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[
{
"content": "Given $$\\log_{a} \\frac {3}{5} < 1$$, the range of values for $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
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"content": "Given the inequality $\\log_{a} \\frac{3}{5} < 1$, we need to determine the values of $a$ th... |
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"content": "Given $\\sin \\left(\\alpha +\\frac{\\pi }{6}\\right)=\\frac{1}{3}$, then the value of $\\cos \\left(\\frac{\\pi }{3}-\\alpha \\right)$ is _______. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "We need to find the value of $\\co... |
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"content": "Distribute 7 doctors, including Doctors A and B, into three medical teams with one team having 3 members and the other two teams having 2 members each. In how many ways can this be done so that Doctors A and B are not in the same team? Let's think step by step and output the final answer within \\... |
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"content": "A student was determining the interval $D$ where the root $x_0$ of the equation $x^3 = -x + 1$ lies. They set the function $f(x) = x^3 + x - 1$, calculated $f(-1) < 0$, and $f(1) > 0$. In the process, they used the \"bisection method\" and chose 3 more values, which are $x_1$, $x_2$, $x_3$, to fin... |
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"content": "Given the sets $S=\\{ x\\mid | x-2 | > 3 \\}$ and $T=\\{ x\\mid a < x < a+8 \\}$, and their union $S\\cup T=R$, find the range of values for $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To find the range of values for $a$ ... |
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"content": "What is the largest integer \\( y \\) such that \\( \\frac{y}{4} + \\frac{3}{7} < \\frac{2}{3} \\)? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve the problem \\(\\frac{y}{4} + \\frac{3}{7} < \\frac{2}{3}\\), we will fir... |
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"content": "Calculate the positive difference between the two solutions of the equation $|x + 8| = \\sqrt{256}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve the equation \\( |x + 8| = \\sqrt{256} \\), we need to consider the posi... |
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"content": "What is the least prime factor of $5^4 - 5^2$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "First, we need to simplify the expression \\(5^4 - 5^2\\). We can factor out \\(5^2\\) from the expression:\n\n\\[5^4 - 5^2 = 5^2(5^2 -... |
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"content": "What is the distance between the points $(1, 2)$ and $(6, 5)$? Express your answer in simplest radical form. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To find the distance between the points $(1, 2)$ and $(6, 5)$, we can use... |
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"content": "Given the universal set U = {x ∈ Z | 0 ≤ x ≤ 10}, and sets A = {1, 2, 4, 5, 9}, B = {4, 6, 7, 8, 10}, and C = {3, 5, 7}. Find:\n1. A ∪ B\n2. (A ∩ B) ∩ C\n3. (∁<sub>U</sub>A) ∩ (∁<sub>U</sub>B) Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"... |
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"content": "Find the sum of all positive real solutions $x$ to the equation \\[2\\cos2x \\left(\\cos2x - \\cos{\\left( \\frac{1008\\pi^2}{x} \\right)}\\right) = \\cos4x - 1,\\] where $x$ is measured in radians. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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{... |
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"content": "Solve $\\log_8 x + \\log_2 x^3 = 9$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve \\(\\log_8 x + \\log_2 x^3 = 9\\), let's start by expressing all logarithms with the same base.\n\n1. Change \\(\\log_8 x\\) to base 2.\... |
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"content": "If $a$, $b\\in \\{-1,1,2\\}$, then the probability that the function $f\\left(x\\right)=ax^{2}+2x+b$ has a zero point is ______. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve the problem, we need to first understand whe... |
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"content": "In parallelogram $PQRS$, angle $Q$ is trisected by $\\overline{QT}$ and $\\overline{QU}$, where $T$ is on $\\overline{PS}$, $U$ is on $\\overline{PR}$, $ST=3$, and $PU=9$. $PQRS$ is a rhombus, and the angle trisected is $120^\\circ$. Calculate the area of $PQRS$. Let's think step by step and outpu... |
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"content": "Simplify first, then evaluate: $\\left(\\dfrac{a^{2}+1}{a}-2\\right)\\div \\dfrac{\\left(a+2\\right)\\left(a-1\\right)}{a^{2}+2a}$, where $-2\\leqslant a\\leqslant 2$. Choose an integer you like and substitute it to find the value. Let's think step by step and output the final answer within \\boxe... |
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"content": "The sum of the interior angles of a polygon is $1800^{\\circ}$, calculate the number of diagonals drawn from a vertex of this polygon. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "The sum of the interior angles of a polygon is ... |
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"content": "Given lines $l_{1}$: $\\sqrt {3}x-y+2=0$ and $l_{2}$: $3x+ \\sqrt {3}y-5=0$, determine the angle between lines $l_{1}$ and $l_{2}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To determine the angle between the lines \\( l_1 \... |
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"content": "A store owner bought 2000 pencils at $0.20 each. He incurs an additional cost of $0.05 per pencil for packaging. If he sells the pencils for $0.50 each, how many pencils must he sell to make a profit of exactly $200.00?\nA) 1250\nB) 1300\nC) 1350\nD) 1400\nE) 1450 Let's think step by step and outp... |
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"content": "Given that $2^{12} \\cdot 3^3 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To find the smallest possible value of \\( a + b \\), ... |
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"content": "Solve the inequality\n\n$$\n\\left(\\sqrt{x^{2}-4x+3}+1\\right) \\log_{2} \\frac{x}{5} + \\frac{1}{x}\\left(\\sqrt{8x-2x^{2}-6}+1\\right) \\leq 0\n$$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve the inequality \\((\\sq... |
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"content": "Given the original triangle, if each of its two including sides is doubled, determine the factor by which the area is multiplied. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "Given the original triangle, if each of its two incl... |
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"content": "Find the smallest constant M, so that for any real numbers $a_1, a_2, \\dots a_{2023} \\in [4, 6]$ and $b_1, b_2, \\dots b_{2023} \\in [9, 12] $ following inequality holds: $$ \\sqrt{a_1^2 + a_2^2 + \\dots + a_{2023}^2} \\cdot \\sqrt{b_1^2 + b_2^2 + \\dots + b_{2023}^2} \\leq M \\cdot \\lef... |
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"content": "The solution set of the inequality $\\dfrac{x}{x+1} < 0$ is . Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "The solution set of the inequality $\\dfrac{x}{x+1} < 0$ is, the sign changes every time we go past a zero:\n - The n... |
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"content": "An uncrossed belt runs without slack around two circular pulleys with radii of $10$ inches and $6$ inches. If the distance between the points of contact of the belt with the pulleys is $26$ inches, calculate the distance between the centers of the pulleys in inches. Let's think step by step and ou... |
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{
"content": "How many nonempty subsets of {1, 2, ..., 10} have the property that the sum of their largest element and smallest element is 11? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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{
"content": " To find the number of nonempty subsets \\( S \\) of ... |
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"content": "How many positive divisors of $180$ are not divisible by 3? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "First, factorise $180$ into its prime factors. \n\n \\begin{align*}\n 180 &= 2 \\times 90 \\\\\n &= 2 \\times 2 \\time... |
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"content": "What are the two smallest prime factors of $462$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "The only correct answer is \\boxed{\\dfrac{\\begin{tikzpicture}\n \\node at (1,1) [draw, circle] {\\textbf{(C)}};\n \\node (A)... |
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"content": "As shown in the diagram, two lines intersect with two circles at 9 points. Choose 4 points from these 9 points such that any 3 points among these 4 are neither collinear nor lie on the same circle. How many different selections can be made? Let's think step by step and output the final answer with... |
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"content": "Every day, Patrick the dog chews up one slipper from the available stock at home. There is a strict 0.5 probability that Patrick wants to chew on the left slipper and a 0.5 probability that he wants to chew on the right slipper. If the desired slipper is not available, Patrick gets upset. How many... |
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{
"content": "There is a uniformly growing grassland. If 20 cows are grazed, they will just finish eating all the grass in 60 days. If 30 cows are grazed, they will just finish eating all the grass in 35 days. Now, 6 cows are grazing on the grassland. After a month, 10 more cows are added. How many more days wi... |
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"content": "Calculate the value of $\\left(2^{0}-1+5^{2}-0\\right)^{-1} \\times 5$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "First, let's simplify the expression inside the parentheses:\n\n\\[\n2^0 - 1 + 5^2 - 0\n\\]\n\nWe know that \\... |
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"content": "For how many integer values of $n$ between 1 and 419 inclusive does the decimal representation of $\\frac{n}{420}$ terminate? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "The decimal representation of a fraction terminates if a... |
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"content": "Let \\(m\\) be a positive integer such that \\((m+2)! + (m+3)! = m! \\cdot 1080\\). Calculate the sum of the digits of \\(m\\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve the problem, we need to start by expressing t... |
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"content": "In a right-angled triangle $ABC$ with $\\angle ABC = 90^\\circ$, two of the altitudes have lengths $6$ and $18$. If the length of the third altitude is also an integer, what is the largest it can be? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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... |
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"content": "In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$ . If the right triangle has area $T$ and is inscribed in a circle of area $C$ , find $ab$ in terms of $T$ and $C$ . Let's think step by step and output the fi... |
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"content": "In a right triangle \\( ABC \\) with a right angle at vertex \\( B \\), median \\( BM \\) is drawn. The incircle of triangle \\( ABM \\) touches sides \\( AB \\) and \\( AM \\) at points \\( K \\) and \\( L \\), respectively. It is known that lines \\( KL \\) and \\( BM \\) are parallel. Find the ... |
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"content": "Find all values of the parameter \\( a \\) for which there exists a value of the parameter \\( b \\) such that the system\n\n$$\n\\left\\{\\begin{array}{l}\n\\arcsin \\left(\\frac{a-y}{3}\\right)=\\arcsin \\left(\\frac{4-x}{4}\\right) \\\\\nx^{2}+y^{2}-8 x-8 y=b\n\\end{array}\\right.\n$$\n\nhas ex... |
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"content": "Given the function $f(x) = x^{\\alpha}$ whose graph passes through the point $(2, \\frac{\\sqrt{2}}{2})$, calculate the value of $f(4)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "First, we need to determine the value of $\\a... |
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"content": "In the geometric sequence $\\{a\\_n\\}$, if $a\\_4$ and $a\\_8$ are the two roots of the equation $x^{2}+11x+9=0$, then the value of $a\\_6$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "In the geometric... |
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"content": "In a circle with a radius of 1.4, determine the distance from the center to a chord that subtends an arc of $120^{\\circ}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "In a circle with a radius of 1.4, let's determine the dist... |
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"content": "Given the expectation \\( m_{x}(t) = 3t^2 + 1 \\) of the random function \\( X(t) \\), find the expectation of the integral \\( Y(t) = \\int_{0}^t X(s) \\, \\mathrm{d}s \\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "The expe... |
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"content": "Find the greatest common factor of $7!$ and $8!$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "Let's begin by calculating $7!$ and $8!$.\n\n\\[\n7! = 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 5040\n\\]\n\n... |
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"content": "Determine how many integer values of $x$ satisfy $-100 < x^4 < 100$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To determine how many integer values of \\(x\\) satisfy the inequality \\(-100 < x^4 < 100\\), we need to analyze... |
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"content": "A line is defined by passing through vectors $\\mathbf{a}$ and $\\mathbf{b}.$ For a certain scalar \\(k\\), the vector\n\\[k \\mathbf{a} + \\frac{5}{7} \\mathbf{b}\\] is also on the line. Determine the value of \\(k\\). Let's think step by step and output the final answer within \\boxed{}",
"r... |
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"content": "Given the function $f(x)=|x+1|-|x-2|$.\n\n$(1)$ Find the solution set of the inequality $f(x) \\geqslant 1$;\n\n$(2)$ If the solution set of the inequality $f(x) \\geqslant x^2-x+m$ is non-empty, find the range of values for $m$. Let's think step by step and output the final answer within \\boxed{... |
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{
"content": "The maximum value of the function $f(x) = -x^2 + 4x - 1$ on the interval $[t, t+1]$ is denoted as $g(t)$. Find the maximum value of $g(t)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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{
"content": "To find the maximum value of the function ... |
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{
"content": "Given a rectangle with vertices at $(0,0)$, $(4,0)$, $(0,2)$, and $(4,2)$, and a slanted line extending from $(2,0)$ to $(4,4)$, determine the value of $d$ if the line $(d,0)$ to $(4,4)$ divides the entire region into two regions of equal area. Let's think step by step and output the final answer ... |
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{
"content": "Given that there are 10 screws in the box, 3 of which are defective, calculate the probability of drawing exactly 1 defective screw when 4 screws are randomly drawn. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve this pr... |
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"content": "Given that $\\alpha$ and $\\beta$ are acute angles, $\\sin\\alpha=\\frac{1}{2}$, $\\cos\\beta=\\frac{1}{2}$, calculate the value of $\\sin(\\alpha+\\beta)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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{
"content": "Firstly, we need to find ... |
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{
"content": "Find the largest value of $c$ such that $3$ is in the range of $f(x) = x^2 - 6x + c$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To determine the largest value of $c$ such that $3$ is in the range of the function $f(x) = x^2 ... |
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{
"content": "The equation of an ellipse with foci on the x-axis is given by $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. A triangle is formed by connecting one endpoint of the minor axis and both foci. The inscribed circle of this triangle has a radius of $\\frac{b}{3}$. Determine the eccentricity of... |
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{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\\frac{-b + \\sqrt{2}c}{\\cos B} = \\frac{a}{\\cos A}$, \n$(I)$ Find the size of angle $A$; \n$(II)$ If $a=2$, find the maximum value of the area $S$. Let's think step by ste... |
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{
"content": "A right-angled triangle has an area of 150 cm² and a perimeter of 60 cm. What are the lengths of its sides? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve this problem, we need to find the lengths of the sides of a right... |
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{
"content": "Given that a ticket to a concert costs $x$ dollars, where $x$ is a whole number, a group of 11th graders buys tickets costing a total of $60, and a group of 12th graders buys tickets costing a total of $90, determine the number of values for $x$ that are possible. Let's think step by step and outp... |
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{
"content": "Given a Cartesian coordinate system in space with points O(0, 0, 0), A(1, 0, 1), B(1, 1, 0), and C(0, 1, 1), determine the volume of the tetrahedron O-ABC. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To find the volume of the ... |
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{
"content": "Define an operation: $|\\begin{array}{l}{a}&{c}\\\\{b}&{d}\\end{array}|=ad-bc$. For example, if $|\\begin{array}{l}{1}&{3}\\\\{-2}&{0}\\end{array}|=1\\times 0-\\left(-2\\right)\\times 3=6$, then when $a=-1^{2}$, $b=\\left(-2\\right)^{2}-1$, $c=-3^{2}+5$, $d=\\frac{3}{4}÷(-\\frac{1}{4})$, find the ... |
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{
"content": "Given that $α∈(0， \\frac {π}{2}), β∈(0， \\frac {π}{2}), \\text{and }sin(2α+β)= \\frac {3}{2}sinβ, \\text{ find the value of } \\frac {tan(α+β)}{tanα }$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "Given the equation:\n\\[ \\si... |
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"content": "Simplify\n\\[\\left( \\frac{2 + i}{2 - i} \\right)^{200}.\\] Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To simplify \\(\\left( \\frac{2 + i}{2 - i} \\right)^{200}\\), we can use properties of complex numbers. Here's the step-... |
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{
"content": "Find the maximum value of the function \\( f(x)=\\sqrt{x-5}+\\sqrt{24-3x} \\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "Reasoning\nTo find the maximum value of the function \\( f(x) = \\sqrt{x-5} + \\sqrt{24-3x} \\), we nee... |
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{
"content": "Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve this problem, we need to determine the... |
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{
"content": "Vanya received three sets of candies for New Year. Each set contains three types of candies: hard candies, chocolates, and gummy candies. The total number of hard candies in all three sets is equal to the total number of chocolates in all three sets, and also to the total number of gummy candies i... |
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"content": "Synthesis and Exploration \n[Reading Comprehension] \"Holistic Thinking\" is an important mathematical thinking method, widely used in the simplification and evaluation of polynomials. \nFor example, $4x-2x+x=\\left(4-2+1\\right)x=3x$. Similarly, if we consider $\\left(a-b\\right)$ as a whole, t... |
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{
"content": "Given the right focus $F_2$ of the ellipse $C: \\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ intersects the ellipse at points $A$ and $B$. Let $F_1$ be the left focus, the perimeter of $\\triangle AF_{1}B$ is $4\\sqrt{3}$, and the eccentricity of the ellipse is $\\frac{\\sqrt{6}}{3}$.\... |
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{
"content": "Let the sequence $\\{a_n\\}$ satisfy $a_1=3$, $a_{n+1}=a_n^2-2na_n+2$ for $n=1,2,3,\\ldots$. By calculating $a_2$, $a_3$, $a_4$, try to deduce the general formula for this sequence, $a_n= \\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user... |
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"content": "Given a random variable $X\\sim N(2, \\sigma ^{2})$, if $P(X\\leq 4)=0.8$, calculate the value of $P(2\\leq X\\leq 4)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve this problem, we need to use the properties of the no... |
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"content": "Evaluate the expression $3^{-2k} - 3^{-(2k-2)} + 3^{-(2k+1)}$.\nA) $-\\frac{23}{3} \\cdot 3^{-2k}$ \nB) $-\\frac{23}{3} \\cdot 3^{-(2k-1)}$ \nC) $-\\frac{23}{3} \\cdot 3^{2k}$ \nD) $-\\frac{23}{3} \\cdot 3^{-2k+1}$ Let's think step by step and output the final answer within \\boxed{}",
"rol... |
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"content": "Given a parabola with its vertex at the origin and focus on the y-axis, which passes through point P(2,1).\n\n1. Find the standard equation of the parabola.\n2. Determine the equation of the line l through point P that intersects the parabola at only one point. Let's think step by step and output ... |
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"content": "\nLet \\( n \\) be a given positive integer. The sum\n\n$$\n\\begin{aligned}\n\\sum_{1 \\leqslant i<j \\leqslant n}\\left| x_i - x_j \\right| = & \\left| x_1 - x_2 \\right| + \\left| x_1 - x_3 \\right| + \\cdots + \\left| x_1 - x_n \\right| + \\\\\n& \\left| x_2 - x_3 \\right| + \\left| x_2 - x_4 ... |
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"content": "Find the equation of the line that is parallel to the line $x+2y+1=0$ and tangent to the circle $x^2+y^2=5$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "To solve this problem, we need to find the equation of a line that is par... |
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"content": "Given two lines $l_1: A_1x + B_1y = 1$ and $l_2: A_2x + B_2y = 1$ intersect at point $P(2, 3)$, the equation of the line passing through points $P_1(A_1, B_1)$ and $P_2(A_2, B_2)$ is . Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"cont... |
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"content": "If $f(x)$ is a linear function, and $f[f(x)] = 4x - 1$, then $f(x) = $ ______. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "If $f(x)$ is a linear function, we can represent it as $f(x) = ax + b$, where $a$ and $b$ are constants... |
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"content": "(1) Calculate: $2\\sin30^\\circ-(2015-\\pi)^0+|1-\\tan60^\\circ|$; \n(2) Solve the equation: $(x-2)^2=3(x-2)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "(1) Calculate: $2\\sin30^\\circ-(2015-\\pi)^0+|1-\\tan60^\\circ|$; \n... |
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"content": "What is the 52nd digit after the decimal point in the decimal representation of $\\frac{1}{7}$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
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"content": "The decimal representation of $\\frac{1}{7}$ is a repeating decimal. To determine the ... |
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"content": "Given a geometric sequence $\\{a_n\\}$ with the sum of the first $n$ terms $S_n = 2 \\cdot 3^n + a$ (where $a$ is a constant), find the value of $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, we need to... |
[
{
"content": "In an opaque box, there are three white balls and two black balls with distinct identifiers. The white balls are labeled $a_1$, $a_2$, $a_3$, and the black balls are labeled $b_1$, $b_2$. Two balls are drawn at random from the box.\n(1) List all the possible different outcomes;\n(2) Calculate the ... |
[
{
"content": "If $i$ is the imaginary unit, then $\\dfrac{i}{1+i}=$ ______. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, let's first simplify the expression $\\frac{i}{1+i}$. To do this, we can multiply the numerator an... |
[
{
"content": "Given that the graph of the power function $y=x^{\\alpha}$ passes through the point $(2, \\sqrt{2})$, find the value of $\\alpha$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The power function is given by $y = x^{\\alpha}$. W... |
[
{
"content": "Given that $f(\\sin\\alpha + \\cos\\alpha) = \\sin\\alpha \\cdot \\cos\\alpha$, find the value of $f\\left(\\sin \\frac{\\pi}{6}\\right)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this problem, we need to determine... |
[
{
"content": "The issue of food safety is increasingly attracting people's attention. The overuse of pesticides and fertilizers poses certain health risks to the public. To provide consumers with reassuring vegetables, a certain rural cooperative invests 2 million yuan annually to build two pollution-free veget... |
[
{
"content": "Find all positive integers \\( k \\) such that the equation\n\\[ \nx^2 + y^2 + z^2 = kxyz \n\\]\nhas positive integer solutions \\((x, y, z)\\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "We need to find all positive integers... |
[
{
"content": "Given the complex number $z=(1+i)(2a-i)$ ($i$ is the imaginary unit) corresponds to a point on the imaginary axis in the complex plane, express $z$ in terms of $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the give... |
[
{
"content": "Find all integer values that the fraction $\\frac{8n + 157}{4n + 7}$ can take for natural numbers \\(n\\). In the answer, write down the sum of the values found. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To determine the int... |
[
{
"content": "Let $a_1, a_2, \\dots, a_{2023}$ be the roots of the polynomial\n\\[x^{2023} + x^{2022} + \\dots + x^2 + x - 1387 = 0.\\] Compute\n\\[\\sum_{n = 1}^{2023} \\frac{1}{1 - a_n}.\\] Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Let'... |
[
{
"content": "A trapezoid has one base equal to four times its height, $x$, and the other base is six times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$. Let's think step by step and output the final answer within \\boxed{}",
"role... |
[
{
"content": "A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probabilities of the arrow stopping on regions $A$ and $B$ are $\\frac{1}{4}$ for $A$ and $\\frac{1}{3}$ for $B$ respectively. Also, the minimum probability of the spinner stopping on $D$ is said to be $\\frac{1}... |
[
{
"content": "In the Cartesian coordinate system $xOy$, the graph of the parabola $y=ax^2 - 3x + 3 \\ (a \\neq 0)$ is symmetric with the graph of the parabola $y^2 = 2px \\ (p > 0)$ with respect to the line $y = x + m$. Find the product of the real numbers $a$, $p$, and $m$. Let's think step by step and output ... |
[
{
"content": "Find the solution set for the inequality $(3-2x)(x+1) \\leqslant 0$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "[Jingxuan, the code should be formatted as a code block without special formatting.]\nFrom the previous question,... |
[
{
"content": "For a cubic function $f\\left(x\\right)=ax^{3}+bx^{2}+cx+d\\left(a\\neq 0\\right)$, the following definition is given: Let ${f'}\\left(x\\right)$ be the derivative of $y=f\\left(x\\right)$, and $\\varphi \\left(x\\right)$ be the derivative of $y={f'}\\left(x\\right)$. If the equation $\\varphi \\l... |
[
{
"content": "The lines $y=4x+2$, $y=-3x-18$, and $y=2x+m$ intersect at the same point. What is the value of $m$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Let's find the intersection point of the lines $y=4x+2$ and $y=-3x-18$. We can do ... |
[
{
"content": "The general term formula for the given sequence 3, 5, 7, 9, …, determine the formula for the nth term. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The sequence given is 3, 5, 7, 9, ..., and we need to find the formula for the ... |
[
{
"content": "Determine the value of \\[3^{2+3+4} - (3^2 + 3^3 + 3^4).\\] Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Give the value of \\[3^{2+3+4} - (3^2 + 3^3 + 3^4)\\]\nLet's substitute values first, and then output the final answer wit... |
[
{
"content": "Consider 4 complex numbers $2a+bi$, $c+2di$, $e+fi$, and $g+2hi$. Given that $b = 6$, $g = -2a - c - e$, and the sum of these complex numbers is $8i$, find the value of $d+f+h$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To s... |
[
{
"content": "The absolute value of $5$ is $|5|$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To complete the sentence, we need to think about another real-world situation where the value \"5\" would make sense. For example, a basket is pla... |
[
{
"content": "For which values of \\(a\\) does the equation \\(|x| = ax - 2\\) have no solutions? Determine the length of the interval of the parameter \\(a\\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, let's first c... |
[
{
"content": "A sequence $b_1, b_2, b_3, \\dots$ is defined recursively by $b_1 = 2, b_2 = 3,$ and for $k \\ge 3,$\n\\[ b_k = \\frac{1}{2} b_{k - 1} + \\frac{1}{3} b_{k - 2}. \\]\nEvaluate $b_1 + b_2 + b_3 + \\dots$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},... |
[
{
"content": "In the circle with center $O$, the measure of $\\angle QIS$ is $45^\\circ$ and $OQ=15$ cm. Find the number of centimeters in the length of arc $QS$. Express your answer in terms of $\\pi$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"cont... |
[
{
"content": "If $g(x) = 4x + 6$ and $f(x) = 6x - 10$, what is the value of $f(g(10))$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve for the value of \\( f(g(10)) \\), we need to follow these steps:\n\n1. First, we need to find the ... |
[
{
"content": "The graph of a certain quadratic $y = ax^2 + bx + c$ is a parabola with vertex $(2, -3)$ which passes through the point $(6, -63)$. What is the value of $a$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given the vertex form of... |
[
{
"content": "A city is hosting a badminton competition for high school students. The total cost $y (\\text{in yuan})$ consists of two parts: the venue fee $b (\\text{in yuan})$ and the cost of materials, which is directly proportional to the number of participants $x (\\text{people})$. The data in the table be... |
[
{
"content": "How many positive factors of 180 are also multiples of 15? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "First, find the factors of 180 and then filter them out to find the factors that are also multiples of 15. We will do this ... |
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