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"content": "On a soccer tournament with $n\\ge 3$ teams taking part, several matches are played in such a way that among any three teams, some two play a match. $(a)$ If $n=7$ , find the smallest number of matches that must be played. $(b)$ Find the smallest number of matches in terms of $n$ .",
"ro... |
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"content": "Evan's dog weighs 63 pounds, which is 7 times as much as Ivan's dog. Kara's cat weighs 5 times the combined weight of Evan's dog and Ivan's dog. What is the total weight of Evan's dog, Ivan's dog, and Kara's cat?",
"role": "user"
},
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"content": "To determine the total weight of Evan's... |
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{
"content": "Let \\( PQR \\) be a triangle with \\( N \\) as the midpoint of side \\( QR \\). The line segment \\( AP \\) intersects the circumcircle of triangle \\( PQR \\) again at point \\( S \\). A line through \\( S \\) that is parallel to \\( QR \\) intersects the circumcircle of triangle \\( PQR \\) aga... |
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"content": "Rewrite the expression $8k^2 - 12k + 20$ in the form $c(k + p)^2 + q$, where $c$, $p$, and $q$ are constants. What is $\\frac{q}{p}$?",
"role": "user"
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"content": "To rewrite the expression \\(8k^2 - 12k + 20\\) in the form \\(c(k + p)^2 + q\\), we need to complete the square. Here... |
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"content": "Consider a rectangular parallelepiped with dimensions 2 by 3 by 6 units. Calculate the volume of the space which consists of points that are within or exactly one unit from the surface of this parallelepiped.",
"role": "user"
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"content": "To find the volume of the space which consi... |
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{
"content": "For every integer $n \\ge 1$ , the function $f_n : \\left\\{ 0, 1, \\cdots, n \\right\\} \\to \\mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \\[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \\] for each $1 \\le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2) + \\cdots + f_{2N} ... |
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"content": "Consider a set of $m \\geq 1$ triangles in the plane, where each triangle may share edges with one or more other triangles. Prove that it is possible to color the plane using three colors such that no two adjacent triangles (triangles that share a side) are of the same color.",
"role": "user"
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"content": "The positive numbers $x$ and $y$ satisfy the equation $\\frac{1}{x} + \\frac{1}{y} = 1$. Find the minimum value of $\\frac{3x}{x-1} + \\frac{8y}{y-1}$.",
"role": "user"
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"content": "Given the equation \\(\\frac{1}{x} + \\frac{1}{y} = 1\\), we can rewrite it as \\(y = \\frac{x}{x-1}... |
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"content": "Consider three variables \\( A \\), \\( B \\), and \\( C \\) that are in the ratio of 2 to 1 to 4, respectively. Determine the result of the expression \\( \\frac{3A + 2B}{4C - A} \\) and provide your answer as a fraction in simplest form.\nPlease give a step-by-step answer and use a \\boxed comma... |
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"content": "Which of the following operations is correct?\n\nA: $(-2mn)^{2}=4m^{2}n^{2}$\n\nB: $y^{2}+y^{2}=2y^{4}$\n\nC: $(a-b)^{2}=a^{2}-b^{2}$\n\nD: $m^{2}+m=m^{3}$",
"role": "user"
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"content": "To determine which of the given operations is correct, we will evaluate each option step by step... |
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"content": "The equation of the directrix of the parabola $y=x^{2}$ is:\n\nA: $y=-\\dfrac{1}{4}$\n\nB: $y=-\\dfrac{1}{2}$\n\nC: $x=-\\dfrac{1}{4}$\n\nD: $x=-\\dfrac{1}{2}$",
"role": "user"
},
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"content": "To determine the equation of the directrix of the parabola given by the equation \\( y = x^2... |
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"content": "Let \\( f\\left(x\\right)=cx^{4}+dx^{2}-3 \\), where \\( c \\) and \\( d \\) are constants. If \\( f\\left(1\\right)=1 \\), then \\( f\\left(-1\\right)=\\_\\_\\_$\n\nA: $-5$\n\nB: $-3$\n\nC: $-1$\n\nD: $1$",
"role": "user"
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"content": "To determine the value of \\( f(-1) \\) given ... |
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"content": "Convert \\\\(-300^\\circ\\\\) to radians. The answer is:\nA: \\\\(-\\dfrac{4\\pi}{3}\\\\)\nB: \\\\(-\\dfrac{5\\pi}{3}\\\\)\nC: \\\\(-\\dfrac{7\\pi}{6}\\\\)\nD: \\\\(-\\dfrac{7\\pi}{4}\\\\)",
"role": "user"
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"content": "To convert the angle \\(-300^\\circ\\) to radians, we use the c... |
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"content": "If a grocery store sells 4 types of jelly, with grape jelly being sold twice as much as strawberry jelly, and raspberry jelly being sold twice as much as plum jelly, and raspberry jelly being sold a third as much as grape jelly, how many jars of strawberry jelly did they sell if they sold 6 jars o... |
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{
"content": "If you throw four dice at the same time, can you find the probability of getting a sum of 10 from the numbers shown on all four dice?",
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"content": "To find the probability of getting a sum of 10 when throwing four dice, we need to determine the number of favorable o... |
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"content": "Calculate: $-8+(-10)-3-(-6)$.",
"role": "user"
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"content": "To calculate the expression \\(-8 + (-10) - 3 - (-6)\\), we will follow these steps:\n\n1. Simplify the expression by removing the parentheses and converting the subtraction of a negative number into addition.\n2. Combine ... |
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"content": "Show that for every natural number \\( m \\), there exists a polynomial of degree \\( 3m + 2 \\) with coefficients in \\( \\mathbb{Q}[x] \\) that has exactly \\( 3 \\) complex zeros and is irreducible in \\( \\mathbb{Q}[x] \\). (20 points)",
"role": "user"
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"content": "To show that... |
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"content": "What is the units digit of the product $1! \\times 2! \\times 3! \\times 4! \\times 5! \\times \\cdots \\times 10!$?",
"role": "user"
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"content": "To determine the units digit of the product \\(1! \\times 2! \\times 3! \\times 4! \\times 5! \\times \\cdots \\times 10!\\), we need t... |
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"content": "Consider the following statements regarding the properties of circles and ellipses: \n① For a circle with center O and radius r, the distance from any point P on the circle to the center O is always equal to r; \n② If a point P moves such that the distance from P to a fixed point F is always twi... |
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"content": "There are 20 points distributed on a circle. Using 10 chords that do not share common endpoints and do not intersect with each other, how many different ways can these points be connected?\n\n(13th Moscow Olympiad problem)",
"role": "user"
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"content": "To determine the number of di... |
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"content": "Given $3(x+1) = 2(y-4) - 5$, express $y$ as an algebraic expression in terms of $x$.",
"role": "user"
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"content": "To express \\( y \\) as an algebraic expression in terms of \\( x \\) given the equation \\( 3(x+1) = 2(y-4) - 5 \\), we will follow these steps:\n\n1. Expand both sid... |
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"content": "In the complex plane, the coordinate of the point corresponding to the complex number $\\frac {3i-1}{1+3i}$ is ( )\nA: $\\left( \\frac {4}{5}, \\frac {3}{5}\\right)$\nB: $\\left(-1, \\frac {3}{5}\\right)$\nC: $\\left( \\frac {3}{5}, \\frac {4}{5}\\right)$\nD: $\\left( \\frac {3}{5},1\\right)$",
... |
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"content": "The imaginary part of the complex number $$\\frac {1}{i-2}+ \\frac {1}{1-2i}$$ is ( )\n\nA: $$\\frac {1}{5}$$ \nB: $$\\frac {1}{5}i$$ \nC: \\- $$\\frac {1}{5}$$ \nD: $$- \\frac {1}{5}i$$",
"role": "user"
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"content": "To find the imaginary part of the complex number \\(\\frac{... |
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"content": "Let \\( m \\) be an odd number. From the integers \\( 1, 2, \\ldots, m \\), five distinct numbers \\( p, q, r, s, t \\) are chosen such that\n\n\\[ p + q + r = s + t . \\]\n\nDemonstrate that the number of different ways to select these integers (without considering the order of \\( p, q, r, s, t ... |
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{
"content": "A certain galaxy is known to comprise approximately 4 x 10^12 stars. Of every 50 million of these stars, one is larger in mass than our sun. Approximately how many stars in this galaxy are larger than the sun?\nAnswer Choices: (A) 800 (B) 1,250 (C) 80,000 (D) 12,000 (E) 80,000",
"role": "user"... |
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{
"content": "In a cricket match held in the US, a batting team scored at a rate of 3.2 runs per over in the first 10 overs. They lost 2 wickets in the process. The team's target is 222 runs. Due to fielding restrictions and considering the team can only lose 3 more wickets without affecting the required run ra... |
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{
"content": "An apple tree produces 40 apples in its first year. The second year the apple tree produces x more than double the amount of apples that it produced the first year, and the third year production went down by a fourth due to an insect infestation. The tree produced 194 apples in total in the first... |
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{
"content": "In right triangle $DEF$, where $DE=15$, $DF=9$, and $EF=12$ units. What is the distance from $F$ to the midpoint of segment $DE$?",
"role": "user"
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"content": "To find the distance from point \\( F \\) to the midpoint of segment \\( DE \\) in the right triangle \\( DEF \\) with \\(... |
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{
"content": "How many positive integers less than 600 are congruent to 3 (mod 9)?",
"role": "user"
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"content": "To determine how many positive integers less than 600 are congruent to \\(3 \\pmod{9}\\), we start by identifying the general form of these integers. An integer \\(n\\) that is congru... |
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{
"content": "What is the smallest positive integer that is both a multiple of $7$ and a multiple of X$?\nIf we know the answer to the above question is 28, what is the value of unknown variable X?",
"role": "user"
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{
"content": "To determine the value of the unknown variable \\( X \\), we need to... |
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{
"content": "In a triangle \\( \\triangle XYZ \\), let \\( P \\) and \\( Q \\) be points on sides \\( XY \\) and \\( XZ \\) respectively, such that the segments \\( XP \\) and \\( XQ \\) are drawn perpendicular to \\( YZ \\). If the area of triangle \\( \\triangle XPQ \\) is equal to half the area of triangle ... |
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{
"content": "What is the greatest common divisor of $876543$ and $765432$?",
"role": "user"
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"content": "To find the greatest common divisor (GCD) of \\(876543\\) and \\(765432\\), we can use the Euclidean algorithm. The Euclidean algorithm involves repeated division and taking remainders until... |
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{
"content": "Mark collects money for the homeless. He visits x households a day for 5 days and half of those households give him a pair of 20s. He collected 2000. What is the value of unknown variable x? What is the value of unknown variable x?",
"role": "user"
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{
"content": "To determine the va... |
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{
"content": "A library has 8 distinct novels that need to be allocated in the following ways. How many different distributions can be made for each scenario: \n(1) Distributed to four friends, X, Y, Z, and W, with each friend receiving 2 novels; \n(2) Distributed to four friends, X, Y, Z, and W, where X rece... |
[
{
"content": "If $a \\diamond b$ is defined as $a \\diamond b = 4a - 2b$, what is the value of $7 \\diamond 3$?",
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"content": "To find the value of \\(7 \\diamond 3\\) given the operation \\(a \\diamond b = 4a - 2b\\), we need to substitute \\(a = 7\\) and \\(b = 3\\) into the exp... |
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{
"content": "Let \\( x_{1}, x_{2}, \\ldots, x_{n} \\) be real numbers satisfying the conditions\n\\[ \\left| x_{1} + x_{2} + \\cdots + x_{n} \\right| = 1 \\quad \\text{and} \\quad \\left| x_{i} \\right| \\leq \\frac{n+1}{2} \\quad \\text{for} \\quad i = 1, 2, \\ldots, n. \\]\nShow that there exists a permutati... |
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{
"content": "A sum was invested at a simple interest for a period of 5 years. Had it been invested at a 4% higher rate, it would have fetched $200 more over the same time period. If the same sum was invested at a 7% higher rate, it would have fetched $350 more. Determine the original sum that was invested.",
... |
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{
"content": "Place six balls numbered 1, 2, 3, 4, 5, and 6 in a row, with the requirement that ball number 1 must be adjacent to ball number 2, and ball number 5 must not be adjacent to ball number 6. The number of different arrangements is ( )\nA: 36\nB: 142\nC: 48\nD: 144",
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{
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"content": "Braden started with $400 in his savings. He made a wager with Bryan on the Raptors defeating the Bulls. If Braden's prediction came true, and they had decided that the winner would receive double the amount Braden originally had in his savings, how much money would Braden have after winning the wa... |
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{
"content": "A right rectangular pyramid, where the base has dimensions $10\\sqrt{2}$ units by $6\\sqrt{2}$ units, and each slant edge is 12 units long, is cut by a plane that is parallel to its base and 4 units above the base. What is the volume, in cubic units, of the new pyramid that is cut off by this plan... |
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{
"content": "x% of Toby's friends are boys and the rest are girls. If he has 33 friends who are boys, He has 27 friends who are girls. What is the value of unknown variable x? What is the value of unknown variable x?",
"role": "user"
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{
"content": "To determine the value of the unknown variable \... |
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{
"content": "When $3x^2$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 9. When $x^2$ was subtracted from it, its minimum value decreased by 9. How will the minimum value of $f(x)$ change if $x^2$ is added to it?",
"role": "user"
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{
"content": "Let's denote the quadr... |
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{
"content": "Let \\( y_{1} \\) and \\( y_{2} \\) be two distinct numbers such that \\( 4y_{i}^{2} + ky_{i} - c = 0 \\) for \\( i = 1, 2 \\). What is \\( y_{1} + y_{2} \\)?\n(A) \\( -\\frac{k}{4} \\).\n(B) \\( \\frac{k}{4} \\).\n(C) \\( \\frac{c}{4} \\).\n(D) \\( 2c \\).\n(E) \\( -\\frac{c}{4} \\).",
"role"... |
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{
"content": "In a small town, two artists were both born within the last 200 years. Each artist has a lifespan of 75 years. If their birth years are uniformly distributed across this 200-year period, what is the probability that they shared at least 10 years of their lives as contemporaries?\nPlease give a ste... |
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{
"content": "Two players engage in a game on a circular board consisting of 150 squares. Each player, on their turn, places either A or B in an empty square. The game concludes when four consecutive squares contain A, B, A, B in that specific order, and the player who made the last move wins. If all squares ar... |
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{
"content": "If $\\left(x+y+z\\right)\\left(y+z-x\\right)=4yz$, and $z\\sin Y=y\\sin Z$, then triangle XYZ is ( )\n\nA: Equilateral triangle\n\nB: Isosceles triangle\n\nC: Right triangle\n\nD: Obtuse triangle",
"role": "user"
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{
"content": "To determine the type of triangle \\(XYZ\\) given the co... |
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{
"content": "Simplify the following expression:\n\n$$7x + 8 - 3x - 4 + 5$$",
"role": "user"
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{
"content": "To simplify the expression \\(7x + 8 - 3x - 4 + 5\\), we need to combine like terms. Like terms are terms that contain the same variable raised to the same power. In this expression, the lik... |
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{
"content": "Given vectors $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$ such that $\\|\\mathbf{a}\\| = 7$, $\\|\\mathbf{b}\\| = 10$, $\\mathbf{a} \\cdot \\mathbf{b} = 30$, $\\|\\mathbf{c}\\| = 4$, and $\\mathbf{a} \\cdot \\mathbf{c} = 12$, find $\\|\\operatorname{proj}_{\\mathbf{b}} \\mathbf{a}\\|$ and $\\|... |
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{
"content": "Jaclyn, Marcelle, Susanna, and Wanda are tutors in the math lab. Their schedule is as follows: Jaclyn works every third school day, Marcelle works every fourth school day, Susanna works every sixth school day, and Wanda works every seventh school day. If they are all currently working in the math ... |
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{
"content": "A manufacturing company wants to determine the maximum profit it can earn by producing two types of products, X and Y. The company has a budget of $200,000 for raw materials and can produce a maximum of 500 units of each product. Each unit of X requires $100 of raw material and 4 hours of labor, w... |
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{
"content": "Let \\( X, Y, Z \\) be the angles of a triangle where \\( X = 60^\\circ \\) and \\( X + Y + Z = 180^\\circ \\). Compute:\n\\[\n\\begin{vmatrix} \n\\sin X & 1 & 1 \\\\ \n1 & \\sin Y & 1 \\\\ \n1 & 1 & \\sin Z \n\\end{vmatrix}.\n\\]\nPlease give a step-by-step answer and use a \\boxed command to den... |
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{
"content": "A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces, $30$ of which are triangular and $14$ of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. How many space diagonals does $Q$ have?",
"role... |
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{
"content": "The sequence $(i_n)$ is defined by $i_0 = 8,$ $i_1 = 6,$ and \\[i_n = i_{n - 1} \\sqrt{9} - i_{n - 2}\\] for all $n \\ge 2.$ Find $i_{35}.$",
"role": "user"
},
{
"content": "To find the value of \\(i_{35}\\) in the sequence defined by \\(i_0 = 8\\), \\(i_1 = 6\\), and \\(i_n = i_{n-1} ... |
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{
"content": "Consider the quadratic expression \\(x^2 + kx + 5\\) where the difference between its roots is \\(\\sqrt{61}\\). Determine the maximum value of the constant \\(k\\).\nPlease give a step-by-step answer and use a \\boxed command to denote the final answer.",
"role": "user"
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{
"content"... |
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"content": "In how many different combinations can a total of 3 euros be formed using coins with denominations of 1, 2, 5, 10, and 20 cents?",
"role": "user"
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"content": "To determine the number of different combinations of coins that sum up to 3 euros, we first convert 3 euros into cents. Sin... |
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{
"content": "In a polar coordinate system, the midpoint of the line segment whose endpoints are $\\left( 10, \\frac{\\pi}{6} \\right)$ and $\\left( 10, \\frac{5\\pi}{6} \\right)$ is the point $(r, \\theta).$ Determine $(r, \\theta),$ where $r > 0$ and $0 \\le \\theta < 2\\pi.$",
"role": "user"
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{
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"content": "Jeannie hikes a certain distance to Mount Overlook at a pace of 4 miles per hour, and then returns at a pace of 6 miles per hour. Her hike took 5 hours. How many miles is the distance to Mount Overlook?",
"role": "user"
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"content": "To determine the distance to Mount Overlook, we n... |
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{
"content": "The lengths of two sides of a triangle are 24 units and 51 units. The third side also has an integral length. What is the least possible number of units in the perimeter of this triangle?",
"role": "user"
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{
"content": "To determine the least possible perimeter of a triangle with sid... |
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{
"content": "Reflect the parabola \\( y = -x^{2} \\) across the x-axis, the resulting parabola is ( )\n\nA: \\( y = x^{2} \\)\n\nB: \\( y = -x^{2} \\)\n\nC: \\( y = -\\left(x-2\\right)^{2} \\)\n\nD: \\( y = \\left(-x\\right)^{2} \\)",
"role": "user"
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{
"content": "To reflect the parabola \\( y = ... |
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{
"content": "Given that the terminal side of angle β passes through the point (-5, 12), find sinβ = ___.",
"role": "user"
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"content": "To find the sine of the angle \\(\\beta\\) whose terminal side passes through the point \\((-5, 12)\\), we need to use the definition of the sine function in te... |
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{
"content": "We are given a combination lock consisting of $6$ rotating discs. Each disc consists of digits $0, 1, 2,\\ldots , 9$ in that order (after digit $9$ comes $0$ ). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock ope... |
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{
"content": "Given the equation of circle C is $(x-1)^2+(y-2)^2=4$. \n(Ⅰ) Find the equation of the tangent line to circle C that passes through point M(3, 1); \n(Ⅱ) Determine the positional relationship between the line $ax-y+3=0$ and circle C.",
"role": "user"
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"content": "To solve the probl... |
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{
"content": "In a blue $12 \\times 12$ grid, on the first move, a $1 \\times 1$ striped rectangle is painted, on the second move a $1 \\times 2$ striped rectangle is painted, on the third move a $1 \\times 3$ striped rectangle is painted, and so forth, continuing until no more rectangles can be painted without... |
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{
"content": "Prove that for any positive integer value of $n$, the number $19 \\cdot 8^{n} + 17$ is composite.",
"role": "user"
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"content": "To prove that the number \\(19 \\cdot 8^n + 17\\) is composite for any positive integer \\(n\\), we will consider the expression modulo 3 and modulo 5.\n\... |
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{
"content": "Let $0 \\le a, b, c, d \\le 1.$ Find the maximum value of\n\\[\\sqrt[4]{abcd} + \\sqrt{(1 - a)(1 - b)(1 - c)(1 - d)}.\\]",
"role": "user"
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{
"content": "To find the maximum value of the expression \\(\\sqrt[4]{abcd} + \\sqrt{(1 - a)(1 - b)(1 - c)(1 - d)}\\), we will analyze the expre... |
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{
"content": "A 25-meter-long ribbon with a thickness of 0.1 mm is tightly wound around a cardboard tube, resulting in a roll with a diameter of 1 dm. What is the diameter of the tube?",
"role": "user"
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"content": "To determine the diameter of the cardboard tube, we need to calculate the total i... |
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{
"content": "Reduce the expression obtained by subtracting the fraction formed by the square root of 175 over the square root of 112 from the fraction formed by the square root of 507 over the square root of 48.\nPlease give a step-by-step answer and use a \\boxed command to denote the final answer.",
"rol... |
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{
"content": "In a scenario where box X contains 3 green marbles and 1 yellow marble, and box Y contains 2 green marbles and 3 yellow marbles, both types of marbles being identical in shape and texture. If one marble is randomly selected from each box, what is the probability that exactly one of the marbles dra... |
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{
"content": "Let \\( g \\) be a function defined on the set of positive rational numbers with the property that \\( g(a\\cdot b) = g(a) + g(b) \\) for all positive rational numbers \\( a \\) and \\( b \\). Suppose that \\( g \\) also has the property that \\( g(p) = p \\) for every prime number \\( p \\). For ... |
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{
"content": "Let \\( f \\) be defined by \n\\[\nf(x) = \\left\\{\n\\begin{array}{cl}\n2x + 1 & \\text{ if } x < 1, \\\\\nx^2 - 4 & \\text{ if } x \\geq 1.\n\\end{array}\n\\right.\n\\]\nCalculate \\( f^{-1}(3) + f^{-1}(5) + f^{-1}(10) \\).",
"role": "user"
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"content": "To find \\( f^{-1}(3) + f^... |
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{
"content": "In a three-dimensional space, let $\\beta$ be a plane and $b$ be a line that lies entirely within plane $\\beta$. Which of the following statements is true regarding the relationship between line $b$ and any line $c$ that is not in plane $\\beta$? \n\nA: Line $c$ must be parallel to line $b$ \nB:... |
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{
"content": "Determine the domain of the function \\( g(x) = \\ln(3 - x^2) + \\frac{1}{\\sqrt{x+5}} \\).\n\nA) \\( (-5, -\\sqrt{3}) \\cup (-\\sqrt{3}, \\sqrt{3}) \\)\nB) \\( (-5, -\\sqrt{3}) \\cup (\\sqrt{3}, 3) \\)\nC) \\( (-\\infty, -5) \\cup (-5, -\\sqrt{3}) \\cup (-\\sqrt{3}, \\sqrt{3}) \\)\nD) \\( (-\\inf... |
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{
"content": "A cement mixture is composed of 3 elements. By weight, 1/4 of the mixture is sand, 2/5 of the mixture is water, and the remaining part of the mixture is gravel. The weight of the entire mixture is 40 pounds. What is the weight of the gravel in the mixture in pounds?",
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{
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[
{
"content": "Can two different circles intersect at three points?\nPlease give a step-by-step answer and use a \\boxed command to denote the final answer.",
"role": "user"
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{
"content": "To determine whether two different circles can intersect at three points, we need to analyze the geometric pro... |
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{
"content": "The roots of the equation $3x^2 - mx + n = 0$ sum to 9 and multiply to 20. Determine the value of $m+n$.",
"role": "user"
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{
"content": "To determine the value of \\( m + n \\) for the quadratic equation \\( 3x^2 - mx + n = 0 \\) given that the roots sum to 9 and multiply to 20, we c... |
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{
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of line $l_1$ is $\\rho =\\dfrac{-1}{{\\sin }\\the... |
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{
"content": "Determine the partial derivatives of the function \\( w = 3x^2y + e^{xy} - \\sin(x + y) \\).",
"role": "user"
},
{
"content": "To determine the partial derivatives of the function \\( w = 3x^2y + e^{xy} - \\sin(x + y) \\), we need to find the partial derivatives with respect to \\( x \... |
[
{
"content": "Solve the differential equation:\n\n$$\ny^{\\prime} - y \\cdot \\cot(x) = \\sin^2(x)\n$$",
"role": "user"
},
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"content": "To solve the differential equation \\( y' - y \\cdot \\cot(x) = \\sin^2(x) \\), we will use the method of integrating factors. This is a first-order linear differe... |
[
{
"content": "What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (8, 6) and the midpoint of the segment with endpoints at (5, 0) and (5, 9)?",
"role": "user"
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{
"content": "To find the slope of the line containing the midpoints of the given segments, ... |
[
{
"content": "Which of the following statements is not a proposition?\n\nA: The shortest line segment between two points.\n\nB: Non-parallel lines have only one intersection point.\n\nC: Is the difference between $x$ and $y$ equal to $x-y$?\n\nD: Equal angles are vertical angles.",
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... |
[
{
"content": "You purchased a vintage book for $80 at a flea market. If you decide to sell it today and agree to increase the price by 15% to make a profit, how much will you sell the book for?\nPlease give a step-by-step answer and use a \\boxed command to denote the final answer.",
"role": "user"
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{... |
[
{
"content": "Find all sequences \\((y_1, y_2, \\ldots, y_{2022})\\) of positive integers such that for every positive integer \\( m \\) there exists an integer \\( b \\) satisfying the equation\n\\[ y_1^m + 3 y_2^m + \\cdots + 2022 y_{2022}^m = b^{m+2} - 1. \\]",
"role": "user"
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{
"content": "To ... |
[
{
"content": "Given the sequence \\( \\left\\{a_{n}\\right\\} \\) defined by:\n\\[ a_{0}=2, \\quad a_{1}=3, \\quad a_{2}=6 \\]\nand for \\( n \\geq 3 \\), the recurrence relation:\n\\[ a_{n}=(n+4) a_{n-1}-4 n a_{n-2}+(4 n-8) a_{n-3}, \\]\nfind the general term of the sequence \\( \\left\\{a_{n}\\right\\} \\).",... |
[
{
"content": "If two distinct members of the set $\\{ 2, 4, 8, 10, 12, 25, 50, 100 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 200? Express your answer as a common fraction.",
"role": "user"
},
{
"content": "To determine the probability that t... |
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