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[
{
"content": "Let the set $A=\\{1,2,3\\}$, and the set $B=\\{0,1,2\\}$. Then, the number of elements in $A\\cup B$ is $(\\quad)$.\n\nA: $2$\n\nB: $3$\n\nC: $4$\n\nD: $5$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"C... | openR1Math_extended |
[
{
"content": "The extreme value points of the function $f(x) = \\frac {1}{3} x^3 - \\frac {1}{4} x^4$ in the interval (0, 3) are __________.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"x = 1\", \"source\": \"openR1M... | openR1Math_extended |
[
{
"content": "Given the power function $f(x) = x^{9-3m}$ $(m \\in N^*)$ whose graph is symmetric about the origin and is monotonically increasing on $\\mathbb{R}$.\n\n$(1)$ Find the expression for $f(x)$; \n$(2)$ Find the range of values for $a$ that satisfies $f(a+1) + f(3a-4) < 0$.\n\nPlease reason step by s... | openR1Math_extended |
[
{
"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 the donation situation of some students, and the collected data is summarize... | openR1Math_extended |
[
{
"content": "In an isosceles triangle, one side length is $6$ and the other side length is $14$. What is the perimeter of the triangle?\n\nA: $26$\n\nB: $26$ or $34$\n\nC: $34$\n\nD: $20$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"refe... | openR1Math_extended |
[
{
"content": "If $a-3b=-3$, then the value of the algebraic expression $5-a+3b$ is ( )\nA: 0\nB: 2\nC: 5\nD: 8\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"D\", \"source\": \"openR1Math_extended\", \"test_case\": nu... | openR1Math_extended |
[
{
"content": "Let's consider the following statements:\n- For statement A: The function $f(x) = |x^2 + mx + n|$ has four monotonic intervals.\n- For statement B: The function $g(x) = \\log(x^2 + mx + n)$ has its range as $\\mathbb{R}$.\n\nThen, statement A is to statement B as ( )\n\nA: A sufficient but not nec... | openR1Math_extended |
[
{
"content": "There are 3 girls and 5 boys standing in a row.\n\n(1) If all the girls stand together, how many different arrangements are there?\n\n(2) If no two girls stand next to each other, how many different arrangements are there?\n\nPlease reason step by step, and put your final answer within \\boxed{}."... | openR1Math_extended |
[
{
"content": "6. Which of the following numbers cannot be expressed as the difference of squares of two natural numbers?\nA. 2021\nB. 2020\nC. 2019\nD. 2018\nE. 2017\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"D\", ... | openR1Math_extended |
[
{
"content": "Let non-zero vectors $\\vec{a}$ and $\\vec{b}$ satisfy $|\\vec{a}| = |\\vec{b}|$. If $(2\\vec{a} + \\vec{b}) \\cdot \\vec{b} = 0$, then the angle between vectors $\\vec{a}$ and $\\vec{b}$ is $\\_\\_\\_\\_\\_\\_\\_\\_$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
... | openR1Math_extended |
[
{
"content": "Given the function $$f(x)=- \\frac {1}{2}x^{2}+1$$, where $x \\in \\mathbb{R}$.\n1. Find the equation of the tangent line to the function's graph that passes through the point (1,1).\n2. Calculate the area of the closed figure formed by the graph of function $f(x)$ and the line $y=-1$.\n\nPlease r... | openR1Math_extended |
[
{
"content": "Given that the derivative of the function $f(x)$ is $f'(x)$, and it satisfies $f(x)=2x{{f'}}(1)+\\frac{1}{x}$, then ${f'}\\left(1\\right)=$____.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"1\", \"sourc... | openR1Math_extended |
[
{
"content": "Given that $S_n$ is the sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$, if $S_3+S_7=37$, then $19a_3+a_{11}=$ ()\nA: $47$\nB: $73$\nC: $37$\nD: $74$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_a... | openR1Math_extended |
[
{
"content": "The number of hyperbolas with asymptotes \\\\(2x \\pm 3y = 0\\\\) is \\\\((\\quad)\\)\nA: \\\\(1\\\\)\nB: \\\\(2\\\\)\nC: \\\\(3\\\\)\nD: Infinite\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"D\", \"sou... | openR1Math_extended |
[
{
"content": "If the complex number $(m^{2}+i)(1-mi)$ is a real number, then the real number $m=$ \nA: $- \\sqrt {2}$\nB: $ \\sqrt {2}$\nC: $-1$\nD: $1$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"D\", \"source\": \... | openR1Math_extended |
[
{
"content": "(2014 Jiangxi Province Key High Schools Alliance Second Mock Exam for Senior High School Students) 21. To us all in China, \\_\\_\\_\\_\\_\\_\\_ experience of taking the college entrance examination seems an important way to judge whether one’s life is \\_\\_\\_\\_\\_\\_ complete one. \nA: the; a... | openR1Math_extended |
[
{
"content": "Given the universal set $U = [0, 2]$ and the set $M = \\{x \\mid x^2 - x \\leq 0\\}$, find the complement of $M$ with respect to $U$, denoted as $\\complement_U M$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answ... | openR1Math_extended |
[
{
"content": "Which of the following calculations is correct?\n\nA: $\\left(a^{3}\\right)^{2}=a^{5}$\n\nB: $ a^{2}+a^{5}=a^{7}$\n\nC: $ \\left(ab\\right)^{3}=ab^{3}$\n\nD: $ a^{2}\\cdot a^{5}=a^{7}$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": nu... | openR1Math_extended |
[
{
"content": "Let $\\{a_n\\}$ be a geometric sequence with a common ratio not equal to 1, and $a_4=16$. The sum of the first $n$ terms is denoted as $S_n$, and $5S_1$, $2S_2$, $S_3$ form an arithmetic sequence. \n(1) Find the general formula for the sequence $\\{a_n\\}$. \n(2) Let $b_n = \\frac{1}{\\log_{2}a_... | openR1Math_extended |
[
{
"content": "Simplify first, then evaluate: $4xy+\\left(2x-y\\right)\\left(2x+y\\right)-\\left(2x+y\\right)^{2}$, where $x=2022$ and $y=-\\sqrt{2}$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"-4\", \"source\": \"o... | openR1Math_extended |
[
{
"content": "Given the function $y=(m-1)x^{m^{2}}+1$ is a linear function, then $m=$ $\\boxed{\\text{\\_\\_\\_\\_\\_\\_}}$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"-1\", \"source\": \"openR1Math_extended\", \"t... | openR1Math_extended |
[
{
"content": "1-2+3-4+5-6+7-8+…+4999-5000= ．\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"-2500\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": null}",
"role": "user"
},
{
... | openR1Math_extended |
[
{
"content": "In $\\triangle ABC$, the sides $a$, $b$, and $c$ opposite to the angles $A$, $B$, and $C$ respectively form a geometric sequence. Given that $\\sin B= \\frac {5}{13}$ and $\\cos B= \\frac {12}{ac}$, find the value of $a+c$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.... | openR1Math_extended |
[
{
"content": "Given $a+b=8$ and $ab=c^2+16$, find the value of $a+2b+3c$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"12\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": null}",
"r... | openR1Math_extended |
[
{
"content": "When a red die and a blue die are thrown at the same time, observe the numbers facing up. Let event A be \"the number facing up on the red die is a multiple of 3\", and event B be \"the sum of the numbers on the two dice is greater than 8\". Then, $P(B|A) = \\_\\_\\_\\_\\_\\_$.\n\nPlease reason st... | openR1Math_extended |
[
{
"content": "In response to the municipal government's call for \"green commuting,\" Xiao Zhang changed his mode of transportation from driving to work to riding a public bicycle. It is known that Xiao Zhang's home is 10 kilometers away from his workplace. The average hourly distance he travels by riding a pub... | openR1Math_extended |
[
{
"content": "Which of the following transformations is correct?\n\nA: From $2+x=3$, we get $x=3+2$\n\nB: From $2x=3x+4$, we get $2x-3x=4$\n\nC: From $3x=2$, we get $x=\\frac{3}{2}$\n\nD: From $x=0$, we get $3x=3$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answe... | openR1Math_extended |
[
{
"content": "The distance from the point (0, b) to the line $x+y-2=0$ is $\\sqrt{2}$. Then, $b$ equals ( )\n\nA: 0 or 4 \nB: 4 \nC: 0 \nD: $\\sqrt{2}$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\text{A}\", ... | openR1Math_extended |
[
{
"content": "Given $m$ and $n$ satisfy $m+n-1=0$, the line $mx+y+n=0$ passes through the fixed point ( )\nA: $(1,-1)$\nB: $(0,-n)$\nC: $(0,0)$\nD: $(-1,1)$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"A\", \"source... | openR1Math_extended |
[
{
"content": "Calculate: $4\\sin 60^{\\circ}-|1-\\sqrt{3}|+(\\frac{1}{3})^{-1}+\\sqrt[3]{{-27}}$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\sqrt{3} + 1\", \"source\": \"openR1Math_extended\", \"test_case\": nu... | openR1Math_extended |
[
{
"content": "11.142. In a regular triangular prism, a plane is drawn through the side of the lower base and the opposite vertex of the upper base, forming an angle of $45^{\\circ}$ with the plane of the lower base. The area of the section is $S$. Find the volume of the prism.\n\nPlease reason step by step, and... | openR1Math_extended |
[
{
"content": "Calculate: $(-2023)^{0}-4\\sin45°+|-\\sqrt{8}|=\\_\\_\\_\\_\\_\\_$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"1\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": null}",... | openR1Math_extended |
[
{
"content": "Which of the following objects can determine a set? ( )\nA: All points in the first quadrant\nB: All students with good grades in a class\nC: All difficult problems in the high school grade one math textbook\nD: All numbers close to 1\n\nPlease reason step by step, and put your final answer withi... | openR1Math_extended |
[
{
"content": "In $\\triangle ABC$, the sides opposite to $\\angle A$, $\\angle B$, and $\\angle C$ are $a$, $b$, and $c$, respectively. If $a$, $b$, and $c$ form an arithmetic sequence, $\\sin B = \\frac{4}{5}$, and the area of $\\triangle ABC$ is $\\frac{3}{2}$, then $b =$ $\\_\\_\\_\\_\\_\\_$ (answer with a n... | openR1Math_extended |
[
{
"content": "10. (1990 American Mathematical Invitational) A fair coin is tossed 10 times, let $\\frac{i}{j}$ be the probability that no two heads appear consecutively, where $i$ and $j$ are relatively prime, find $i+j$.\n\n\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info... | openR1Math_extended |
[
{
"content": "Which of the following functions is both a decreasing function and an odd function?\n\nA: $f(x)=(\\frac{1}{2})^x-1$\n\nB: $f(x)=\\frac{1}{x}$\n\nC: $f\\left(x\\right)=-x$\n\nD: $f\\left(x\\right)=-e^{|x|}$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\... | openR1Math_extended |
[
{
"content": "If the even function $f(x)$ is monotonically decreasing on $(-\\infty, 0]$, and $a=f(\\log_{2}3)$, $b=f(\\log_{4}5)$, $c=f(2^{\\frac{3}{2}})$, then which of the following is true?\nA: $a < b < c$\nB: $b < a < c$\nC: $c < a < b$\nD: $c < b < a$\n\nPlease reason step by step, and put your final answ... | openR1Math_extended |
[
{
"content": "Given the vector $\\overrightarrow{a}=(-1,2,2)$, then $|2\\overrightarrow{a}|$ is equal to:\n\nA: $9$\n\nB: $3$\n\nC: $6$\n\nD: $2\\sqrt{3}$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"C\", \"source\":... | openR1Math_extended |
[
{
"content": "In triangle $\\triangle ABC$, it is known that $AB=6$, $AC=BC=4$. $P$ is the intersection point of the angle bisector of $\\angle ACB$ with side $AB$, and $M$ is a point on $PC$ such that $\\overrightarrow {BM} = \\overrightarrow {BA} + \\lambda \\left( \\frac { \\overrightarrow {AC}}{| \\overrigh... | openR1Math_extended |
[
{
"content": "A batch of equipment is worth $a$ ten thousand yuan. Due to wear and tear from use, its value decreases by $b\\%$ every year. The value of this batch of equipment after $n$ years is ( ).\n\nA: $na(1-b\\%)$\n\nB: $a(1-nb\\%)$\n\nC: $a(1-b\\%)^{n}$\n\nD: $a[1-(b\\%)^{n}]$\n\nPlease reason step by st... | openR1Math_extended |
[
{
"content": "Given the equation $(2a+b)x-1=0$ has no solution with respect to $x$, then the value of $ab$ is ( )\nA: Negative\nB: Positive\nC: Non-negative\nD: Non-positive\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\"... | openR1Math_extended |
[
{
"content": "The solution set of the inequality $\\dfrac{x}{x+1} < 0$ is .\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"(-1,0)\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": null}",
... | openR1Math_extended |
[
{
"content": "Given $x \\gt 0$, $y \\gt 0$, and $x+4y-2xy=0$. Find:$(1)$ the minimum value of $xy$;$(2)$ the minimum value of $x+y$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\frac{9}{2}\", \"source\": \"openR1... | openR1Math_extended |
[
{
"content": "Given a linear regression equation $y = 2 - x$, when the variable $x$ increases by one unit, ( )\n\nA: $y$ increases by an average of 1 unit \nB: $y$ increases by an average of 2 units \nC: $y$ decreases by an average of 1 unit \nD: $y$ decreases by an average of 2 units\n\nPlease reason step ... | openR1Math_extended |
[
{
"content": "A senior high school student participated in the level tests for Physics, Chemistry, and Politics. It is known that the probabilities of this student achieving an A+ in Physics, Chemistry, and Politics are $\\dfrac{7}{8}$, $\\dfrac{3}{4}$, and $\\dfrac{5}{12}$ respectively. The results of these th... | openR1Math_extended |
[
{
"content": "Given the equation of a circle is $x^2+y^2-2x-9=0$, and the equation of a line is $mx+y+m-2=0$, then the positional relationship between the line and the circle is ( ) \nA: Intersect \nB: Separate \nC: Tangent \nD: Uncertain\n\nPlease reason step by step, and put your final answer within \\bo... | openR1Math_extended |
[
{
"content": "To enhance security awareness, a shopping mall plans to randomly select 3 days for emergency evacuation drills in the next consecutive 10 days. The probability that the selected 3 days are consecutive is ___ (express the answer as a simplest fraction).\n\nPlease reason step by step, and put your f... | openR1Math_extended |
[
{
"content": "Given that in $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $\\frac{a-c}{a-b} = \\frac{\\sin A + \\sin B}{\\sin (A+B)}$.\n(I) Find the value of angle $B$;\n(II) If the circumradius of $\\triangle ABC$ is $1$, find the maximum value of the area $S$... | openR1Math_extended |
[
{
"content": "The base radius of a cylinder is 3, and its lateral area is $12\\pi$. The volume of the cylinder is \\_\\_\\_\\_\\_\\_.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"18\\\\pi\", \"source\": \"openR1Math_... | openR1Math_extended |
[
{
"content": "The point P lies on the curve $y = x^2 - \\ln x$. Find the minimum distance from point P to the straight line $y=x-2$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\sqrt{2}\", \"source\": \"openR1Mat... | openR1Math_extended |
[
{
"content": "Which of the following equations is a quadratic equation?\n\nA: $2x^{2}=5x-1$\n\nB: $x+\\frac{1}{x}=2$\n\nC: $(x-3)\\left(x+1\\right)=x^{2}-5$\n\nD: $3x-y=5$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \... | openR1Math_extended |
[
{
"content": "Calculate:$(1)\\sqrt{10} \\times \\sqrt{2} + \\sqrt{15} \\div \\sqrt{3}$;$(2)\\sqrt{27} - (\\sqrt{12} - \\sqrt{\\frac{1}{3}})$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\frac{4}{3}\\\\sqrt{3}\", ... | openR1Math_extended |
[
{
"content": "After running the following program, the output value is ( )\n\n$$\\boxed{\\begin{matrix}i=0 \\\\ DO \\\\ \\quad i=i+1 \\\\ LOOP UNTIL\\quad 2^i > 2 000 \\\\ \\quad i=i-1 \\\\ PRINT\\quad i \\\\ END\\end{matrix}}$$\nA: $8$\nB: $9$\nC: $10$\nD: $11$\n\nPlease reason step by step, and put your fi... | openR1Math_extended |
[
{
"content": "11. (5 points)\nXiao Huhu encountered a problem: Compare the sizes of $\\frac{99}{100}, \\frac{100}{101}, \\frac{199}{201}$. He is very confused, please help him find the correct answer.\nA. $\\frac{99}{100}>\\frac{100}{101}>\\frac{199}{201}$\nB. $\\frac{199}{201}>\\frac{100}{101}>\\frac{99}{100}$... | openR1Math_extended |
[
{
"content": "Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \" n = 2 \", \"source\": \"openR1Math_extended\", \"test_case\": null, \"t... | openR1Math_extended |
[
{
"content": "Given the set A={2, 0, 1, 9}, and B={x|x=2a, a∈A}, find the sum of all elements in A∪B.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"34\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"thi... | openR1Math_extended |
[
{
"content": "If $$\\int_{1}^{a} (2x + \\frac{1}{x})dx = 3 + \\ln 2$$ (where $a > 1$), then the value of $a$ is \\_\\_\\_\\_\\_\\_.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"2\", \"source\": \"openR1Math_extended\... | openR1Math_extended |
[
{
"content": "(Full score: 10 points) Given a complex number $z$ that satisfies $z + |z| = 2 + 8i$, find the complex number $z$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"z = -15 + 8i\", \"source\": \"openR1Math_e... | openR1Math_extended |
[
{
"content": "A math teacher is 176 cm tall. The heights of his grandfather, father, and son are 173 cm, 170 cm, and 182 cm, respectively. Since the son's height is related to the father's height, the teacher uses the method of linear regression analysis to predict his grandson's height as ______ cm.\n\nPlease ... | openR1Math_extended |
[
{
"content": "In the right triangular prism $ABC-A_1B_1C_1$, if $\\overrightarrow{CA} = \\overrightarrow{a}$, $\\overrightarrow{CB} = \\overrightarrow{b}$, and $\\overrightarrow{CC_1} = \\overrightarrow{c}$, then find $\\overrightarrow{A_1B}$.\n\nPlease reason step by step, and put your final answer within \\bo... | openR1Math_extended |
[
{
"content": "Given the functions $f(x)=e^{x}$ ($e$ is the base of the natural logarithm) and $g(x)=\\frac{a}{2}x+b$ ($a,b \\in \\mathbb{R}$).\n\n(1) If $h(x)=f(x)g(x)$, $b=1-\\frac{a}{2}$, and $a=-4$, find the maximum value of $h(x)$ on $[0,1]$.\n\n(2) If $a=4$, find the range of the real number $b$ such that ... | openR1Math_extended |
[
{
"content": "Let P and Q be two sets. The \"difference set\" of P and Q is defined as P - Q = {x | x ∈ P, x ∉ Q}. Given that $$P = \\{x | 1 - \\frac{2}{x} < 0\\}$$ and Q = {x ||x - 2| < 1}, find Q - P.\n\nA: {x | 0 < x < 1}\nB: {x | 0 < x ≤ 1}\nC: {x | 1 ≤ x < 2}\nD: {x | 2 ≤ x < 3}\n\nPlease reason step by st... | openR1Math_extended |
[
{
"content": "Xiao Ming is practicing shooting. In the first stage, he shot $4$ bullets with scores of $8$, $9$, $9$, and $10$ respectively. What is the median of the data $8$, $9$, $9$, $10$?\n\nA: $8$\n\nB: $9$\n\nC: $9.5$\n\nD: $10$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",... | openR1Math_extended |
[
{
"content": "Given that the maximum value of $x$ that satisfies the inequality $(x^2 - 4x + a) + |x - 3| \\leq 5$ is 3, find the value of $a$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"8\", \"source\": \"openR1Ma... | openR1Math_extended |
[
{
"content": "Given real numbers $x$ and $y$ that satisfy the equation $x^{2}+y^{2}-4x+6y+12=0$, find the minimum value of $|2x-y-2|$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"5-\\\\sqrt{5}\", \"source\": \"openR... | openR1Math_extended |
[
{
"content": "A certain project requires 24 days for Team A to complete on their own, and 18 days for Team B to complete on their own. Now, if 6 people from Team B are transferred to Team A, then Team A alone would need 18 days to complete the project, and Team B alone would need 24 days. If Team B starts worki... | openR1Math_extended |
[
{
"content": "Given that the line $l: ax-y+3=0$ is the axis of symmetry of the circle $C: x^{2}+y^{2}+2x-4y-4=0$, a tangent line to circle $C$ passes through point $P(a,-2)$ and touches the circle at point $Q$. Find the length of $|PQ|$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.... | openR1Math_extended |
[
{
"content": "A straight line $l$, whose parametric equations are \n$$\n\\begin{cases}\nx = x_{0} + t\\cos\\theta \\\\\ny = t\\sin\\theta\n\\end{cases}\n$$\n($t$ is the parameter), passes through the left focus $F_1$ of the ellipse $\\frac{x^2}{3} + y^2 = 1$ and intersects the positive y-axis at point $C$. It a... | openR1Math_extended |
[
{
"content": "Use mathematical induction to prove that when \\\\((n+1)(n+2)\\cdots (n+n) = {{2}^{n}}\\cdot 1\\cdot 3\\cdots (2n-1), (n \\in {{N}_{+}})\\\\), from \\\\(n=k\\\\) to \\\\(n=k+1\\\\), the term that should be added to the left side is \\\\((\\quad)\\\\)\nA: \\\\(2k+1\\\\)\nB: \\\\(2k+3\\\\)\nC: \\\\(... | openR1Math_extended |
[
{
"content": "Given sets $A=(-\\infty, 4]$, $B=\\{1,3,5,7\\}$, find $A$ ⋂ $B$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\{1,3\\\\}\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\"... | openR1Math_extended |
[
{
"content": "The real number $x$ satisfies $\\log_{2}x = 3 - 2\\cos^{2}\\theta$. The value of $|x - 2| + |x - 8|$ is ( )\n\nA: 6 \nB: 6 or -6 \nC: 10 \nD: Uncertain\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"A... | openR1Math_extended |
[
{
"content": "Given that $\\{a_n\\}$ is a geometric sequence with the sum of the first $n$ terms denoted as $S_n (n\\in\\mathbb{N^*})$, and $\\frac{1}{a_1}-\\frac{1}{a_2}=\\frac{2}{a_3}$, $(S_6=63)$.\n\n(1) Find the general term formula for $\\{a_n\\}$;\n(2) If for any $n\\in\\mathbb{N^*}$, $b_n$ is the arithme... | openR1Math_extended |
[
{
"content": "The radius and the coordinates of the center of the circle $x^2+y^2-4x+2y+4=0$ are ( )\nA: $r=1$; $(﹣2,1)$\nB: $r=2$; $(﹣2,1)$\nC: $r=1$; $(2,﹣1)$\nD: $r=2$; $(2,﹣1)$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_... | openR1Math_extended |
[
{
"content": "The first term of the sequence ${a_n}$ is $a_1=2$, and $a_{n+1}=3a_{n}+2 (n∈N^{*})$. Let $b_n=\\log_{3}(a_n+1)$, then the sum of the first 2019 terms of the sequence $\\{ \\frac {1}{b_{2n-1}b_{2n+1}}\\}$ is $S_{2019}=\\_\\_\\_\\_\\_\\_.$\n\nPlease reason step by step, and put your final answer wit... | openR1Math_extended |
[
{
"content": "Given that the vertex of the parabola is at the origin, and the focus is on the y-axis, for a point M(m, -3) on the parabola, the distance from point M to the focus of the parabola is 5,\n(1) Find the value of m;\n(2) Determine the equation of the parabola and the equation of its directrix.\n\nPle... | openR1Math_extended |
[
{
"content": "The opposite number of \\\\(-2016\\\\) is \\_\\_\\_\\_\\_\\_ .\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"2016\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": null}",
... | openR1Math_extended |
[
{
"content": "In order to estimate the number of fish in a pond, initially 2,000 fish were caught, marked (without affecting their survival), and then released back into the pond. After some time, 500 fish were caught again, of which 40 were marked. Based on the data provided, estimate the total number of fish ... | openR1Math_extended |
[
{
"content": "Factorize the following expressions: \n(1) $3ax^2+6axy+3ay^2$ \n(2) $a^2(x-y)-b^2(x-y)$ \n(3) $a^4+3a^2-4$; \n(4) $4x^2-y^2-2y-1$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"(2x+y+1)(2x-y-1)\", \"... | openR1Math_extended |
[
{
"content": "A four-digit number is divisible by 9, and the three-digit number obtained by removing the last digit is exactly a multiple of 4. What is the last digit of the largest such four-digit number?\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_conten... | openR1Math_extended |
[
{
"content": "Given $A=\\{1, 2, a^2-3a-1\\}$, $B=\\{1, 3\\}$, and $A \\cap B = \\{3, 1\\}$, then $a=$ ______.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"-1 \\\\text{ or } 4\", \"source\": \"openR1Math_extended\", \... | openR1Math_extended |
[
{
"content": "Given proposition $p$: $\\forall x \\in \\mathbb{N}^{*}$, $\\left( \\frac{1}{2} \\right)^{x} \\geqslant \\left( \\frac{1}{3} \\right)^{x}$, and proposition $q$: $\\exists x \\in \\mathbb{N}^{*}$, $2^{x} + 2^{1-x} = 2 \\sqrt{2}$, then among the following propositions, the true one is $(\\quad)$.\nA... | openR1Math_extended |
[
{
"content": "If the hyperbola $C_{1}$ has the same focal length as the hyperbola ${C_2}:\\frac{{{x^2}}}{7}-{y^2}=1$, and $C_{1}$ passes through the point $\\left(3,1\\right)$, then the standard equation of the hyperbola $C_{1}$ is ( )\n\nA: $\\frac{{{x^2}}}{6}-\\frac{{{y^2}}}{2}=1$\n\nB: $\\frac{{{y^2}}}{{9-\\... | openR1Math_extended |
[
{
"content": "If the eccentricity of the hyperbola $x^{2}+ky^{2}=1$ is $2$, then the value of the real number $k$ is ______.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"-\\\\frac{1}{3}\", \"source\": \"openR1Math_ex... | openR1Math_extended |
[
{
"content": "Given that $\\theta$ is an internal angle of a triangle, and $\\sin \\theta + \\cos \\theta = \\frac{1}{2}$, then the equation $x^{2}\\sin \\theta - y^{2}\\cos \\theta = 1$ represents\nA: An ellipse with foci on the $x$-axis\nB: An ellipse with foci on the $y$-axis\nC: A hyperbola with foci on the... | openR1Math_extended |
[
{
"content": "A certain craft factory plans to produce 2100 handicrafts in a week, with an average daily production of 300. However, the actual daily production deviates from the plan. The table below shows the production situation for a week (positive for overproduction and negative for underproduction):\n\n| ... | openR1Math_extended |
[
{
"content": "In the geometric sequence $\\{a_n\\}$, $a_2$ and $a_6$ are the two roots of the equation $x^2 - 34x + 64 = 0$. What is the value of $a_4$?\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"8\", \"source\": \... | openR1Math_extended |
[
{
"content": "Given the function $y=mx^{2}+x+5$, find the range of values for $m$ that makes the function increasing on the interval $[-2,+\\infty)$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"[0, \\\\frac{1}{4}]\"... | openR1Math_extended |
[
{
"content": "In the rectangular coordinate system $Oxy$, the straight line $\\frac {x}{2}- \\frac {y}{4}=1$ intersects the coordinate axes at points A and B. Determine the standard equation of the circle passing through points O, A, and B.\n\nPlease reason step by step, and put your final answer within \\boxed... | openR1Math_extended |
[
{
"content": "$\\sqrt{12}+\\sqrt{\\frac{1}{3}}=$____.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"\\\\frac{7\\\\sqrt{3}}{3}\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": null}",
... | openR1Math_extended |
[
{
"content": "If the graph of the function $y=kx^{2}-2x-1$ intersects the $x$-axis, then the range of values for $k$ is ( ).\n\nA: $k\\geqslant -1$\n\nB: $k \\gt -1$\n\nC: $k \\gt -1$ and $k\\neq 0$\n\nD: $k\\geqslant -1$ and $k\\neq 0$\n\nPlease reason step by step, and put your final answer within \\boxed{}."... | openR1Math_extended |
[
{
"content": "If $x$ and $y$ are positive numbers, and $\\frac{1}{x} + \\frac{4}{y} = 1$, then $x+y$ has ( ).\n\nA: Minimum value $9$\n\nB: Maximum value $9$\n\nC: Minimum value $5+2\\sqrt{2}$\n\nD: Maximum value $5+2\\sqrt{2}$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"in... | openR1Math_extended |
[
{
"content": "Given that the sum of two monomials $-2a^2b^{m+1}$ and $na^2b^4$ is 0, find the value of $m+n$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"5\", \"source\": \"openR1Math_extended\", \"test_case\": null... | openR1Math_extended |
[
{
"content": "In the ancient Chinese mathematical classic \"The Nine Chapters on the Mathematical Art,\" there is a description: \"Now there are good horses and donkeys traveling from Chang'an to Qi. Qi is 1,125 miles away from Chang'an. The good horse travels 103 miles on the first day and increases by 13 mile... | openR1Math_extended |
[
{
"content": "Given the sequence $\\{a\\_n\\}$ that satisfies $a\\_n= \\frac{a_{n-1}}{a_{n-2}}\\ (n \\in \\mathbb{N}^*, n\\geqslant 3)$, with $a\\_1=2$ and $a\\_5= \\frac{1}{3}$, find the value of $a\\_{2016}$.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_c... | openR1Math_extended |
[
{
"content": "1. In triangle $ABC$, the median $BD$ is drawn, and the centroid $G$ is marked on it. A line parallel to $BC$ and passing through point $G$ intersects $AB$ at point $E$. It turns out that $\\angle AEC = \\angle DGC$. Prove that $\\angle ACB = 90^{\\circ}$.\n\nPlease reason step by step, and put yo... | openR1Math_extended |
[
{
"content": "A certain food retail store has recently introduced a new cold beverage product, with a cost of $8$ dollars per unit. During the sales process, the daily sales quantity $y$ (units) is related to the selling price $x$ (dollars per unit) by the equation $y=-x+30$. If the profit of selling this cold ... | openR1Math_extended |
[
{
"content": "Show that $M=\\sqrt[3]{\\sqrt{5}+2}-\\sqrt[3]{\\sqrt{5}-2}$ is an integer.\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"1\", \"source\": \"openR1Math_extended\", \"test_case\": null, \"think_content\": ... | openR1Math_extended |
[
{
"content": "If $\\lim_{{\\Delta x}\\to0}\\frac{{f({-2+\\Delta x})-f({-2-\\Delta x})}}{{\\Delta x}}=-2$, then ${f'}\\left(-2\\right)=\\left(\\ \\ \\right)$\n\nA: $1$\n\nB: $-1$\n\nC: $2$\n\nD: $-2$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": nu... | openR1Math_extended |
[
{
"content": "Given that point $P(m,2)$ lies on the $y$-axis, the value of $m$ is ( )\n\nA: $2$\n\nB: $0$\n\nC: $-2$\n\nD: $1$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_content\": null, \"reference_answer\": \"B\", \"source\": \"openR1Math_extended\", \"... | openR1Math_extended |
[
{
"content": "Given the sets $M=\\{x|\\lg x^2 = 0\\}$, and $N=\\{x|2^{-1} < 2^{x+1} < 2^2, x \\in \\mathbb{Z}\\}$, then $M \\cap N =$ ( )\n\nA: $\\{-1, 1\\}$\n\nB: $\\{-1\\}$\n\nC: $\\{0\\}$\n\nD: $\\{-1, 0\\}$\n\nPlease reason step by step, and put your final answer within \\boxed{}.",
"info": "{\"answer_c... | openR1Math_extended |