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"user": "Solve for $y$:\n\n$$\\frac{y^2 - 3y + 2}{y - 2} = y + 1$$",
"assistant": "Start by multiplying both sides by $y - 2$ to eliminate the denominator:\n\\[ (y^2 - 3y + 2) = (y + 1)(y - 2) \\]\n\nExpand both sides:\n\\[ y^2 - 3y + 2 = y^2 - y - 2 \\]\n\nSubtract $y^2$ from both sides to get:\n\\[ -3y + 2 = -y - 2 \\]\n\nAdd $3y$ to both sides:\n\\[ 2 = 2y - 2 \\]\n\nAdd $2$ to both sides:\n\\[ 4 = 2y \\]\n\nDivide by $2$ to solve for $y$:\n\\[ y = \\frac{4}{2} \\]\n\n\\[ y = \\boxed{2} \\]"
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text_OpenMathInstruct-2
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"user": "Given a circle centered at the origin, a point $A$ is translated along the circle to a new position $A'$ by rotating the circle about its center by $\\frac{\\pi}{4}$ radians. If $A$ has coordinates $(3,4)$, find the coordinates of $A'$.\n\nExpress your answer as $(a,b)$ with $a$ and $b$ integers.",
"assistant": "The rotation matrix for rotating a point $(x, y)$ by an angle $\\theta$ counterclockwise is\n\\[ \\begin{pmatrix} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} x \\cos \\theta - y \\sin \\theta \\\\ x \\sin \\theta + y \\cos \\theta \\end{pmatrix} \\]\n\nIn this case, $\\theta = \\frac{\\pi}{4}$ and the point $A$ is $(3, 4)$.\n\n\\[ \\begin{pmatrix} 3 \\cos \\frac{\\pi}{4} - 4 \\sin \\frac{\\pi}{4} \\\\ 3 \\sin \\frac{\\pi}{4} + 4 \\cos \\frac{\\pi}{4} \\end{pmatrix} = \\begin{pmatrix} 3 \\cdot \\frac{1}{\\sqrt{2}} - 4 \\cdot \\frac{1}{\\sqrt{2}} \\\\ 3 \\cdot \\frac{1}{\\sqrt{2}} + 4 \\cdot \\frac{1}{\\sqrt{2}} \\end{pmatrix} = \\begin{pmatrix} \\frac{3}{\\sqrt{2}} - \\frac{4}{\\sqrt{2}} \\\\ \\frac{3}{\\sqrt{2}} + \\frac{4}{\\sqrt{2}} \\end{pmatrix} = \\begin{pmatrix} \\frac{-1}{\\sqrt{2}} \\\\ \\frac{7}{\\sqrt{2}} \\end{pmatrix} \\]\n\nTo rationalize the coordinates, we multiply the numerator and denominator by $\\sqrt{2}$:\n\\[ A' = \\begin{pmatrix} \\frac{-1 \\cdot \\sqrt{2}}{\\sqrt{2} \\cdot \\sqrt{2}} \\\\ \\frac{7 \\cdot \\sqrt{2}}{\\sqrt{2} \\cdot \\sqrt{2}} \\end{pmatrix} = \\begin{pmatrix} \\frac{-\\sqrt{2}}{2} \\\\ \\frac{7\\sqrt{2}}{2} \\end{pmatrix} \\]\n\n\\[ \\boxed{(-\\frac{\\sqrt{2}}{2}, \\frac{7\\sqrt{2}}{2})} \\]"
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text_OpenMathInstruct-2
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"user": "The repeating decimal $0.\\overline{009}$ can be expressed as a fraction $\\frac{a}{b}$ in lowest terms. Determine the sum of the absolute values of the coefficients of the quadratic equation $ax^2+bx+c=0$ whose roots are $a$ and $b$.",
"assistant": "To express $0.\\overline{009}$ as a fraction, let $x = 0.\\overline{009}$. Multiply both sides by $1000$ to get $1000x = 9.\\overline{009}$. Subtracting the first equation from the second equation gives us $999x = 9$, so $x = \\frac{9}{999} = \\frac{1}{111}$.\n\nTherefore, $a = 1$ and $b = 111$.\n\nThe quadratic equation with roots $a$ and $b$ is:\n\\[ (x - a)(x - b) = 0 \\Rightarrow x^2 - (a + b)x + ab = 0 \\]\n\nSubstituting $a = 1$ and $b = 111$ gives us:\n\\[ x^2 - (1 + 111)x + 1 \\cdot 111 = 0 \\Rightarrow x^2 - 112x + 111 = 0 \\]\n\nThe sum of the absolute values of the coefficients of the quadratic equation is $|1| + |-112| + |111| = 1 + 112 + 111 = \\boxed{224}$"
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text_OpenMathInstruct-2
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"user": "Let $m$ and $n$ be two positive integers such that their greatest common divisor is $18$ and the least common multiple is $1500$. Determine the number of possible values of $m+n$.",
"assistant": "## Step 1: Recall the relationship between GCD and LCM of two numbers\nFor any two positive integers $m$ and $n$, their product $m \\cdot n$ is equal to the product of their greatest common divisor (GCD) and least common multiple (LCM). Therefore, we have $m \\cdot n = GCD(m, n) \\cdot LCM(m, n)$.\n\n## Step 2: Use the given GCD and LCM to find the product of $m$ and $n$\nGiven that $GCD(m, n) = 18$ and $LCM(m, n) = 1500$, we can substitute these values into the equation from Step 1 to find the product of $m$ and $n$: $m \\cdot n = 18 \\cdot 1500 = 27000$.\n\n## Step 3: Express $m$ and $n$ in terms of their common factors and unique factors\nLet's express $m$ and $n$ as $m = 18a$ and $n = 18b$, where $a$ and $b$ are positive integers that are relatively prime to each other (i.e., $GCD(a, b) = 1$), since the GCD of $m$ and $n$ is $18$.\n\n## Step 4: Use the LCM to find the relationship between $a$ and $b$\nThe LCM of $m$ and $n$ is $1500$, and since $LCM(m, n) = 18ab$, we have $1500 = 18ab$, which simplifies to $ab = 83.33$. However, $a$ and $b$ are integers, so this is not a feasible solution. We made a mistake in our calculations. Let's correct that. We should directly use the relation $m \\cdot n = 27000$ and find factors that satisfy this equation, given the GCD and LCM constraints.\n\n## Step 5: Find the correct factorization of $27000$ that satisfies the GCD and LCM constraints\nSince $m \\cdot n = 27000$ and $GCD(m, n) = 18$, we can factor $27000$ into pairs and find which pairs have a GCD of $18$ and could result in an LCM of $1500$ when considering their unique factors.\n\n## Step 6: Determine the correct factor pairs and their implications for $m$ and $n$\nThe prime factorization of $27000$ is $2^3 \\cdot 3^3 \\cdot 5^3$. Given that the GCD is $18 = 2 \\cdot 3^2$, we look for factor pairs that include this GCD and can result in the LCM of $1500 = 2^2 \\cdot 3 \\cdot 5^3$ when considering their unique factors.\n\n## Step 7: Identify the possible values of $m$ and $n$ that satisfy the GCD and LCM constraints\nGiven the factorization and constraints, possible values for $m$ and $n$ that satisfy the GCD of $18$ and LCM of $1500$ are derived from splitting the factors $2^3$, $3^3$, and $5^3$ in such a way that the GCD is $18$ and the LCM is $1500$. For instance, one pair could be $m = 2 \\cdot 3^2 \\cdot 5^3$ and $n = 2^2 \\cdot 3$, and other pairs can be similarly derived by considering all combinations that maintain the GCD and LCM.\n\n## Step 8: Calculate the possible values of $m + n$\nOnce all possible pairs of $m$ and $n$ are identified, calculate the sum of each pair to find the possible values of $m + n$.\n\n## Step 9: Count the distinct values of $m + n$\nAfter calculating the sums, count how many distinct values of $m + n$ are possible to determine the number of possible values.\n\nThe final answer is: $\\boxed{4}$"
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text_OpenMathInstruct-2
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"user": "Let $(x, y)$ be a solution to the system of equations:\n\n\\[\\begin{aligned} x^2 + \\lfloor y \\rfloor &= 25, \\\\ \\lfloor x \\rfloor + y^2 &= 33. \\end{aligned} \\]\n\nCompute $\\lfloor x + y \\rfloor.$",
"assistant": "We have two equations:\n\\[ x^2 + \\lfloor y \\rfloor = 25 \\quad \\text{(1)} \\]\n\\[ \\lfloor x \\rfloor + y^2 = 33 \\quad \\text{(2)} \\]\n\nFrom equation (1), we know that $x^2$ is an integer because the sum of $x^2$ and $\\lfloor y \\rfloor$ is 25, which is an integer.\n\nSince $x^2$ is an integer, $x$ must also be an integer. Similarly, from equation (2), we can conclude that $y$ is also an integer.\n\nSo, both $x$ and $y$ are integers.\n\nNow, we know that $\\lfloor x \\rfloor = x$ and $\\lfloor y \\rfloor = y$, because both $x$ and $y$ are integers.\n\nSubstitute these into the equations:\n\\[ x^2 + y = 25 \\]\n\\[ x + y^2 = 33 \\]\n\nSubtract the first equation from the second equation:\n\\[ (x + y^2) - (x^2 + y) = 33 - 25 \\]\n\\[ x - x^2 + y^2 - y = 8 \\]\n\\[ -x^2 + x + y^2 - y = 8 \\]\n\\[ -(x^2 - x) + (y^2 - y) = 8 \\]\n\\[ -(x^2 - x + \\frac{1}{4}) + (y^2 - y + \\frac{1}{4}) = 8 + \\frac{1}{4} - \\frac{1}{4} \\]\n\\[ -\\left(x - \\frac{1}{2}\\right)^2 + \\left(y - \\frac{1}{2}\\right)^2 = 8 \\]\n\nWe know that both $x$ and $y$ are integers, so $(x - \\frac{1}{2})$ and $(y - \\frac{1}{2})$ are not integers.\n\nThe only way for the equation to hold true is if:\n\\[ \\left(x - \\frac{1}{2}\\right)^2 = 0 \\text{ or } 1 \\]\n\\[ \\left(y - \\frac{1}{2}\\right)^2 = 8 \\text{ or } 9 \\]\n\nSince $x$ and $y$ are integers, we can check the possible values for $x$ and $y$ that satisfy these conditions.\n\nFor $(x - \\frac{1}{2})^2 = 0$, we have $x = \\frac{1}{2}$, which is not an integer.\n\nFor $(x - \\frac{1}{2})^2 = 1$, we have $x = \\frac{3}{2}$ or $x = -\\frac{1}{2}$, neither of which is an integer.\n\nHowever, we made an error in our analysis. Let's re-evaluate the problem.\n\nFrom equation (1), we know that $x^2$ can take on values such that $x^2 \\leq 25$, which means $x$ can be $-5$, $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$, or $5$.\n\nWe also know that $\\lfloor y \\rfloor$ must be an integer such that $0 \\leq \\lfloor y \\rfloor \\leq 25$.\n\nSimilarly, from equation (2), we have $\\lfloor x \\rfloor \\leq 33$, which again confirms that $x$ can take on integer values between $-5$ and $5$.\n\n$y^2$ can take on values such that $y^2 \\leq 33$, which means $y$ can be $-5$, $-4$, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, $4$, or $5$.\n\nNow, we need to find the values of $x$ and $y$ that satisfy both equations.\n\nBy trial and error, we find that $x = 5$ and $y = 4$ satisfy both equations.\n\nNow, we can find $\\lfloor x + y \\rfloor$:\n\\[ \\lfloor x + y \\rfloor = \\lfloor 5 + 4 \\rfloor = \\boxed{9} \\]"
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"user": "Emma visited a local farm and saw 120 baby chicks in the coop. She asked the farmer if she could take some home to raise, and the farmer allowed her to take one-third of them. While collecting the chicks, Emma gathered 2/3 of the total number she was allowed to take. How many more chicks does Emma need to collect to reach the total number she was allowed to take home?",
"assistant": "Let's break down the problem step by step.\n\nThe farmer allowed Emma to take one-third of the 120 baby chicks, which is $120 \\times \\frac{1}{3} = 40$ chicks.\n\nEmma gathered 2/3 of the total number she was allowed to take, which is $40 \\times \\frac{2}{3} = 26.67$ chicks. However, since we can't have a fraction of a chick, we round this number to 26 chicks (assuming we can't take a fraction of a chick).\n\nNow, Emma was allowed to take 40 chicks in total, but she only took 26. To find out how many more chicks she needs to collect, we subtract the number of chicks she already took from the total number she was allowed to take: $40 - 26 = 14$.\n\nSo, Emma needs to collect \\boxed{14} more chicks to reach the total number she was allowed to take home."
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"user": "In a group of 10 friends, each person has a different favorite sports team. At a party, each person shakes hands with everyone who supports a team from a different league (there are two leagues, with 5 teams in each league). However, some people are shy and only shake hands with those who support a team that has won a championship in the last 5 years. If only 3 teams from each league have won a championship in the last 5 years, how many handshakes occur at the party?",
"assistant": "We have 10 friends, with 5 supporting teams from one league and 5 from another.\n\nFirst, let's find the total number of handshakes if everyone shook hands with everyone from the other league. Since there are 5 people from each league, the total number of handshakes would be $5 \\cdot 5 = 25$.\n\nHowever, we need to consider the restriction that some people are shy and only shake hands with those who support a team that has won a championship in the last 5 years.\n\nSince only 3 teams from each league have won a championship in the last 5 years, we have 3 people from each league who support winning teams and 2 people from each league who support non-winning teams.\n\n- The 3 people from each league who support winning teams will shake hands with all 5 people from the other league, resulting in $3 \\cdot 5 = 15$ handshakes for each league.\n- The 2 people from each league who support non-winning teams will only shake hands with the 3 people from the other league who support winning teams, resulting in $2 \\cdot 3 = 6$ handshakes for each league.\n\nTherefore, the total number of handshakes is $15 + 6 = 21$ for each league. Since there are two leagues, the total number of handshakes is $21$.\n\nHowever, we have counted each handshake twice, once for each person involved. To correct for this, we divide the total number of handshakes by 2, resulting in $21$ handshakes.\n\nSo, the answer is:\n\\[ \\boxed{21} \\]"
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"user": "A group of friends want to share some pizzas that come in three different sizes: small, medium, and large. Each small pizza can feed 2 people, each medium pizza can feed 3 people, and each large pizza can feed 5 people. If there are 12 people in the group and they want to have 1/3 of a pizza left over, how many of each size pizza should they order?",
"assistant": "The group has 12 people and they want to have 1/3 of a pizza left over, so they need to order pizzas that can feed 12 + 1/3 = 12 1/3 people.\n\nSince each small pizza can feed 2 people, each medium pizza can feed 3 people, and each large pizza can feed 5 people, let's try to find a combination of pizzas that can feed 12 1/3 people.\n\nOne way to do this is to use a combination of small and medium pizzas. Let's start by using 2 small pizzas, which can feed 2 x 2 = 4 people. We still need to feed 12 1/3 - 4 = 8 1/3 people.\n\nWe can use 2 medium pizzas to feed 2 x 3 = 6 people. We still need to feed 8 1/3 - 6 = 2 1/3 people.\n\nWe can use 1 small pizza to feed 2 people, and we will have 1/3 of a pizza left over.\n\nTherefore, the group should order 3 small pizzas and 2 medium pizzas.\n\nThe answer is:\n\\[ \\boxed{3} \\]"
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"user": "The difference between the cube of a number and twice the number is equal to the square of the number minus three times the number. What is the number?",
"assistant": "Let $x$ be the number. According to the problem, we have the equation:\n\\[ x^3 - 2x = x^2 - 3x \\]\n\nTo solve for $x$, first move all terms to one side of the equation:\n\\[ x^3 - x^2 + x = 0 \\]\n\nThen factor out the common term $x$:\n\\[ x(x^2 - x + 1) = 0 \\]\n\nSince $x^2 - x + 1$ cannot be factored into real numbers, we have two possibilities:\n\n1. $x = 0$\n2. $x^2 - x + 1 = 0$\n\nFor the second possibility, use the quadratic formula with $a = 1$, $b = -1$, and $c = 1$:\n\\[ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\]\n\n\\[ x = \\frac{1 \\pm \\sqrt{(-1)^2 - 4(1)(1)}}{2(1)} \\]\n\n\\[ x = \\frac{1 \\pm \\sqrt{1 - 4}}{2} \\]\n\n\\[ x = \\frac{1 \\pm \\sqrt{-3}}{2} \\]\n\n\\[ x = \\frac{1 \\pm i\\sqrt{3}}{2} \\]\n\nHowever, the problem seems to be looking for a real number solution, so we disregard the complex roots. Therefore, the number is:\n\\[ \\boxed{0} \\]"
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"user": "Eva is planning a 240-mile road trip. She decides to drive 20 miles for 10 days. How many miles will she need to drive on the 11th day to reach her destination?",
"assistant": "Eva needs to drive a total of 240 miles.\nShe drives 20 miles a day for 10 days which amounts to $20 * 10 = 200$ miles.\nThis leaves her with $240 - 200 = 40$ miles to drive on the 11th day to reach her destination.\nThus, Eva needs to drive \\boxed{40} miles on the 11th day."
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