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"content": "Example 2. Given: $f(x)=2 x+3,|x| \\leqslant 2$, and $x \\in Z$. Find the range of the function.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or nu... | 5C |
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"content": "Example 3 Solve the equation $3^{x}+4^{x}=5^{x}$.\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. \n\nHowever, since the provided text is already in English, here is the confirmation of the text with the ... | 5C |
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"content": "Example 3. Calculate the mass of the surface $z=x y$, located inside the cylinder $x^{2}+\\frac{y^{2}}{4}=1$, if the density is $\\rho=\\frac{|z|}{\\sqrt{1+x^{2}+y^{2}}}$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to so... | 5C |
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"content": "Example 4. Find a curve passing through the point $(0,-2)$ such that the tangent of the angle of inclination of the tangent at any point on the curve equals the ordinate of that point increased by three units.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question makin... | 5C |
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"content": "Exercise 5. How many ways are there to form a team of 4 people from 5 boys and 4 girls, so that the team includes at least one boy and at least one girl?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think... | 5C |
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"content": "Express the average diameter of $2019-nCoV$ coronavirus particles in scientific notation, which is approximately $0.00000012m$. The number $0.00000012$ in scientific notation is ____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is ... | 5C |
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"content": "Factorization: $6ab-a^{2}-9b^{2}=$____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n</an... | 5C |
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"content": "Factorize the expression: $a^{2}-4b^{2}-2a+4b$.",
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"content": "<think>\n<sample>\nRewritten Question: Factorize the expression: $a^2 - 4b^2 - 2a + 4b$.\n\nStep 1: Group the terms to identify common factors.\n- We can group the first two terms together and the last tw... | 5C |
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"content": "Find all real numbers $a$ for which the following equation has a unique real solution:\n$$|x-1|+|x-2|+\\ldots+|x-99|=a.$$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your... | 5C |
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"content": "Find the area of a triangle, two sides of which are 6 and 8, and the median drawn to the third side is 5.\n\n#\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting ... | 5C |
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"content": "Find the eccentricity of the ellipse $\\dfrac{x^2}{25}+\\dfrac{y^2}{9}=1$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in ... | 5C |
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"content": "Find the equation of the circle whose center is on the line $2x-y-3=0$ and passes through points A(5, 2) and B(3, -2).",
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"content": "<think>\n<sample>\nTo find the equation of the circle, we need to determine its center \\((h, k)\\) and radius \\(r\\).\n\n1. **Deter... | 5C |
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"content": "Find the equation of the tangent line to the curve $y=x^{3}+1$ at $x=-1$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in t... | 5C |
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"content": "Find the sum of all the real values of x satisfying $\u0012(x+\\frac{1}{x}-17)^2$ $= x + \\frac{1}{x} + 17.$",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by writing out a plan and go really deep into detail about how I should sol... | 5C |
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"content": "For an arbitrary real number $ x$, $ \\lfloor x\\rfloor$ denotes the greatest integer not exceeding $ x$. Prove that there is exactly one integer $ m$ which satisfy $ \\displaystyle m\\minus{}\\left\\lfloor \\frac{m}{2005}\\right\\rfloor\\equal{}2005$.\n\nBegin your response with \"Rewritten Quest... | 5C |
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"content": "For any two vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$, which of the following propositions is correct?\n\nA: If $|\\overrightarrow{a}| > |\\overrightarrow{b}|$ and $\\overrightarrow{a}$ is in the same direction as $\\overrightarrow{b}$, then $\\overrightarrow{a} > \\overrightarrow{b}... | 5C |
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"content": "Return your final response within <answer>...</answer>. For how many positive integers $x$ is $100 \\leq x^2 \\leq 200$?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your ... | 5C |
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"content": "G1.1 Let $x * y=x+y-x y$, where $x, y$ are real numbers. If $a=1 *(0 * 1)$, find the value of $a$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or ... | 5C |
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"content": "Given $M=\\{1,2,4\\}$, $A=\\{y|y=x^2, x\\in M\\}$, $B=\\{y|y=\\log_2 x^2, x\\in M\\}$, then $Card(A\\cup B)=$ ___.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your result... | 5C |
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"content": "Given $x+y=4$ and $x-y=6$, find $2x^{2}-2y^{2}$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answ... | 5C |
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"content": "Given a cone with a base circle of radius 1, the side surface, when unfolded, forms a sector with a central angle of $\\frac{2\\pi}{3}$. What is the side surface area of this cone?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is nee... | 5C |
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"content": "Given a point $A(1,0)$ on the parabola $y=f(x)$, and the slope of the tangent line at point $A$ is $45^{\\circ}$, then $f′(1)=$ ______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and ... | 5C |
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"content": "Given a straight line $l$, another line $m$, and a plane $\\alpha$, which of the following statements is correct?\nA: If $l \\parallel m$ and $m \\subset \\alpha$, then $l \\parallel \\alpha$.\nB: If $l \\parallel \\alpha$ and $m \\subset \\alpha$, then $l \\parallel m$.\nC: If $l \\perp m$ and $l... | 5C |
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"content": "Given an isosceles triangle with one side length of $4$ and another side length of $8$, the perimeter of this isosceles triangle is ____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step an... | 5C |
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"content": "Given lines $l\\_1$: $x-3y+1=0$, and $l\\_2$: $2x+my-1=0$. If $l\\_1$ is parallel to $l\\_2$, find the value of the real number $m$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and the... | 5C |
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"content": "Given proposition p: \"There exists some $x_0 \\in [1, +\\infty)$ such that $(\\log_2 3)^{x_0} \\geq 1$\", what would be the negation of proposition p?\n\nA: There exists some $x_0 \\in [1, +\\infty)$ such that $(\\log_2 3)^{x_0} < 1$ \nB: There exists some $x_0 \\in [1, +\\infty)$ such that $(\\... | 5C |
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"content": "Given real numbers $a$, $b$, $c$ satisfying $2^{a}=5$, $2^{b}=10$, $2^{c}=80$, find the value of $2019a-4039b+2020c$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your res... | 5C |
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"content": "Given that $S_n$ is the sum of the first $n$ terms of the sequence ${a_n}$, with $a_1=1$, $a_2=2$, and $a_{n+2} - a_n = 1 + (-1)^n$, find the value of $S_{100}$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, th... | 5C |
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"content": "Given that $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ are two non-collinear vectors, and $\\overrightarrow{AB} = 2\\overrightarrow{e_{1}} + m\\overrightarrow{e_{2}}$, $\\overrightarrow{BC} = \\overrightarrow{e_{1}} + 3\\overrightarrow{e_{2}}$, if points $A$, $B$, and $C$ are collinear... | 5C |
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"content": "Given that $a > 0$ and $a \\neq 1$, the function $f(x) = \\begin{cases} 2x^{2}, &x\\geqslant 0 \\\\ a^{x}-1, &x < 0 \\end{cases}$ is a monotonic function on $R$, and $f(a)=5a-2$. Find the value of the real number $a$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the questi... | 5C |
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"content": "Given that $a_n > 0$ ($n\\in\\mathbb{N}^*$), \n\n(1) If $a_1+a_2=1$, then $\\dfrac{1}{a_1}+ \\dfrac{1}{a_2} =(a_1+a_2)( \\dfrac{1}{a_1}+ \\dfrac{1}{a_2}) \\geqslant 4$;\n\n(2) If $a_1+a_2+a_3=1$, then $\\dfrac{1}{a_1}+ \\dfrac{1}{a_2}+ \\dfrac{1}{a_3} =(a_1+a_2+a_3)( \\dfrac{1}{a_1}+ \\dfrac{1}{a_... | 5C |
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"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$ and $f(x+4) = f(x)$, when $x \\in (0, 2)$, $f(x) = 2x^2$, find $f(7)$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give y... | 5C |
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"content": "Given that $f(x)$ is an odd function on $\\mathbb{R}$, and $f(x)=3^{x}$ when $x > 0$, find the value of $f(\\log_{\\frac{1}{2}}{4})$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and th... | 5C |
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"content": "Given that $x > 0, y > 0$, and $2x + 8y = xy$, find the minimum value of $x + y$.",
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"content": "<think>\n<sample>\nTo find the minimum value of \\( x + y \\) given the equation \\( 2x + 8y = xy \\), we start by manipulating this equation algebraically.\n\nFirst, rew... | 5C |
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"content": "Given that in the polar coordinate system, points $A(2,\\frac{\\pi }{6})$ and $B(4,\\frac{2\\pi }{3})$, and $O$ is the pole. Calculate the area of triangle $AOB$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, t... | 5C |
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"content": "Given that line \\\\(l\\\\) is perpendicular to plane \\\\(\\alpha\\\\), and line \\\\(m\\\\) is contained in plane \\\\(\\beta\\\\), consider the following three propositions: \\\\(① \\alpha \\parallel \\beta \\Rightarrow l \\perp m; ② \\alpha \\perp \\beta \\Rightarrow l \\parallel m; ③ l \\para... | 5C |
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"content": "Given that point P moves on the circle $x^2+y^2=1$, the minimum distance from P to the line $3x+4y+15=0$ is ____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulti... | 5C |
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"content": "Given that the complex number $z$ satisfies $(1+i)z=2i$ ($i$ is the imaginary unit), find the conjugate of $z$, denoted as $\\bar{z}$ = $\\underline{\\quad\\quad}$ .\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it... | 5C |
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"content": "Given that the complex number $z$ satisfies the condition $z-3= \\frac {3+i}{i}$, find the value of $|z|$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equa... | 5C |
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"content": "Given that the first four terms of the sequence $\\{a_n\\}$ are 2, 0, 2, 0, which of the following expressions cannot serve as a term in the general formula for the sequence $\\{a_n\\}$? \nA: $a_n = 1 + (-1)^{n+1}$ \nB: $a_n = 2\\sin(n\\pi)$ \nC: $a_n = 1 - \\cos(n\\pi)$ \nD: $a_n = 2\\sin^2\\... | 5C |
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"content": "Given that the function $f(x)$ is an odd function and is monotonically increasing on $(0,+\\infty)$, which of the following statements is correct? $(\\quad)$\nA: The function $|f(x)|$ is an even function and is monotonically increasing on $(-\\infty,0)$.\nB: The function $|f(x)|$ is an odd functio... | 5C |
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"content": "Given that the random variable $\\xi$ follows a binomial distribution $\\xi \\sim B(6, \\frac{1}{3})$, then its expected value $E\\xi$ is $______$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step ... | 5C |
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"content": "Given that the solution to the equation about $y$, $2 - 3(1 - y) = 2y$, is the same as the solution to the equation about $x$, $m(x - 3) - 2 = -8$, find the value of $m$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to sol... | 5C |
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"content": "Given the arithmetic sequence $\\{a\\_n\\}$, where $a\\_7+a\\_9=16$ and $a\\_4=1$, find the value of $a\\_12$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting ... | 5C |
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"content": "Given the function $f(x) = \\log\\_2 \\frac{2(1+x)}{x-1}$, if $f(a) = 2$, find $f(-a)$ ___.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number ... | 5C |
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"content": "Given the function $f(x) = ax^2 + (b-3)x + 3$, where $x \\in [2a-3, 4-a]$ is an even function, find the value of $a+b$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your r... | 5C |
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"content": "Given the function $f(x)= \\begin{cases} x(x+1), & x\\geqslant 0 \\\\ x(1-x), & x < 0\\end{cases}$, then $f(-3)=$ \\_\\_\\_\\_\\_\\_.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and th... | 5C |
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"content": "Given the function $f(x)=2x+1$, if $f_{1}(x)=f(x)$, $f_{n+1}(x)=f[f_{n}(x)]$, $n\\in\\mathbb{N}^{*}$. Then the expression for $f_{5}(x)$ is ______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step ... | 5C |
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"content": "Given the inequality system $\\left\\{\\begin{array}{l}{x+2a>4}\\\\{2x<b}\\end{array}\\right.$ and its solution set is $0 < x < 2$, find $a+b$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by s... | 5C |
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"content": "Given the power function $f(x)=x^{-m^{2}+2m+3}$ $(m∈\\mathbb{Z})$ is an increasing function on $(0,+∞)$, and it is an even function within its domain, determine the value of $m$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is neede... | 5C |
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"content": "Given the quadratic equation $x^{2}-2x-2=0$, after completing the square, we get $\\left(x-1\\right)^{2}=n$. Find the value of $n$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then... | 5C |
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"content": "Given the unit vectors $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ with an angle of $30^{\\circ}$ between them, find the value of $|\\overrightarrow{e_{1}} - \\sqrt{3}\\overrightarrow{e_{2}}|$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it... | 5C |
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"content": "Given the universal set $U={2,4,1-a}$, set $A={2,a^{2}-a+2}$, if the complement of set $A$ with respect to $U$ is ${-1}$, find the value of $a$.",
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"content": "<think>\n<sample>\nTo solve this problem, we need to determine which elements belong to set \\( A \\) and w... | 5C |
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"content": "Given vectors $a$ and $b$ with an angle of $\\frac{\\pi}{6}$ between them, $a=(x,1)$, $|b|=2$, $c=(2,\\sqrt{2})$, and $a\\parallel c$, then $a\\cdot(a+b)=$\\_\\_\\_\\_\\_\\_\\_.",
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"content": "<think>\n<sample>\nFirst, I'll try to understand the problem better by wri... | 5C |
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"content": "Given ○+○+○+○=△+△+○, and △=63, then ○= .\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(your answer)\n... | 5C |
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"content": "Given: $2^{a+1}=3$, ${2}^{b-3}=\\frac{1}{3}$, which of the following statements is incorrect?\n\nA: $a+b=2$\n\nB: $1＜b＜\\frac{3}{2}$\n\nC: $b-a \\lt 1$\n\nD: $ab \\gt 1$",
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"content": "<think>\n<sample>\nRewritten Question: Given \\(2^{a+1}=3\\) and \\(2^{b-3}=\\frac... | 5C |
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"content": "Given: $x=\\sqrt{2}-1$, find the value of $x^{2}+2x+3$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<answer>\n(yo... | 5C |
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"content": "Hope Primary School has a total of 1996 students, approximately __________ people.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the an... | 5C |
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"content": "How many sets $A$ satisfy $A \\cup \\{1, -1\\} = \\{1, 0, -1\\}$?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \"<an... | 5C |
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"content": "I1.4 Given that the difference between two 3-digit numbers $\\overline{x y z}$ and $\\overline{z y x}$ is $700-c$, where $x>z$. If $d$ is the greatest value of $x+z$, find the value of $d$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only wh... | 5C |
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"content": "If \n$$\\sum_{k=1}^{40} \\left( \\sqrt{1 + \\frac{1}{k^{2}} + \\frac{1}{(k + 1)^{2}}}\\right) = a + \\frac {b}{c}$$\nwhere $a, b, c \\in \\mathbb{N}, b < c, gcd(b,c) =1 ,$ then what is the value of $a+ b ?$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making i... | 5C |
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"content": "If \"$a > b$\", then \"$a^{3} > b^{3}$\" is a ______ proposition (Fill in: True or False).\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting statement as the ans... | 5C |
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"content": "If $(1+x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\\cdots+a_{5}x^{5}$, then $a_{1}+a_{2}+\\cdots+a_{5}=$ ______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation o... | 5C |
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"content": "If $(x^2-4) + (x^2+3x+2)i$ is a pure imaginary number, then the value of the real number $x$ is \\_\\_\\_\\_\\_\\_.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resul... | 5C |
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"content": "If $A\\div 2ab=1-4a^{2}$, then the polynomial $A$ is equal to ______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the answer in the \... | 5C |
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"content": "If \\\\((1+x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\\ldots+a_{5}x^{5}\\\\), then \\\\(a_{1}+a_{2}+\\ldots+a_{5}=\\\\) \\_\\_\\_\\_\\_\\_.",
"role": "user"
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"content": "<think>\n<sample>\nTo find the value of \\(a_1 + a_2 + \\ldots + a_5\\) for the expansion of \\((1+x)^5\\), we start by con... | 5C |
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"content": "If the coefficient of $x^3$ in the expansion of $(ax-1)^5$ is 80, then the value of the real number $a$ is.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equ... | 5C |
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"content": "If the constant term in the expansion of $(2x+ \\frac {a}{x})^{4}$ ($a>0$) is 96, then the real number $a$ equals \\_\\_\\_\\_\\_\\_.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and th... | 5C |
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"content": "If the function $f(x) = (x+a)^3$ holds true for any $x \\in \\mathbb{R}$ such that $f(1+x) = -f(1-x)$, find the value of $f(2) + f(-2)$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and... | 5C |
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"content": "If the geometric sequence $\\{a_n\\}$ satisfies $a_2+a_4=20, a_3+a_5=40$, then the common ratio $q=$_______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting eq... | 5C |
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"content": "If the line $y = x + t$ is tangent to the curve $y = e^x$, then $t = \\_\\_\\_\\_\\_\\_$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as... | 5C |
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"content": "If the radius of the base of a cone is $2$ cm, and the unfolded side of the cone forms a sector with a central angle of $120^{\\circ}$, then the length of the slant height of this cone is ______ $cm$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it cont... | 5C |
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"content": "If the system of equations $\\left\\{\\begin{array}{l}x+2y=4\\\\ 3x+ay=6\\end{array}\\right.$ has no solution, then the real number $a=$____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by ste... | 5C |
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"content": "If the value of the algebraic expression $2x^{2}+3x+7$ is $8$, then the value of the algebraic expression $2x^{2}+3x-7$ is ______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then ... | 5C |
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"content": "In 2015, Paul is 11 years old and his sister Anne is 14. How old will Anne be when Paul is 38 years old?\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equati... | 5C |
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"content": "Return your final response within <answer>...</answer>. In a can, there were red and green candies. Čeněk ate $\\frac{2}{5}$ of all the red candies, and Zuzka ate $\\frac{3}{5}$ of all the green candies. Now, the red candies make up $\\frac{3}{8}$ of all the candies in the can.\n\nWhat is the mini... | 5C |
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{
"content": "Return your final response within <answer>...</answer>. In a football championship, there are 16 teams that play against each other once. What is the minimum number of games that must be played so that among any three teams, there are two teams that have already played against each other?\n\nBegin... | 5C |
[
{
"content": "In a math and intelligence competition, the average score of the whole class is $90$ points. If Xiaohong scored $92$ points, which can be denoted as $+2$ points, then Xiaoming scored $85$ points, which can be denoted as ______ points.\n\nBegin your response with \"Rewritten Question: \" and by rew... | 5C |
[
{
"content": "In a math class, the teacher asked four students, A, B, C, and D, to solve a rational number operation problem. Who do you think got the correct answer?\n\nA: A\n\nB: B\n\nC: C\n\nD: D\n\nGiven calculations:\n\nA: $9-3^{2}\\div 8=0\\div 8=0$.\n\nB: $24-(4\\times 3^{2})=24-4\\times 6=0$.\n\nC: $\\l... | 5C |
[
{
"content": "In a reading room, there were originally 20 books on a bookshelf. It is specified that returning books is considered positive and borrowing books is considered negative. After two days, the borrowing situation is as follows: $(-3,+1)$, $(-1,+2)$. Then the number of books currently on the bookshelf... | 5C |
[
{
"content": "In order for the square root $\\sqrt{x-3}$ to be meaningful, the value of $x$ can be\n\nA: $0$\n\nB: $1$\n\nC: $2$\n\nD: $4$\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give y... | 5C |
[
{
"content": "In right triangle $\\triangle ABC$, $\\angle B=90^\\circ$, $AB=BC=3$, if $\\overrightarrow {BD}$$=2 $$\\overrightarrow {DC}$, point $M$ is the midpoint of line segment $AD$, then the value of $\\overrightarrow {AM}$$\\cdot \\overrightarrow {BC}$ is $\\_\\_\\_\\_\\_\\_$.",
"role": "user"
},
... | 5C |
[
{
"content": "In the Cartesian coordinate system, if point $A(3,-2)$ is translated $2$ units to the right, then the coordinates of the corresponding point $A'$ are ____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then thi... | 5C |
[
{
"content": "In the arithmetic sequence $\\{a_n\\}$, if $a_1+a_2+a_3+a_4=30$, then $a_2+a_3=$ .\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equation or number as the... | 5C |
[
{
"content": "In the arithmetic sequence $\\{a_{n}\\}$, it is known that $a_{4}+a_{8}=16$. Find the sum of the first $11$ terms, $S_{11}=$ ___.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then g... | 5C |
[
{
"content": "In the binomial expansion of $(a+b)^n$, if the sum of the binomial coefficients of the odd terms is 128, then the maximum value of the binomial coefficients is __________ (Answer in numerals).\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only... | 5C |
[
{
"content": "In the expansion of $(x^{3}+\\frac{1}{x})^{12}$, the coefficient of the term containing $\\frac{1}{{{x^4}}}$ is ____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your res... | 5C |
[
{
"content": "In the positive term geometric sequence $\\\\{a\\_n\\\\}$, it is known that $a\\_1a\\_2a\\_3=4$, $a\\_4a\\_5a\\_6=12$, $a\\_{n-1}a\\_n\\_a\\_{n+1}=324$, then $n=$____.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve ... | 5C |
[
{
"content": "In the three-dimensional Cartesian coordinate system $O-xyz$, a normal vector of the plane $OAB$ is $n=(2, -2, 1)$. Given the point $P(-1, 3, 2)$, the distance $d$ from point $P$ to the plane $OAB$ is ______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making... | 5C |
[
{
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $A=60^{\\circ}$, and the area of $\\triangle ABC$ is $\\frac{{3\\sqrt{3}}}{2}$. If $b+c=3\\sqrt{3}$, then $a=\\_\\_\\_\\_\\_\\_$.\n\nBegin your response with \"Rewritten Question: ... | 5C |
[
{
"content": "Let $$\\overrightarrow {a} = (2k+2, 4)$$, $$\\overrightarrow {b} = (k+1, 8)$$, if $$\\overrightarrow {a}$$ is parallel to $$\\overrightarrow {b}$$, then the value of $k$ is \\_\\_\\_\\_\\_\\_.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only... | 5C |
[
{
"content": "Let $S_n$ be the sum of the first $n$ terms of the arithmetic sequence $\\{a_n\\}$, where $a_{12}=-8$ and $S_9=-9$. Find $S_{16}$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then ... | 5C |
[
{
"content": "Return your final response within <answer>...</answer>. Let $\\mathbf{a}$ and $\\mathbf{b}$ be unit vectors such that $\\mathbf{a} + 2 \\mathbf{b}$ and $5 \\mathbf{a} - 4 \\mathbf{b}$ are orthogonal. Find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees.\n\nNote: A unit vector is a v... | 5C |
[
{
"content": "Let $\\overrightarrow{e_{1}}$ and $\\overrightarrow{e_{2}}$ be two non-collinear, non-zero vectors. If $\\overrightarrow{AB} = 3\\overrightarrow{e_{1}} + k\\overrightarrow{e_{2}}$, $\\overrightarrow{BC} = 4\\overrightarrow{e_{1}} + \\overrightarrow{e_{2}}$, $\\overrightarrow{CD} = 8\\overrightarro... | 5C |
[
{
"content": "Let $\\triangle ABC$ be an equilateral triangle and $P$ a point on $\\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give yo... | 5C |
[
{
"content": "Let $f(x)$ be an odd function defined on $\\mathbb{R}$, and when $x > 0$, $f(x) = x^2 - 3$. Find $f(-2)$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your resulting equat... | 5C |
[
{
"content": "Let $f(x)$ be an odd function defined on $\\mathbb{R}$. If for $x \\leq 0$, $f(x) = 2x^2 - x$, then $f(1) =$ _______.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then think step by step and then give your res... | 5C |
[
{
"content": "Let $i$ be the imaginary unit. If the complex number $(2+i)(a-2i)$ is a pure imaginary number, then the value of the real number $a$ is $\\_\\_\\_\\_\\_\\_$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it, then t... | 5C |
[
{
"content": "Let ${a_n}$ be a sequence with the sum of its first $n$ terms denoted as $S_n$. If $S_2=7$ and $a_{n+1}=2S_n+1$ for $n\\in \\mathbb{N}^*$, then find the value of $S_5$.\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve... | 5C |
[
{
"content": "Return your final response within <answer>...</answer>. Let \\( a \\) and \\( b \\) be such that \\( a + b = 13 \\) and \\( ab = 41 \\). Calculate \\( a^3 + b^3 \\).\n\nBegin your response with \"Rewritten Question: \" and by rewriting the question making it contain only what is needed to solve it... | 5C |
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