messages listlengths 3 3 | conversation_id int64 0 50k |
|---|---|
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Define a function $f(x)$ on $\\mathbb{R}$ satisfying $f(x+2) + f(x) = 0$, and the function $f(x+1)$ is an odd function. Consider the following statements:\n\n① The smallest positive period of the function $f(x)$ is 4; \n... | 712 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "5. Tang僧 and his four disciples arrived at the territory of the Flaming Mountain, where the weather was scorching, and they were all thirsty. Wukong and Bajie went to a mountain village 10 li away to find watermelons. The... | 24,774 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "Let $f(x)=\\sqrt{3}\\cos 2x+2\\sin x\\cos x$. The following conclusions are given:① The smallest positive period of $f(x)$ is $\\pi$;② The graph of $y=f(x)$ is symmetric about the line $x=\\frac{π}{{12}}$;③ $f(x)$ is mono... | 45,142 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "Ha az előbbi feladatban azt is tudni akarjuk, hogy a hamis pénz könnyebb-e, vagy nehezebb a többinél, hány pénzdarab a legtöbb, amennyi közül még három mérés elegendő ennek a kérdésnek az eldöntéséhez?\n\nIf in the previo... | 27,119 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "11、Given the ellipse $C: \\frac{x^{2}}{3}+y^{2}=1$ with its upper vertex $A$, a line $l$ that does not pass through $A$ intersects the ellipse $C$ at points $P$ and $Q$, and $A P \\perp A Q$. Find the maximum value of the... | 21,139 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Let's determine $m$ such that the expression\n\n$$\n(4-m) x^{2}-3 x+4+m\n$$\n\nis positive for all values of $x$.\n\n---\n\nDetermine $m$ so that the expression\n\n$$\n(4-m) x^{2}-3 x+4+m\n$$\n\nis positive for all values... | 19,129 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "The 3rd problem of the mathematical class competition:\n\nFind the six-digit square numbers such that, when the first three digits are erased, the square root of the number is obtained. Let's think step by step and output... | 6,433 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "\n1. Find all real numbers $p$ such that the system\n\n$$\n\\begin{aligned}\n& x^{2}+(p-1) x+p \\leqslant 0 \\\\\n& x^{2}-(p-1) x+p \\leqslant 0\n\\end{aligned}\n$$\n\nof inequalities has at least one solution $x \\in \\m... | 38,854 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f(x) = |x-1| + |x-2|$, \n(1) Solve the inequality $f(x+2) \\geq 2$. \n(2) If the solution set of the inequality $f(x) < a$ with respect to $x$ is empty, find the range of values for $a$. Let's think ... | 22,489 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "XXX OM - I - Task 8\n\nAmong the cones inscribed in the sphere $ B $, the cone $ S_1 $ was chosen such that the sphere $ K_1 $ inscribed in the cone $ S_1 $ has the maximum volume. Then, a cone $ S_2 $ of maximum volume w... | 31,104 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "Example 9. Simplify $1-\\frac{1}{4} \\sin ^{2} 2 \\alpha-\\sin ^{2} \\beta-\\cos ^{4} \\alpha$ into a product form of trigonometric functions. (84 College Entrance Examination for Liberal Arts)\n\n(84年高考文科试题) \n\nNote: Th... | 21,349 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "G3.2 Let $p$ be a prime and $m$ be an integer. If $p(p+m)+2 p=(m+2)^{3}$, find the greatest possible value of $m$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
... | 29,116 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that $M(a,3)$ and $N(4,b)$ are symmetric with respect to the $y$-axis, the value of $a+b$ is ______. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content... | 10,772 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "3. Given a right-angled triangle with the two legs $a, b$ $(a \\leqslant b)$ and the hypotenuse $c$ all being integers, and the radius of its inscribed circle $r=3$. Then the number of such right-angled triangles is ( ).\... | 18,332 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "7. The real solution $(x, y, z)=$ $\\qquad$ of the equation $2 \\sqrt{x-4}+3 \\sqrt{y-9}+4 \\sqrt{z-16}=\\frac{1}{2}(x+y+z)$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
}... | 1,003 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "1. Around a round table, 50 schoolchildren are sitting: blondes, brunettes, and redheads. It is known that in any group of schoolchildren sitting in a row, between any two blondes there is at least one brunette, and betwe... | 22,156 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given functions $f(x)=x$ and $g(x)=\\ln x$. \n$(1)$ Let $h(x)=af(x)-g(x)$, discuss the extremum of $h(x)$. \n$(2)$ If $x>0$, $af(x)-g(x+1)>0$ always holds, find the range of positive real number $a$. Let's think step by... | 26,453 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "$17 \\cdot 60$ In an isosceles triangle, the angle bisector of one of the base angles divides the perimeter into two parts, 63 and 35. Then the length of the leg must be\n(A) 26.4.\n(B) 33.\n(C) 38.5.\n(D) None of the abo... | 17,039 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the set $R_{n}=\\{X|X=(x_{1},x_{2},\\ldots,x_{n})\\}$, where $x_{i}\\in\\{0,1\\}$, $i=1, 2, \\ldots, n$ $(n\\geqslant 2)$. For $A=(a_{1},a_{2},\\ldots,a_{n})\\in R_{n}$ and $B=(b_{1},b_{2},\\ldots,b_{n})\\in R_{n}$,... | 7,166 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given vectors $\\overrightarrow {a} = (4, -2, -4)$ and $\\overrightarrow {b} = (6, -3, 2)$, find the magnitude of $\\overrightarrow {a}$, the magnitude of $\\overrightarrow {b}$, and the dot product of $(2\\overrightarrow... | 48,310 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "How many distinct four letter arrangements can be formed by rearranging the letters found in the word [b]FLUFFY[/b]? For example, FLYF and ULFY are two possible arrangements. Let's think step by step and output the final ... | 12,187 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "13. (15 points) From the sequence of consecutive natural numbers $1,2,3, \\cdots, 2014$, select $n$ numbers such that these $n$ numbers satisfy: taking any two of them, one number will not be 7 times the other. Find the m... | 20,284 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The graph of the linear function $y=-x-2$ does not pass through which quadrant?\n\nA: First quadrant\n\nB: Second quadrant\n\nC: Third quadrant\n\nD: Fourth quadrant Let's think step by step and output the final answer wi... | 1,052 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "25) Which of the following numbers does NOT divide 100! (recall that 100! $=100 \\cdot 99 \\cdot$ $98 \\cdots 3 \\cdot 2 \\cdot 1) ?$\n(A) 1968\n(B) 1988\n(C) 1998\n(D) 2008\n(E) 2048 Let's think step by step and output t... | 13,752 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Arrange the numbers 1, 2, 3, 4, 5, 6 in a row, denoting the $i$-th number as $a_i$ ($i=1,2,\\ldots,6$). If $a_1 \\neq 1$, $a_3 \\neq 3$, $a_5 \\neq 5$, and $a_1 < a_3 < a_5$, then the number of different arrangements is _... | 37,139 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Which of the following statements is correct?\n\nA: In the formula for the area of a circle $S=\\pi r^{2}$, the constants are $2$, and the variables are $S$, $\\pi$, $r$.\n\nB: In the formula for uniform motion $S=vt$, th... | 29,727 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "A company has hired 4 outstanding university graduates and has 5 departments. The human resources department is tasked with assigning these graduates to work in any 3 of the departments, with at least one person in each d... | 41,854 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In triangle $\\triangle ABC$, $AB=AC$, $\\angle B=60^{\\circ}$, and $BC=4$. Find the perimeter of $\\triangle ABC$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
... | 40,333 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "13. Test (from 8th grade. 3 points) Vasya and Misha were taking a test, which had 20 questions of the same difficulty, and for each question, answer options were provided. Since neither Vasya nor Misha knew anything, they... | 33,248 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the terms of the geometric sequence {an} are all positive, a6+a5=4, a4+a3-a2-a1=1, find the value of a1. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
... | 9,680 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "Illustrate a semicircle with diameter $A G$. The arc of the semicircle is divided into six equal parts by points $B$, $C$, $D$, $E$, and $F$. $D F$ and $C G$ are both straight line segments. Given that the area of the sem... | 16,805 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "6. Given that the function $f(x)$ is defined and increasing on $(0,+\\infty)$, and satisfies $f(x) \\cdot f\\left(f(x)+\\frac{1}{x}\\right)=1$. Then $f(1)=$ ).\n(A) 1\n(B) 0\n(C) $\\frac{1+\\sqrt{5}}{2}$\n(D) $\\frac{1-\\... | 14,652 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5. In the border desert area, patrol vehicles travel 200 kilometers per day, and each vehicle can carry enough gasoline to travel for 14 days. There are 5 patrol vehicles that set out from base $A$ simultaneously, complet... | 15,793 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In Squareland, only squares live. Except for two exceptions, each of them has two friends, one of whom has a perimeter that is $8 \\, \\text{cm}$ smaller and the other $8 \\, \\text{cm}$ larger. The average side length of... | 30,210 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "## Problem Statement\n\nCalculate the lengths of the arcs of the curves given by the equations in polar coordinates.\n\n$$\n\\rho=5(1-\\cos \\varphi),-\\frac{\\pi}{3} \\leq \\varphi \\leq 0\n$$ Let's think step by step an... | 47,426 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In the table, the numbers in each row form an arithmetic sequence when read from left to right. Similarly, the numbers in each column form an arithmetic sequence when read from top to bottom. What is the sum of the digits... | 20,875 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Simplify the expression $S_n = n + (n-1) \\times 2 + (n-2) \\times 2^2 + \\ldots + 2 \\times 2^{n-2} + 2^{n-1}$ to get ( )\n\nA: $2^{n+1} + n - 2$\n\nB: $2^{n+1} - n + 2$\n\nC: $2^n - n - 2$\n\nD: $2^{n+1} - n - 2$ Let's ... | 4,161 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Reading comprehension: If $x$ satisfies $\\left(210-x\\right)\\left(x-200\\right)=-204$, find the value of $\\left(210-x\\right)^{2}+\\left(x-200\\right)^{2}$. Solution: Let $210-x=a$, $x-200=b$, then $ab=-204$, and $a+b=... | 40,340 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "239. Twenty Questions. I want to offer you a variation of the well-known game of \"twenty questions,\" which works as follows. I think of a natural number, and you have to guess it by asking no more than twenty questions,... | 28,286 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "In a coordinate plane, given that the coordinates of point $A$ are $(2,-3)$, line segment $AB$ is parallel to the $x$-axis, and $AB=4$, then the coordinates of point $B$ are ( ).\n\nA: $(-2,3)$\n\nB: $(6,3)$\n\nC: $(-2,-3... | 13,256 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5. As shown in Figure 1, given the hyperbola $\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1$ $(a>0, b>0), A A^{\\prime}$ is a chord passing through the right focus $F$ and perpendicular to the real axis $x$, $M$ is the righ... | 48,934 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$.\n. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"cont... | 11,935 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "3. Determine the largest positive integer $N$, such that there exists a $6 \\times N$ table $T$ satisfying:\n(1) Each column is a permutation of $1,2, \\cdots, 6$;\n(2) For any two columns $i \\neq j$, there exists a row ... | 48,447 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The slope angle of the line $\\sqrt {3}x - y + a = 0$ (where $a$ is a constant) is ( )\nA: 30°\nB: 60°\nC: 150°\nD: 120° Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
... | 15,492 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Let the functions $f(x)$ and $g(x)$ have the domain of $\\mathbb{R}$, and $f(x)$ is an odd function, $g(x)$ is an even function. Among the following conclusions, the correct one is $\\boxed{\\text{B}}$.\n\nA: $f(x)g(x)$ i... | 32,513 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "If line $l_1: 2x-ay-1=0$ is perpendicular to line $l_2: x+2y=0$, then $a= \\boxed{1}$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To determine the... | 36,099 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "7. Given the function $f(x)=x^{3}-2 x^{2}-3 x+4$, if $f(a)=f(b)=f(c)$, where $a<b<c$, then $a^{2}+b^{2}+c^{2}=$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"co... | 39,260 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the set $U=\\{x|x^2-2x<3\\}$, and $P=\\{x|-1<x\\leq2\\}$, then $\\complement_U P=$ ? Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "Given the se... | 46,517 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f(x)=ax^2+bx+c (a\\neq 0)$, it is known that $f(0)=2$ and $f(x+1)-f(x)=2x-1$.\n\n(1) Find the analytical expression of the function $f(x)$;\n\n(2) The function $g(x)=2^{f(x)+\\lambda x}$ is monotonic i... | 33,095 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that are not collinear, and $t$ is a real number.\n(I) If $\\overrightarrow{OA} = \\overrightarrow{a}$, $\\overrightarrow{OB} = t\\overrightarrow{b}$, and $\\o... | 3 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "2. Let $A_{n}$ denote the product of multiples of $n$ not exceeding 100, where $n$ is a positive integer. For example, $A_{3}=3 \\times 6 \\times 9 \\times \\cdots \\times 99$. Then the greatest common divisor of $A_{2}, ... | 21,683 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "12. A person bought a total of 12 packs of black and white sunflower seeds, spending 9 yuan and 9 jiao. Each pack of white sunflower seeds is 3 jiao more expensive than each pack of black sunflower seeds, and the number o... | 13,446 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "8. Each permutation of $1,2, \\cdots, 8$ is considered as an eight-digit number. Then the number of eight-digit numbers that are multiples of 11 is $\\qquad$. Let's think step by step and output the final answer within \\... | 49,515 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "A certain school initiated a charity book donation activity called \"A Dream of Reading for Children in Poor Areas\". Within a short week, a large number of books were donated by students. A random survey was conducted on... | 27,943 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "28. Let $a$ be an integer, and $n, r$ be integers greater than 1. $p$ is an odd prime, and $(n, p-1)=1$. Find the number of solutions to the following congruence equation:\n$$x_{1}^{n}+x_{2}^{n}+\\cdots+x_{r}^{n} \\equiv ... | 29,059 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "6. Given that $P$ is a point on the hyperbola $C: \\frac{x^{2}}{4}-\\frac{y^{2}}{12}=1$, $F_{1}, F_{2}$ are the left and right foci of the hyperbola $C$, $M, I$ are the centroid and incenter of $\\triangle P F_{1} F_{2}$,... | 32,738 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Let $U=\\mathbb{R}$, and define the sets $A=\\{x|x^2+3x+2=0\\}$, $B=\\{x|x^2+(m+1)x+m=0\\}$. If $B \\subseteq A$, find the value of $m$. Let's think step by step and output the final answer within \\boxed{{}}",
"role"... | 445 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that all terms of the sequence $\\\\{a\\_n\\\\}$ are positive numbers, and for any $n∈N^{*}$, $4S\\_n=a\\_n^2+2a\\_n$ holds, where $S\\_n$ is the sum of the first $n$ terms of the sequence $\\\\{a\\_n\\\\}$, find th... | 32,216 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The strict decreasing interval of the function $y={e^{{x^2}-2x-3}}$ is ____. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To determine the strict de... | 18,250 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "For the polynomial $f$, it holds that $f\\left(x^{2}+1\\right)-f\\left(x^{2}-1\\right)=4 x^{2}+6$. Determine the polynomial $f\\left(x^{2}+1\\right)-f\\left(x^{2}\\right)$. Let's think step by step and output the final an... | 1,684 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "11.2. On the board, the expression $\\frac{a}{b} \\cdot \\frac{c}{d} \\cdot \\frac{e}{f}$ is written, where $a, b, c, d, e, f$ are natural numbers. If the number $a$ is increased by 1, then the value of this expression in... | 2,899 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Calculate: $2\\tan 60^{\\circ}+\\tan 45^{\\circ}-4\\cos 30^{\\circ}=$____. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To calculate \\(2\\tan 60^{\... | 39,428 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Problem 10.3. On the side $AD$ of rectangle $ABCD$, a point $E$ is marked. On the segment $EC$, there is a point $M$ such that $AB = BM, AE = EM$. Find the length of side $BC$, given that $ED = 16, CD = 12$.\n\n![](https:... | 40,758 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "8. For positive integer $a$ and integers $b$, $c$, the quadratic equation $a x^{2}+b x+c=0$ has two roots $\\alpha, \\beta$. And it satisfies $0<\\alpha<\\beta<$ 1. Find the minimum value of $a$. Let's think step by step ... | 29,154 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "13.091. The 300 kg of goods in stock were sold in unequal quantities to two organizations at a price of 37.5 rubles per kg. The first organization transports the purchased goods a distance of 20 km, while the second organ... | 43,958 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "Reading material: After studying square roots, Kang Kang found that some expressions containing square roots can be written as the square of another expression, such as $3+2\\sqrt{2}={({1+\\sqrt{2}})^2}$. With his good th... | 33,814 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
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{
"content": "Example 2.60. Calculate the surface area formed by the rotation of the arc of the circle $x^{2}+(y-a)^{2}=R^{2}$ around the $O Y$ axis over the segment $0<y_{1} \\leqslant y \\leqslant y_{2}<R$. Let's think step by step a... | 34,153 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "10. Given $A_{1}, A_{2}, \\cdots, A_{n}$ are $n$ non-empty subsets of the set $A=\\{1,2,3, \\cdots, 10\\}$, if for any $i, j \\in$ $\\{1,2,3, \\cdots, n\\}$, we have $A_{i} \\cup A_{j} \\neq A$, then the maximum value of ... | 43,609 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Find the sum of the squares of the distances from the vertices of a regular $n$-gon inscribed in a circle of radius $R$ to an arbitrary line passing through the center of the polygon.\n\n# Let's think step by step and out... | 10,762 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Example 5 As shown in Figure 3, let $P\\left(x_{0}, y_{0}\\right)$ be a fixed point, and $x_{0}>0, y_{0}>0, x_{0} y_{0}>$ 1. A moving line through $P$ intersects the curve $C: x y=1(x>0)$ at points $A\\left(x_{1}, y_{1}\\... | 17,898 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "24 Find the value of $\\frac{\\frac{1}{2}-\\frac{1}{3}}{\\frac{1}{3}-\\frac{1}{4}} \\times \\frac{\\frac{1}{4}-\\frac{1}{5}}{\\frac{1}{5}-\\frac{1}{6}} \\times \\frac{\\frac{1}{6}-\\frac{1}{7}}{\\frac{1}{7}-\\frac{1}{8}} ... | 43,653 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Let $a, b, c, d$ be a permutation of the numbers $1, 9, 8,4$ and let $n = (10a + b)^{10c+d}$. Find the probability that $1984!$ is divisible by $n.$ Let's think step by step and output the final answer within \\boxed{{}}"... | 14,233 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "After expanding the domain of the exponential function to complex numbers, there is a formula: $e^{ix}=\\cos x+i\\sin x$, where $i$ is the imaginary unit and $e$ is the base of the natural logarithm. It establishes the re... | 39,520 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f(x) = a - \\frac{2}{2^x + 1}$ ($a \\in \\mathbb{R}$), \n(Ⅰ) Determine the monotonicity of the function $f(x)$ on $\\mathbb{R}$ and prove it using the definition of monotonic functions; \n(Ⅱ) Is ther... | 18,138 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Three, (25 points) A chemical plant, starting from January this year, if it does not improve its production environment and continues to produce as it is, will earn 700,000 yuan per month. At the same time, it will receiv... | 9,212 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The surface area of a sphere with a diameter of 4 is equal to \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To find the surface a... | 6,456 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "4. The sequence $\\left(a_{n}\\right)$ satisfies $a_{1}=2, a_{n+1}=1-\\frac{1}{a_{n}}$, then $a_{2023}=$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content":... | 22,948 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Calculate the value of $$\\sqrt[3]{(2-\\pi )^{3}} + \\sqrt {(3-\\pi)^{2}}$$\n\nOptions:\n\nA: 5 \nB: -1 \nC: 2π-5 \nD: 5-2π Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
... | 7,591 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "8. For positive integer $a$ and integers $b$, $c$, the quadratic equation $a x^{2}+b x+c=0$ has two roots $\\alpha$, $\\beta$. And it satisfies $0<\\alpha<\\beta<$ 1. Find the minimum value of $a$. Let's think step by ste... | 29,996 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "4B. Given $y=x^{2}-2(k-1) x+k^{2}$, where $k$ is a real number, is a set of parabolas (each number $k$ defines one parabola). Determine the geometric place of the points which are the vertices of the given parabolas. Let'... | 10,378 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "7. If the decimal part of $\\log _{c} b$ is zero, and $\\log _{a} \\frac{1}{b}>\\log _{a} \\sqrt{b}>\\log _{b} a^{2}$, then among the following four conclusions:\n(1) $\\frac{1}{b}>\\sqrt{b}>a^{2}$;\n(2) $\\log b+\\log _{... | 27,337 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given a sector with a central angle of $\\frac{π}{9}$ and an arc length of $\\frac{π}{3}$, the area of the sector is ______. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
}... | 19,965 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "G4.1 If $P=2 \\sqrt[4]{2007 \\cdot 2009 \\cdot 2011 \\cdot 2013+10 \\cdot 2010 \\cdot 2010-9}-4000$, find the value of $P$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},... | 24,405 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"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???? Let's think step by step and output the final an... | 45,135 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "19 The sequence of positive integers $\\left\\{a_{n}\\right\\}$ satisfies: for any positive integers $m, n$, if $m \\mid n, m<n$, then $a_{m} \\mid a_{n}$, and $a_{m}<a_{n}$. Find the minimum possible value of $a_{2000}$.... | 11,274 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Example 8 When $s$ and $t$ take all real numbers, then the minimum value that $(s+5-3|\\cos t|)^{2}+(s-2|\\sin t|)^{2}$ can achieve is $\\qquad$\n(1989, National High School Mathematics Competition) Let's think step by st... | 1,941 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the terminal side of acute angle $a$ passes through point $P$ with coordinates $(4\\sin3, -4\\cos3)$, then $a$ equals ( )\n\nA: 3 \nB: $-3$ \nC: $3- \\frac{\\pi}{2}$ \nD: $\\frac{\\pi}{2} - 3$ Let's think s... | 48,940 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the universal set U = R, the complement of A in U is ∁UA = {x | x ≥ 1}, and B = {x | x < -2}, find A ∪ B = ( )\n\nA: {x | x ≥ 1}\nB: {x | -2 < x ≤ 1}\nC: {x | x < 1}\nD: {x | x < -2} Let's think step by step and o... | 36,794 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the lengths of two sides of a triangle are $4$ and $9$, which of the following could be the length of the third side?\n\nA: $13$\n\nB: $6$\n\nC: $5$\n\nD: $4$ Let's think step by step and output the final answe... | 21,994 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the graph of the logarithmic function $f\\left(x\\right)$ passes through the point $({3,\\frac{1}{2}})$. \n$(1)$ Find the analytical expression of $f\\left(x\\right)$; \n$(2)$ The equation $f\\left(x\\right)-... | 49,944 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given vectors $\\overrightarrow{a}=(3, -2)$, $\\overrightarrow{b}=(-2, 1)$, $\\overrightarrow{c}=(7, -4)$, express vector $\\overrightarrow{c}$ in terms of $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step... | 43,442 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, with $a=2$, $\\sin \\frac{B}{2}= \\frac{\\sqrt{5}}{5}$, and the area of $\\triangle ABC$ is $4$.\n(I) Find the value of $\\co... | 45,096 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? \n\n[i](A. Golovanov)[/i] Let's think step by step and outp... | 42,430 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Problem 6. A wooden block was cut into eight smaller blocks with three cuts. On Fig. 5, the surface area of seven blocks is indicated. What is the surface area of the invisible block?\n\n$[8$ points] (A. V. Shapovalov) Le... | 42,255 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Example 4 If $x \\neq 0$, find the maximum value of\n$$\n\\frac{\\sqrt{1+x^{2}+x^{4}}-\\sqrt{1+x^{4}}}{x}\n$$\n\n(1992, National Junior High School Mathematics League) Let's think step by step and output the final answer ... | 5,771 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In the arithmetic sequence $\\left\\{a_{n}\\right\\}$ and the geometric sequence $\\left\\{b_{n}\\right\\}$, $a_{1}=b_{1}=1$, $a_{2}=b_{2}$, and $2+a_{4}=b_{3}$.\n\n1. Find the general formulas for $\\left\\{a_{n}\\right\... | 8,320 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Example 3 Given $x \\in \\mathbf{R}$, let $M=\\max \\{1-x, 2 x-$ $3, x\\}$. Find $M_{\\min }$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "Given \\... | 48,807 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "A particle projected vertically upward reaches, at the end of $t$ seconds, an elevation of $s$ feet where $s = 160 t - 16t^2$. The highest elevation is:\n$\\textbf{(A)}\\ 800 \\qquad \\textbf{(B)}\\ 640\\qquad \\textbf{(C... | 2,675 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "## Problem Statement\n\nCalculate the limit of the function:\n\n$$\n\\lim _{x \\rightarrow 1}\\left(\\frac{2-x}{x}\\right)^{\\frac{1}{\\ln (2-x)}}\n$$ Let's think step by step and output the final answer within \\boxed{{}... | 37,658 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Problem 2. Show that if the numbers $a, b, c \\in (0, \\infty)$ satisfy the relation $a+b+c=3$, then:\n\n$$\n\\frac{a}{3a+bc+12} + \\frac{b}{3b+ca+12} + \\frac{c}{3c+ab+12} \\leq \\frac{3}{16}\n$$ Let's think step by step... | 45,710 |
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