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"content": "If the equation $(5x+ \\frac {5}{x})-|4x- \\frac {4}{x}|=m$ has exactly four distinct real roots in $(0,+\\infty)$, then the range of the real number $m$ is \\_\\_\\_\\_\\_\\_.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Wh... |
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"content": "Given the function $f(x) = e^x(x-b) \\ (b \\in \\mathbb{R})$. If there exists $x \\in \\left[ \\frac{1}{2},2 \\right]$ such that $f(x) + xf''(x) > 0$, then the range of the real number $b$ is $(\\ )$.\nA: $(-\\infty, \\frac{5}{6} )$\nB: $(-\\infty, \\frac{8}{3} )$\nC: $(- \\frac{3}{2} , \\frac{5}{... |
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"content": "Given that the initial side of angle $\\alpha$ coincides with the positive half of the x-axis, and its terminal side lies on the ray defined by $3x-4y=0$ (where $x<0$), find the value of $\\sin\\alpha - \\cos\\alpha$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"... |
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"content": "Calculate: \n\\\\((1) \\sqrt{48}+ \\sqrt{8}- \\sqrt{18}- \\sqrt{12} \\\\) \\\\((2) 2( \\sqrt{2}+ \\sqrt{3})-( \\sqrt{3}- \\sqrt{2})^{2} \\\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "This problem examines the ... |
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"content": "Let $f(x) = x^3 - 1$ and $g(x) = 3x^2 + 3x + 1$. What is $g(f(-3))$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
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"content": "First, compute $f(-3)$:\n$f(-3) = (-3)^3 - 1 = -27 - 1 = -28$.\nNext, substitute $-28$ into $g(x)$:\n$g(-28) =... |
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"content": "Which of the following statements about algorithms and the three types of logical structures (sequential structure, conditional structure, loop structure) is correct?\nA: An algorithm can contain at most two types of logical structures.\nB: If the algorithms for the same problem are different, the... |
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"content": "Selene has 120 cards numbered from 1 to 120, inclusive, and she places them in a box. Selene then chooses a card from the box at random. What is the probability that the number on the card she chooses is a multiple of 2, 4, or 5? Express your answer as a common fraction.\nPlease reason step by ste... |
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"content": "What is the area enclosed by the region defined by the equation $x^2 + y^2 - 8x + 6y = -9$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "1. **Complete the square for $x$:** \n \\( x^2 - 8x \\rightarrow (x^2 - 8x + 16) - 1... |
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"content": "Given the function $f(x)=\\ln x+ \\frac{1-x}{ax}$, where $a$ is a positive constant.\n(1) Find the range of values for $a$ such that the function $f(x)$ is monotonically increasing in the interval $[1,+\\infty)$;\n(2) Find the minimum value of the function $f(x)$ in the interval $[1,2]$.\nPlease r... |
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"content": "Let $U={1,2,...,100}$. For the sequence ${a_n}$ ($n∈N^*$) and the subset $T$ of $U$, if $T=∅$, define $S_T=0$; if $T={t_1,t_2,...,t_k}$, define $S_T=a_{t_1}+a_{t_2}+...+a_{t_k}$. For example, when $T={1,3,66}$, $S_T=a_1+a_3+a_66$. Now suppose ${a_n}$ ($n∈N^{*}$) is a geometric sequence with a comm... |
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"content": "Buratino calculated the time accurately and left Papa Carlo's house at 13:40 to reach the Field of Miracles and plant 4 coins exactly at sunset. If he had walked $25 \\%$ faster, he would have arrived at the Field of Miracles 1.5 hours earlier and waited. At what time did Buratino arrive at the Fi... |
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"content": "Given five distinct real numbers, consider all their pairwise sums. The smallest two sums are 20 and 200, while the largest two sums are 2014 and 2000. Determine the range of the smallest number, \\( a \\), among these five real numbers.\nPlease reason step by step, and put your final answer withi... |
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"content": "Let \\( C \\) and \\( D \\) be two distinct points on a semicircle of diameter \\([A B]\\). The lines \\((A C)\\) and \\((B D)\\) intersect at \\( F \\), and the lines \\((A D)\\) and \\((B C)\\) intersect at \\( E \\). Show that the midpoints of segments \\([A B]\\), \\([C D]\\), and \\([E F]\\) ... |
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"content": "The expression $2y^2 - 5y - 12$ can be written as $(2y + a)(y + b),$ where $a$ and $b$ are integers. What is $a - b$?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
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"content": "We start by expressing the quadratic $2y^2 - 5y - 12$ in the ... |
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"content": "Let set $U = \\{ x \\mid x \\text{ is a positive integer less than } 9\\}$, set $A = \\{1, 2, 3\\}$, and set $B = \\{3, 4, 5, 6\\}$. Then the intersection of the complements of $A$ and $B$ with respect to $U$, denoted $\\complement_U A \\cap \\complement_U B$, equals to ( )\n\nA: $\\{3\\}$ \nB: $... |
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"content": "Simplify first, then evaluate: $3(a^{2}-ab+7)-2(3ab-a^{2}+1)+3$, where $a=2$ and $b=\\frac{1}{3}$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
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"content": "To simplify and then evaluate the given expression $3(a^{2}-ab+7)-2(3ab-a^{2}+1)... |
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"content": "Which of the following statements is correct? ( )\nA: The square of number $a$ is positive.\nB: The absolute value of number $a$ is positive.\nC: The square root of 16 equals 4.\nD: 3 is the square root of 9.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "... |
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"content": "Given that the graph of a linear function passes through the points (3, 5) and (-4, -9),\n(1) Find the equation of this linear function;\n(2) Determine whether the point (-1, -3) is on the graph of this linear function.\nPlease reason step by step, and put your final answer within \\boxed{}.",
... |
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"content": "Consider the function \\( g(x) = \\log \\left(\\frac{2+x}{2-x}\\right) \\) for \\( -2 < x < 2 \\). Evaluate \\( g\\left(\\frac{4x+x^2}{2+2x}\\right) \\) in terms of \\( g(x) \\).\nA) \\( g(x) \\)\nB) \\( 2g(x) \\)\nC) \\( 3g(x) \\)\nD) \\( 4g(x) \\)\nE) \\( \\left[g(x)\\right]^2 \\)\nPlease reason... |
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"content": "On a board, nine quadratic trinomials are written: \\(x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \\ldots, x^{2}+a_{9} x+b_{9}\\). It is known that the sequences \\(a_{1}, a_{2}, \\ldots, a_{9}\\) and \\(b_{1}, b_{2}, \\ldots, b_{9}\\) are arithmetic progressions. It turns out that the sum of all ni... |
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"content": "A list of five positive integers has mean $15$ and range $20$. The mode and median are both $10$. Determine how many different values are possible for the second largest element of the list.\n$\\mathrm{(A) \\ 3 } \\qquad \\mathrm{(B) \\ 5 } \\qquad \\mathrm{(C) \\ 6 } \\qquad \\mathrm{(D) \\ 8 } \... |
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"content": "Ray writes a two-digit number. He sees that the number exceeds 4 times the sum of its digits by a certain value. If the number is increased by 18, the result is the same as the number formed by reversing the digits. The number is 57. What is the value by which the number exceeds 4 times the sum of... |
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"content": "Find the length of the longest pole that can be placed in a room 12 m long, 8 m broad, and 9 m high.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "The longest pole that can be placed in a room would be the diagonal of the roo... |
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"content": "$a=0$ is the condition for the complex number $z=a+bi$ ($a, b \\in \\mathbb{R}$) to be a purely imaginary number ( ).\n\nA: Sufficient but not necessary condition \nB: Necessary but not sufficient condition \nC: Necessary and sufficient condition \nD: Neither sufficient nor necessary condition\... |
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"content": "Let $a$ and $b$ be the roots of the equation $x^2 - 6x + 8 = 0$. Compute:\n\\[a^2 + a^5 b^3 + a^3 b^5 + b^2.\\]\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "First, by Vieta's formulas:\n- $a + b = 6$\n- $ab = 8$\n\nCompute $... |
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"content": "Find the variance of the discrete random variable $X$ that is distributed according to the Poisson distribution:\n\n$$\n\\begin{array}{ccccccc}\nX & 0 & 1 & 2 & \\cdots & k & \\cdots \\\\\nP & \\mathrm{e}^{-\\lambda} & \\lambda \\mathrm{e}^{-\\lambda} / 1! & \\lambda^{2} \\mathrm{e}^{-\\lambda} / ... |
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"content": "Two stations p and q are 200 km apart on a straight track. One train starts from p at 7 a.m. and travels towards q at 20 kmph. Another train starts from q at a certain time and travels towards p at a speed of 25 kmph. They meet at 12. At what time did the second train start from station q?\nPlease... |
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"content": "Given that $m$ and $n$ are opposites, $a$ and $b$ are negative reciprocals, and the absolute value of $x$ equals 3, then the value of the expression $x^3 - (1+m+n+ab)x^2 + (m+n)x^{2004} + (ab)^{2005}$ equals to.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role":... |
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"content": "Tom and Tim both brought 4, six-sided dice to school. How many total sides are there?\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "To solve the problem, we start by calculating the total number of dice Tom and Tim brought to... |
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"content": "For all positive integer $n$ , define $f_n(x)$ such that $f_n(x) = \\sum_{k=1}^n{|x - k|}$ .\nDetermine all solution from the inequality $f_n(x) < 41$ for all positive $2$ -digit integers $n$ (in decimal notation).\nPlease reason step by step, and put your final answer within \\boxed{}.",... |
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"content": "Evaluate $\\cos 420^{\\circ} + \\sin 330^{\\circ}$.\n\nA: $1$\n\nB: $-1$\n\nC: $\\dfrac{1}{2}$\n\nD: $0$\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "We have $\\cos 420^{\\circ} + \\sin 330^{\\circ} = \\cos (360^{\\circ} + 6... |
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"content": "Prove that\n\n$$\n\\int_{0}^{\\pi}|\\ln \\sin x| \\, dx \\leq \\frac{\\pi^{3}}{12}\n$$\n\nwhere $\\ln$ denotes the natural logarithm with base $e = 2.71\\ldots$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "\n1. **Understand... |
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"content": "Prove that in a right-angled triangle, the distance between the incenter and the circumcenter is at least $(\\sqrt{2}-1)$ times the radius of the circumcircle.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "\n1. **Identificati... |
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"content": "Given the polynomial expression \\[2a(b - c)^3 + 3b(c - a)^3 + 2c(a - b)^3,\\] it is known to be factorable into the form $(a - b)(b - c)(c - a) p(a,b,c)$. Find the polynomial $p(a,b,c)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"c... |
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"content": "Two pipes A and B can fill a tank in 10 hours and some hours respectively. If both the pipes are opened simultaneously, the tank will be filled in approximately 6 hours. How much time will pipe B take to fill the tank alone?\nPlease reason step by step, and put your final answer within \\boxed{}."... |
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"content": "The line $y=kx-2$ intersects the parabola $y^2=8x$ at points A and B, and the x-coordinate of the midpoint of AB is 2. Find the value of $k$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Since the line $y=kx-2$ and the parab... |
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"content": "In the book \"Nine Chapters on the Mathematical Art,\" there is a mathematical problem stated as follows: During the Qingming Festival outing to the park, a total of eight boats were taken. Each large boat holds six people, each small boat holds four people. There were thirty-eight students in tot... |
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"content": "Find the maximum value of $3 \\cos x + 4 \\sin x$ over all angles $x$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "We start by identifying that the expression $3 \\cos x + 4 \\sin x$ can be rewritten in the form of $R \\cos... |
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"content": "Calculate the volume of the body bounded above by the surface \\(z = xy^2\\) and below by the rectangle \\(0 \\leq x \\leq 1\\), \\(0 \\leq y \\leq 2\\).\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
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"content": "\n1. **Identify the probl... |
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"content": "\nVasya cut a triangle out of cardboard and numbered its vertices with the digits $1, 2, 3$. It turned out that if Vasya rotates the triangle 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya rotates the... |
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"content": "Harry, Tom, and Dick start a 150-mile journey together. Tom and Dick initially travel by car at 30 mph, while Harry walks at 4 mph. After 50 miles, Harry continues walking, and Tom goes back to pick up Dick who had started walking at 4 mph after 50 miles. Tom picks up Dick and they continue at 30 ... |
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"content": "Jackson is making dinner. He makes a salad out of lettuce (50 calories), carrots (twice the calories of the lettuce) and dressing (210 calories). He also makes a pizza with 600 calories for the crust, 1/3 the crust's calories for the pepperoni, and 400 calories for the cheese. If Jackson eats some... |
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"content": "A unit has 200 employees. Now, 40 employees need to be selected as a sample using the systematic sampling method. All employees are randomly numbered from 1 to 200 and evenly divided into 40 groups according to their numbers in order (1-5, 6-10, ..., 196-200). If the number drawn from the 5th grou... |
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"content": "Given sets $A=\\{x|(x-2)(x-(a+1)) < 0\\}$ and $B=\\left\\{x\\middle| \\frac{x-2a}{x-(a^{2}+1)} < 0\\right\\}$. Find the range of real values of $a$ for which $B \\subseteq A$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Whe... |
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"content": "At North High School, 2200 students have choices among basketball, football, and baseball. At South Academy, there are 2600 students. In the two schools combined, the percentage of students who prefer basketball is:\n$\\text{(A)}\\ 26\\% \\qquad \\text{(B)}\\ 28\\% \\qquad \\text{(C)}\\ 30\\% \\qq... |
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"content": "Consider two equiangular polygons $P_1$ and $P_2$. Each angle of $P_1$ is $x$ degrees, and each angle of $P_2$ is $\\frac{x}{2}$ degrees. Determine the number of possibilities for the integer number of sides $n_1$ and $n_2$ for polygons $P_1$ and $P_2$, respectively.\nA) 0\nB) 1\nC) 2\nD) 3\nE) Mo... |
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"content": "Given that $α\\in\\mathbb{R}$ and $\\sin α + 2\\cos α = \\frac{\\sqrt{10}}{2}$, find the value of $\\tan α$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Since we are given that $\\sin α + 2\\cos α = \\frac{\\sqrt{10}}{2}$, ... |
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"content": "A team of 60 workers constructed a bridge in 5 days working at the same efficiency. If the subsequent team worked at 80% efficiency and consisted of 40 workers, how many days would it have taken this team to complete an identical bridge?\nPlease reason step by step, and put your final answer withi... |
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"content": "The slope angle of the tangent line to the graph of the function $f(x) = \\ln(x^2+1)$ at the point $(1, f(1))$ is \nA: 0\nB: $\\frac{\\pi}{2}$\nC: $\\frac{\\pi}{3}$\nD: $\\frac{\\pi}{4}$\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"con... |
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"content": "In the given figure, semicircles with centers at points \\( A \\) and \\( B \\) and radii 2 and 1, respectively, are inscribed in a larger semicircle with diameter \\( JK \\). The two smaller semicircles are also tangent to each other. \\( \\odot P \\) is a circle that is tangent to both smaller s... |
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"content": "Expand $(3t^3 + 2t^2 - 4t + 3)(-2t^2 + 3t - 4)$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Using the distributive property, the expansion is given by:\n\\[\n(3t^3 + 2t^2 - 4t + 3)(-2t^2 + 3t - 4) = 3t^3(-2t^2 + 3t - 4) + ... |
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"content": "Find the remainder when $$123456 + 123457 + 123458 + 123459 + 123460 + 123461$$ is divided by 11.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "First, we reduce each number modulo 11:\n\\[\n123456 \\equiv 1, \\quad 123457 \\e... |
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"content": "Among all the roots of\n\\[ z^6 - z^4 + z^2 - 1 = 0, \\]\nthe maximum imaginary part of a root can be expressed as $\\cos \\phi$, where $-90^\\circ \\le \\phi \\le 90^\\circ$. Find $\\phi$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"... |
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"content": "The tower in King Arthur's castle is topped with a roof that represents a triangular pyramid in which all plane angles at the apex are right angles. The three roof slopes are painted in different colors. The red roof slope is inclined to the horizontal at an angle $\\alpha$, and the blue slope at ... |
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"content": "m = { - 9 , - 7 , - 5 , - 3 , 0, 5 , 7 } \nt = { - 8 , - 6 , - 2 , 0, 3 , 4 , 6 , 7 }\n\nIf an integer is to be randomly selected from set m above and an integer is to be randomly selected from set t above, what is the probability that the product of the two integers will be negative?\nPlease reas... |
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"content": "Let the sequence \\\\(\\{a_n\\}\\) have a sum of the first \\\\(n\\\\) terms denoted by \\\\(S_n\\\\), and it is known that \\\\(S_n = 2a_n - 2^{n+1} (n \\in \\mathbb{N}^*)\\).\n\\\\((1)\\\\) Find the general formula for the sequence \\\\(\\{a_n\\}\\).\n\\\\((2)\\\\) Let \\\\(b_n = \\log_{\\frac{a... |
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"content": "Given sets $M=\\{2, 4\\}$, $N=\\{1, 2\\}$, and $P=\\{x|x= \\frac{a}{b}, a\\in M, b\\in N\\}$, then the number of subsets of set $P$ is ( )\n\nA: 3 \nB: 4 \nC: 8 \nD: 16\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Since ... |
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"content": "Let \\( S_{n} \\) be the number of sequences \\((a_{1}, a_{2}, \\ldots, a_{n})\\), where \\( a_{i} \\in \\{0,1\\} \\), in which no six consecutive blocks are equal. Prove that \\( S_{m} \\rightarrow \\infty \\) as \\( n \\rightarrow \\infty \\).\nPlease reason step by step, and put your final answ... |
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"content": "If $(X-2)^8 = a + a_1(x-1) + \\ldots + a_8(x-1)^8$, then the value of $\\left(a_2 + a_4 + \\ldots + a_8\\right)^2 - \\left(a_1 + a_3 + \\ldots + a_7\\right)^2$ is (Answer in digits).\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content... |
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"content": "Every day, Lou works out by running three miles on a circular track that is one-quarter of a mile long and has a constant incline of 5%. His wife, Rosie, also runs on the same track at the same time as her husband, but with an adjusted pace. Due to the incline, Lou and Rosie slow down: Lou maintai... |
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"content": "Jim spends 8 hours scuba diving. In that time he finds a treasure chest with 100 gold coins in it. He also finds some smaller bags that have half as much gold each. He finds 25 gold coins per hour. How many smaller bags did he find?\nPlease reason step by step, and put your final answer within \\b... |
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"content": "The zero of the function $f(x) = 1 - \\log_{2}x$ is ( ) \nA: $(1, 1)$ \nB: $1$ \nC: $(2, 0)$ \nD: $2$\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Solution: Let $f(x) = 1 - \\log_{2}x = 0$, we can get $x = 2$. \nTheref... |
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"content": "A student was asked to find 4 / 5 of a number. But the student divided the number by 4 / 5, thus the student got a certain amount more than the correct answer. The number is 60.00000000000002. How much more did the student get than the correct answer?\nPlease reason step by step, and put your fina... |
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"content": "The price of rice falls by 20%. With the money that was sufficient to buy a certain amount of rice previously, now 25 kg of rice can be bought. How much rice could be bought previously with the same amount of money?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"ro... |
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"content": "The sides of an isosceles triangle are $\\sin x,$ $\\sin x,$ and $\\sin 5x,$ and its vertex angle is $2x.$ (All angle measurements are in degrees.) Enter all possible values of $x,$ separated by commas.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"... |
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"content": "Let $z$ be a complex number such that\n\\[|z - 8| + |z + 6i| = 17.\\] Find the smallest possible value of $|z|$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "First, by the Triangle Inequality,\n\\[ |z - 8| + |z + 6i| = |z - ... |
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"content": "Among the following numbers, the positive integers are:\n\nA: $3$\n\nB: $2.1$\n\nC: $0$\n\nD: $-2$\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "To determine which of the given options are positive integers, we evaluate each ... |
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"content": "In the plane rectangular coordinate system $xOy$, it is known that the initial side of angle $\\alpha$ is the non-negative half-axis of the $x$-axis, and the terminal side passes through the point $P(-1,2)$. Then $\\cos(\\pi -\\alpha)=$\n\nA: $\\frac{\\sqrt{5}}{5}$\n\nB: $\\frac{2\\sqrt{5}}{5}$\n\... |
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"content": "How many positive integers $n$ satisfy \\[\\dfrac{n+800}{80} = \\lfloor \\sqrt{n} \\rfloor?\\]\n(Recall that $\\lfloor x \\rfloor$ is the greatest integer not exceeding $x$.)\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "1. *... |
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"content": "Given the sets $A=\\{-2,-1,0,1,2\\}$ and $B=\\{x|x^{2}\\leqslant 1\\}$, find the intersection $A\\cap B=$ .\n\nA: $\\{-2,-1,0,1\\}$\n\nB: $\\{-1,1\\}$\n\nC: $\\{-1,0\\}$\n\nD: $\\{-1,0,1\\}$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
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... |
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"content": "Let $a_1, a_2, \\ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \\ldots ,2007$ . Define an **operation** to be 22 consecutive applications of the following steps on the sequence $S$ : on $i$ -th step, choose a number from the ... |
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"content": "One fourth of one third of two fifth of a number is 16. What is the value of two fifth of that number?\nPlease reason step by step, and put your final answer within \\boxed{}.",
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"content": "Let's call the number we're looking for \"N\". \n\nAccording to the problem,... |
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"content": "It is known that the length of the segment representing the product of two segments depends on the choice of the unit segment. How does the length of the product segment change if the unit segment is halved?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "us... |
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"content": "A lecture was given for 90 minutes to an audience of 200 people. Thirty percent of the audience listened to the full lecture, while fifteen percent were not attentive throughout the lecture. Of the remaining audience, 40% listened to half of the lecture, and the rest listened to three quarters of ... |
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"content": "Edward spent $250 on clothes, then spent 35% of his remaining money on food. After buying food, he used 50% of the remaining money to purchase electronics, and at that point had $200 left. How much did Edward have before he spent his money on clothes, food, and electronics?\nPlease reason step by ... |
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"content": "A firecracker was thrown vertically upward with a speed of \\(20 \\text{ m/s}\\). One second after the flight began, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of 48 m/s. Find the magnitude of the speed of the sec... |
[
{
"content": "Find the sum \\( S_{n} = \\sin \\alpha \\cdot \\sin ^{2} \\frac{\\alpha}{2} + 2 \\sin \\frac{\\alpha}{2} \\cdot \\sin ^{2} \\frac{\\alpha}{4} + \\cdots + 2^{n-1} \\cdot \\sin \\frac{\\alpha}{2^{n-1}} \\cdot \\sin ^{2} \\frac{\\alpha}{2^{n}} \\).\nPlease reason step by step, and put your final answ... |
[
{
"content": "Given the function $f(x)=ax^{2}+bx+c$, where $a\\in\\mathbb{N}^{*}$, $b\\in\\mathbb{N}$, $c\\in\\mathbb{Z}$.\n\n$(1)$ If $b > 2a$, and $f(\\sin x)$ ($x\\in\\mathbb{R}$) has a maximum value of $2$ and a minimum value of $-4$, find the minimum value of the function $f(x)$.\n\n$(2)$ For the $f(x)$ in... |
[
{
"content": "Given that the function $y=f(x)$ has a domain of $(0,+\\infty)$, and when $x > 1$, $f(x) > 0$. For any $x$, $y\\in(0,+\\infty)$, the equation $f(x)+f(y)=f(xy)$ holds. If the sequence $\\{a_n\\}$ satisfies $a_1=f(1)$, and $f(a_{n+1})=f(2a_n+1)$ for $n\\in\\mathbb{N}^*$, then the value of $a_{2017}$... |
[
{
"content": "The sum of the first and third of three consecutive integers is 118. What is the value of the second integer?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Let's denote the second integer as $n$. Given that these are consecut... |
[
{
"content": "Let $a,$ $b,$ $c$ be three terms in an arithmetic series where all terms are positive and the common difference is $d$. If $abc = 125$, find the smallest possible value of $b$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Gi... |
[
{
"content": "The graph of the function $y = a^{x+1} - 1$ ($a > 0$ and $a \\neq 1$) must pass through the point $\\_\\_\\_\\_\\_\\_$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "For the function $y = a^{x+1} - 1$ ($a > 0$ and $a \\neq 1$... |
[
{
"content": "In $\\triangle ABC$, point $D$ lies on segment $BC$ and does not coincide with $B$ or $C$. The circumcircle of $\\triangle ABD$ intersects segment $AC$ at another point $E$, and the circumcircle of $\\triangle ACD$ intersects segment $AB$ at another point $F$. Let $A'$ be the reflection of point $... |
[
{
"content": "Dana has 15 more pencils than Jayden, who has twice as much as Marcus. How many more pencils does Dana have than Marcus, if Jayden has 20 pencils?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "If Jayden has 20 pencils, and Da... |
[
{
"content": "After spending Rs. 5000 on rent, Rs. 1500 on milk, Rs. 4500 on groceries, Rs. 2500 on children's education, some amount on petrol, and Rs. 5650 on miscellaneous expenses, Mr. Kishore saved 10% of his monthly salary. He saved Rs. 2350. How much did he spend on petrol?\nPlease reason step by step, a... |
[
{
"content": "Given a circle O with equation $x^2+y^2=13$. A perpendicular line to the y-axis is drawn through any point P on circle O, with Q as the foot of the perpendicular. Find the equation of the locus of the midpoint M of segment PQ.\nPlease reason step by step, and put your final answer within \\boxed{}... |
[
{
"content": "If $7$ lunks can be traded for $4$ kunks, and $3$ kunks will buy $5$ apples, how many lunks are needed to purchase two dozen apples?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "1. First, determine the number of kunks needed... |
[
{
"content": "Given that the inequality $x^2 - ax + 2a > 0$ is always true for all $x$ in $\\mathbb{R}$, find the range of the real number $a$.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "Since the inequality $x^2 - ax + 2a > 0$ is alway... |
[
{
"content": "On the sides of a convex centrally symmetric hexagon \\( A B C D E F \\), equilateral triangles are constructed externally. Prove that the midpoints of the segments connecting the vertices of adjacent triangles form a regular hexagon.\nPlease reason step by step, and put your final answer within \... |
[
{
"content": "There are 50 boys and 50 girls lined up in some order. In this row, there exists exactly one group of 30 consecutive children where the number of boys and girls is equal. Prove that there exists a group of 70 consecutive children where the number of boys and girls is also equal.\nPlease reason ste... |
[
{
"content": "Given the sequence $\\{a_n\\}$ with the sum of the first $n$ terms $S_n=3^n-2$, find the general formula for the term $a_n=$ ______.\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "To solve, we know the sum of the first $n$ ter... |
[
{
"content": "In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \\in \\{70,85,88,90,98,100\\}$ and it satisfies $f(1)<f(2) \\leq f(3)<f(4)$, then the total number of possible combinations of test scores for these 4 students is \\_\\_\\_\\_\\_... |
[
{
"content": "A person writes \\( n \\) letters for \\( n \\) individuals, each receiving one letter, and prepares \\( n \\) envelopes labeled with the recipients' names and addresses. How many different ways are there to place the letters into the envelopes (one letter per envelope) such that no letter is plac... |
[
{
"content": "If the ellipse $\\dfrac {x^{2}}{a^{2}}+y^{2}=1$ passes through the point $P(1, \\dfrac {\\sqrt {6}}{3})$, then the eccentricity $e$ of the ellipse is\nA: $\\dfrac {\\sqrt {3}}{2}$\nB: $\\sqrt {3}-1$\nC: $\\dfrac {\\sqrt {3}}{3}$\nD: $\\dfrac {\\sqrt {6}}{3}$\nPlease reason step by step, and put yo... |
[
{
"content": "A car drives 60 miles on local roads at a certain speed, and 120 miles on the highway at 60 mph. The average speed of the entire trip is 36 mph. What is the speed of the car on local roads?\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"... |
[
{
"content": "Let $a, b, c$ be positive integers whose greatest common divisor is $1$ . Determine whether there always exists a positive integer $n$ such that, for every positive integer $k$ , the number $2^n$ is not a divisor of $a^k+b^k+c^k$ .\nPlease reason step by step, and put your final answer wi... |
[
{
"content": "Evaluate the infinite geometric series: $$\\frac{1}{4} + \\frac{1}{12} + \\frac{1}{36} + \\frac{1}{108} + \\dots$$\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "1. **Identify the first term and common ratio**: The first term ... |
[
{
"content": "During the baseball season, Judy had $50$ hits. Among her hits were $2$ home runs, $3$ triples, and $8$ doubles. The rest of her hits were singles. What percent of her hits were singles?\nA) 68%\nB) 70%\nC) 74%\nD) 76%\nE) 78%\nPlease reason step by step, and put your final answer within \\boxed{}... |
[
{
"content": "A jacket is originally priced at \\$50 and is on sale for 15% off. John has a special member card that grants an additional 30% discount on the already reduced price. What single percent discount would result in the same final price as these two successive discounts?\nPlease reason step by step, a... |
[
{
"content": "Given $x$ and $y$ as positive real numbers such that $x \\leq y$, show that $\\frac{1}{y^{2}} \\leq \\frac{1}{x^{2}}\nPlease reason step by step, and put your final answer within \\boxed{}.",
"role": "user"
},
{
"content": "\n1. **Given**: We have two positive real numbers \\(x\\) and ... |
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