer. All roots of the equation \\( \\log_{a}(2x) + x - 2 = 0 \\) (where \\( a > 0 \\) and \\( a \\neq 1 \\)) are ... | 342 |
[
{
"content": "Given sets A={1, m-2} and B={2, 3}, and their intersection A∩B={2}, find the value of the real number m.",
"role": "user"
},
{
"content": "To solve the problem, we need to find the value of \\( m \\) such that the intersection of sets \\( A \\) and \\( B \\) is \\(\\{2\\}\\).\n\nGiven:... | 343 |
[
{
"content": "What is the remainder when $2^{19}$ is divided by $7$?",
"role": "user"
},
{
"content": "To solve for the remainder when \\( 2^{19} \\) is divided by 7, we can use modular arithmetic. One effective way to solve this problem is by repeatedly squaring and reducing modulo 7.\n\nWe can app... | 344 |
[
{
"content": "Adam has just bought some new books so he has decided to clean up his bookcase. After he adds his new books to his bookshelf he realizes that he has bought enough books to finish completely filling his bookcase with 2 books left over. Before his shopping trip, Adam had 56 books in his bookcase. If... | 345 |
[
{
"content": "$\\log _{2}\\sin 15^{\\circ}-\\log_{\\frac{1}{2}}\\cos 345^{\\circ}=$____.",
"role": "user"
},
{
"content": "To solve the problem, we need to evaluate the expression \\(\\log_{2} \\sin 15^\\circ - \\log_{\\frac{1}{2}} \\cos 345^\\circ\\). \n\nLet's break it down:\n\n1. **Evaluate the a... | 346 |
[
{
"content": "A piece of cheese is located at $(12,10)$ in a coordinate plane. A mouse is at $(4,-2)$ and is running up the line $y=-5x+18$. At the point $(a,b)$ the mouse starts getting farther from the cheese rather than closer to it. What is $a + b$?",
"role": "user"
},
{
"content": "To find ... | 347 |
[
{
"content": "There are 9 bicycles and 16 cars in the garage. How many wheels are there in the garage?",
"role": "user"
},
{
"content": "To solve this problem, we need to consider the number of wheels for both bicycles and cars.\n\n1. **Bicycles**: Each bicycle has 2 wheels.\n2. **Cars**: Each car h... | 348 |
[
{
"content": "Compute\n\\[\\sum_{k = 1}^\\infty \\frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})}.\\]",
"role": "user"
},
{
"content": "To solve the infinite series\n\n\\[\n\\sum_{k = 1}^\\infty \\frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})},\n\\]\n\nwe can start by analyzing the general term and loo... | 349 |
[
{
"content": "Compute the value of $2\\lg \\sqrt{2}+\\lg 5=$ \\_\\_\\_\\_\\_.",
"role": "user"
},
{
"content": "To solve the problem \\(2\\lg \\sqrt{2} + \\lg 5\\) using Python and SymPy, we need to follow these steps:\n\n1. Express the logarithmic terms correctly using SymPy's logarithmic functions... | 350 |
[
{
"content": "In the larger of two concentric circles, a chord of 32 units is drawn that is tangent to the smaller circle. Find the radius of each of the circles if the width of the resulting ring is 8 units.",
"role": "user"
},
{
"content": "Sure, let's solve the problem step-by-step using Python a... | 351 |
[
{
"content": "Alice rolls two octahedral dice with the numbers 2, 3, 4, 5, 6, 7, 8, and 9. What is the probability that the sum of the numbers on the two dice is 11?",
"role": "user"
},
{
"content": "To solve this problem, we need to calculate the probability that the sum of the numbers on two octah... | 352 |
[
{
"content": "A certain residential area needs to lay a sewage pipeline with a total length of 720 meters in order to discharge sewage. To reduce the impact of construction on residents' lives, the construction time needs to be shortened. During the actual construction, the length of pipeline laid each day is 1... | 353 |
[
{
"content": "There are six letter cards, $A$, $B$, $C$, $D$, $E$, and $F$, arranged in a row. Card $A$ is required to be at the left end and card $F$ at the right end. How many different arrangements are possible?",
"role": "user"
},
{
"content": "To determine the number of different arrangements o... | 354 |
[
{
"content": "In the coordinate plane, there are two regions, \\( M \\) and \\( N \\). \\( M \\) is determined by the inequalities \\( y \\geqslant 0 \\), \\( y \\leqslant x \\), and \\( y \\leqslant 2 - x \\). \\( N \\) is a region varying with \\( t \\), determined by the inequality \\( t \\leqslant x \\leqsl... | 355 |
[
{
"content": "Along the shore of a circular lake, there are apple trees. Petya and Vasya start walking from point $A$ on the shore in opposite directions along the shore, counting all the apple trees they encounter and all the apples growing on the trees. When they meet at some point $B$, they compare their res... | 356 |
[
{
"content": "A two-player game is played on a chessboard with $ m$ columns and $ n$ rows. Each player has only one piece. At the beginning of the game, the piece of the first player is on the upper left corner, and the piece of the second player is on the lower right corner. If two squares have a common ed... | 357 |
[
{
"content": "In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have????",
"role": "user"
},
{
"content": "To solve this problem, we can use a combination of combinatorial reasoning ... | 358 |
[
{
"content": "How many integers are between $(10.2)^3$ and $(10.3)^3$?",
"role": "user"
},
{
"content": "Let's break down the problem into its steps carefully:\n\n1. Calculate the value of \\( (10.2)^3 \\).\n2. Calculate the value of \\( (10.3)^3 \\).\n3. Find the integers between these two values.\... | 359 |
[
{
"content": "Given the parabola \\( y = ax^2 \\) with \\( a > 0 \\), a line through the focus \\( F \\) intersects the parabola at points \\( P \\) and \\( Q \\). If the lengths of segments \\( PF \\) and \\( FQ \\) are \\( p \\) and \\( q \\) respectively, then \\( \\frac{1}{p} + \\frac{1}{q} \\) is equal to ... | 360 |
[
{
"content": "Let $f(x)$ be a function defined on the set of integers. Given that $f(1) = 1$, $f(2) = 0$, $f(-1) \\lt 0$, and for any integers $x$ and $y$, $f(x+y) = f(x)f(1-y) + f(1-x)f(y)$. Find $f(55)$.",
"role": "user"
},
{
"content": "To solve for \\( f(55) \\) given the function \\( f(x) \\) d... | 361 |
[
{
"content": "Simplify \\( \\frac{\\sqrt{6} + 4\\sqrt{3} + 3\\sqrt{2}}{(\\sqrt{6} + \\sqrt{3})(\\sqrt{3} + \\sqrt{2})} \\). The result is:\n(A) \\( \\frac{2}{3} \\).\n(B) \\( \\frac{2\\sqrt{3}}{3} \\).\n(C) \\( \\sqrt{6} + \\sqrt{2} \\).\n(D) \\( \\sqrt{6} - \\sqrt{2} \\).",
"role": "user"
},
{
"con... | 362 |
[
{
"content": "Given that the sum of the first $m$ terms of a geometric sequence $a_n$ is $S_m = 10$, and $S_{2m} = 30$, find $S_{3m}$.",
"role": "user"
},
{
"content": "To solve this problem, let's break it down step-by-step and use Sympy for calculations:\n\n1. **Understand the Geometric Sequence S... | 363 |
[
{
"content": "Simplify the following expressions: \n(1) $$\\sqrt {6 \\frac {1}{4}}+ \\sqrt {3 \\frac {3}{8}}+ \\sqrt {0.0625}+(\\sqrt {\\pi})^{0}-2^{-1}$$ \n(2) $$\\lg14-2\\lg \\frac {7}{3}+\\lg7-\\lg18$$.",
"role": "user"
},
{
"content": "To simplify the given expressions, we'll use the Python li... | 364 |
[
{
"content": "A natural number is divided by 2 with a remainder of 1, divided by 3 with a remainder of 2, and it is also a multiple of 7. There are how many such natural numbers within 100?",
"role": "user"
},
{
"content": "To solve this problem, we need to find natural numbers that satisfy the foll... | 365 |
[
{
"content": "Five people, numbered $1$, $2$, $3$, $4$, $5$, are to sit in five seats, also numbered $1$, $2$, $3$, $4$, $5$. There is exactly one arrangement where only two people sit in seats that match their numbers. Determine the number of such arrangements. (Answer with a number only)",
"role": "user"
... | 366 |
[
{
"content": "Ken created a care package to send to his brother, who was away at boarding school. Ken placed a box on a scale, and then he poured into the box enough jelly beans to bring the weight to 2 pounds. Then, he added enough brownies to cause the weight to triple. Next, he added another 2 pounds of j... | 367 |
[
{
"content": "If vector $a = (x-1, 2)$ and vector $b = (4, y)$ are perpendicular, then the minimum value of $9^x + 3^y$ is $\\boxed{6}$.",
"role": "user"
},
{
"content": "To solve this problem, we need to follow these steps:\n\n1. **Understand the Condition for Perpendicular Vectors**: Two vectors \... | 368 |
[
{
"content": "A subset \\( X \\) of the set \\(\\{00, 01, \\ldots, 98, 99\\}\\) satisfies the property that in any infinite sequence of digits, there are two adjacent digits that form an element of \\( X \\). What is the minimum number of elements that \\( X \\) should contain?",
"role": "user"
},
{
... | 369 |
[
{
"content": "Among the following statements:\n① After adding or subtracting the same constant from each data point in a dataset, the variance remains unchanged;\n② Given a regression equation $\\hat y = 3-5x$, when the variable $x$ increases by one unit, $y$ increases by an average of 5 units;\n③ A linear regr... | 370 |
[
{
"content": "The result of the calculation \\(2014 \\times\\left(\\frac{1}{19}-\\frac{1}{53}\\right)\\) is ( )\nA. 34\nB. 68\nC. 144\nD. 72",
"role": "user"
},
{
"content": "To solve the problem \\(2014 \\times\\left(\\frac{1}{19}-\\frac{1}{53}\\right)\\), we can follow these steps:\n\n1. Compute t... | 371 |
[
{
"content": "$23-|-6|-\\left(+23\\right)=\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
{
"content": "To solve the given mathematical expression \\(23 - |-6| - \\left(+23\\right)\\), we can break it down into smaller components and evaluate each part step-by-step.\n\n1. Calculate the absolute value of \\... | 372 |
[
{
"content": "If \\(x=4\\) and \\(x+y=0\\), what is the value of \\(y\\)? \n(A) 0 \n(B) -2 \n(C) -3 \n(D) -1 \n(E) -4",
"role": "user"
},
{
"content": "To find the value of \\(y\\), we start with the given equations:\n\n1. \\(x = 4\\)\n2. \\(x + y = 0\\)\n\nWe substitute \\(x = 4\\) into the second ... | 373 |
[
{
"content": "Inside the parallelogram \\(ABCD\\), a point \\(E\\) is chosen such that \\(AE = DE\\) and \\(\\angle ABE = 90^\\circ\\). Point \\(M\\) is the midpoint of segment \\(BC\\). Find the angle \\(DME\\).",
"role": "user"
},
{
"content": "To solve this problem, we need to perform the followi... | 374 |
[
{
"content": "The digits in a two-digit positive integer are reversed. The new two-digit integer minus the original integer equals 54. What is the positive difference between the two digits of the original integer?\n\n(A) 5\n(B) 7\n(C) 6\n(D) 8\n(E) 9",
"role": "user"
},
{
"content": "Let's represen... | 375 |
[
{
"content": "There is a sequence of one thousand random digits. A zero and a decimal point are placed in front of this sequence. As a result, a random number \\( X \\) is formed, written as a decimal fraction: \\( X = 0.x_1 x_2 x_3 \\ldots x_{1000} \\). Find \\( \\mathrm{E}X \\) (the expected value of \\( X \\... | 376 |
[
{
"content": "Given a complex number $z$ satisfies $\\overline{z}i=3+4i$ (where $i$ is the imaginary unit), then $|z|=$____",
"role": "user"
},
{
"content": "To solve this problem, we need to find the magnitude (or modulus) of the complex number \\( z \\) that satisfies \\( \\overline{z} \\cdot i = ... | 377 |
[
{
"content": "If six cards labeled 1, 2, 3, 4, 5, and 6 are to be placed into three different envelopes with each envelope containing two cards, and the cards labeled 1 and 2 must be placed in the same envelope, how many different placement methods are there?",
"role": "user"
},
{
"content": "To sol... | 378 |
[
{
"content": "In the ancient Chinese mathematical treatise \"Nine Chapters on the Mathematical Art,\" it is recorded: \"There is a field in the shape of a sector, with a circumference of 30 steps and a diameter of 16 steps. What is the area of the field?\" Note: \"宛田\" refers to a field in the shape of a sector... | 379 |
[
{
"content": "Let the function\n$$\nf(x, y) = \\sqrt{x^{2}+y^{2}-6y+9} + \\sqrt{x^{2}+y^{2}+2\\sqrt{3}x+3} + \\sqrt{x^{2}+y^{2}-2\\sqrt{3}x+3}\n$$\n\nFind the minimum value of \\( f(x, y) \\).",
"role": "user"
},
{
"content": "To find the minimum value of \\( f(x, y) \\), we can start by analyzing t... | 380 |
[
{
"content": "The first tourist travels for 1.5 hours on a bicycle at a speed of 16 km/h, then takes a break for 1.5 hours, and then continues at the initial speed. Four hours after the first tourist starts, a second tourist starts chasing the first tourist on a motorcycle at a speed of 56 km/h. What distance w... | 381 |
[
{
"content": "Find the smallest positive integer $n$ such that\n\\[\\begin{pmatrix} \\cos 170^\\circ & -\\sin 170^\\circ \\\\ \\sin 170^\\circ & \\cos 170^\\circ \\end{pmatrix}^n = \\mathbf{I}.\\]",
"role": "user"
},
{
"content": "To solve the problem of finding the smallest positive integer \\( n \... | 382 |
[
{
"content": "A fault occurred in the transmission line from the substation to the Wuyang Hospital, which cannot be detected from the outside. It requires sending people to climb to the top of the utility poles for inspection. Each inspection can determine whether the fault is at the beginning, middle, or end o... | 383 |
[
{
"content": "In the example where a common measure fits into one segment 15 times and into another segment 12 times, can a larger segment, which will also be a common measure, be found?",
"role": "user"
},
{
"content": "To solve this problem, we need to find the greatest common divisor (GCD) of the... | 384 |
[
{
"content": "If \\( x = \\frac{a}{b} \\), where \\( a \\neq b \\) and \\( b \\neq 0 \\), then \\( \\frac{a + b}{a - b} \\) equals:\n(A) \\(\\frac{x}{x + 1}\\).\n(B) \\(\\frac{x + 1}{x - 1}\\).\n(C) 1.\n(D) \\( x - \\frac{1}{x} \\).\n(E) \\( x + \\frac{1}{x} \\).\n",
"role": "user"
},
{
"content": "... | 385 |
[
{
"content": "At the feline sanctuary, there were 12 lions, 14 tigers, and several cougars. If there were half as many cougars as lions and tigers combined, then what was the total number of big cats at the feline sanctuary?",
"role": "user"
},
{
"content": "Let's start by translating the problem s... | 386 |
[
{
"content": "If the curve $y= \\frac {1}{2e}x^2$ and the curve $y=alnx$ have a common tangent at their common point $P(s, t)$, then the real number $a=$ ______.",
"role": "user"
},
{
"content": "To solve for the real number \\( a \\) such that the curve \\( y = \\frac{1}{2e}x^2 \\) and the curve \\... | 387 |
[
{
"content": "A parking lot has a row of $8$ parking spaces. There are $4$ different car models that need to be parked, with the requirement that the remaining $4$ empty parking spaces be connected. The number of different parking methods is:",
"role": "user"
},
{
"content": "To solve this problem, ... | 388 |
[
{
"content": "In an equilateral triangle, mark each point whose distance from the nearest vertex is less than the distance from the centroid. What percentage of the area of the equilateral triangle do the points with this property occupy?",
"role": "user"
},
{
"content": "To solve this problem, let'... | 389 |
[
{
"content": "Jessie won 3 times as many athletic awards as his buddy Scott, who won 4 awards. The best athlete at the rival high school won twice as many awards as Jessie. How many awards did the rival win?",
"role": "user"
},
{
"content": "Firstly, let's break down the problem step by step:\n\n1... | 390 |
[
{
"content": "Given that the square root of $a+1$ is $\\pm 2$, and the cube root of $2a+b-2$ is $2$, find the arithmetic square root of $a^{2}+b^{2}$.",
"role": "user"
},
{
"content": "Let's solve the given problem step-by-step using Python and SymPy.\n\n**Problem Breakdown:**\n\n1. We are given tha... | 391 |
[
{
"content": "The maximum value of the function $f(x) = \\frac{-x^{2} + x - 4}{x}$ (where $x > 0$) is _______, and this value occurs when $x$ is equal to _______.",
"role": "user"
},
{
"content": "To solve this problem, we need to find the maximum value of the function \\( f(x) = \\frac{-x^{2} + x -... | 392 |
[
{
"content": "A regular coin is flipped 4 times. What is the probability of getting heads exactly two times?",
"role": "user"
},
{
"content": "To solve this problem, we need to determine the probability of getting exactly two heads when flipping a fair coin four times. This is a classic problem of c... | 393 |
[
{
"content": "Define an even function $y = f(x)$ on $\\mathbb{R}$. When $x > 0$, $y = f(x)$ is monotonically increasing, and $f(1) \\cdot f(2) < 0$. Then, the number of intersection points between the graph of the function $y = f(x)$ and the $x$-axis is ______.",
"role": "user"
},
{
"content": "To s... | 394 |
[
{
"content": "The outdoor scouts went on a hike to see a waterfall. To get to the hike, the club members took 3 cars, 6 taxis and 2 vans. There were 4 people in each car, 6 people in each taxis and 5 people in each van. How many people went on the hike?",
"role": "user"
},
{
"content": "To determine... | 395 |
[
{
"content": "If $C_{15}^{n+1} = C_{15}^{2n-1}$, find the value(s) of $n$.",
"role": "user"
},
{
"content": "To solve the given problem, we'll use the properties of binomial coefficients. The problem translates to solving the equation:\n\n\\[ \\binom{15}{n+1} = \\binom{15}{2n-1} \\]\n\n### Steps to ... | 396 |
[
{
"content": "Calculate the length of the arc of the curve given by the polar equation\n\n$$\n\\varrho = 6 \\sin \\varphi, \\quad 0 \\leq \\varphi \\leq \\pi / 3\n$$",
"role": "user"
},
{
"content": "To calculate the length of the arc for a curve given by a polar equation \\( \\varrho = f(\\varphi) ... | 397 |
[
{
"content": "Let $x$ be the least real number greater than $1$ such that $\\sin(x)= \\sin(x^2)$, where the arguments are in degrees. What is $x$ rounded up to the closest integer?\n$\\textbf{(A) } 10 \\qquad \\textbf{(B) } 13 \\qquad \\textbf{(C) } 14 \\qquad \\textbf{(D) } 19 \\qquad \\textbf{(E) } 20$",
... | 398 |
[
{
"content": "Divide 20 numbers into two groups of equal size. The first group has a mean of 50 and a variance of 33, while the second group has a mean of 40 and a variance of 45. Find the standard deviation of the entire set of numbers.",
"role": "user"
},
{
"content": "To solve this problem, we'll... | 399 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.