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2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$. | 2. Solution. Since the desired integer solution $x$ is divisible by 10 and 12, it is divisible by 60, hence it can be written in the form $x=60 t, t \in Z$. Substitute $x=60 t$ into the original equation:
$3600 t^{2}+60 b t-9600=0$. Express $b: b=\frac{160}{t}-60 t$. For $b$ to be an integer, $t$ must be a divisor of ... | 9599 | Algebra | math-word-problem | olympiads | 622e4608-b0d7-5071-90bd-fa39367cb1ac | [
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"content": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.",
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"content": "<think>\nOkay, let's see. The problem is asking how many integers \\( b \\) exist s... | <think>
Okay, let's see. The problem is asking how many integers \( b \) exist such that the quadratic equation \( x^2 + b x - 9600 = 0 \) has an integer solution that is a multiple of both 10 and 12. Also, we need to specify the largest possible \( b \).
First, let's parse the question. We need to find integers \( b ... | 27,577 | 44 | 80 | 0.536131 | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.<|im_end|>
<|im_start|>assistant
<|im... |
"18.3.19 $\\star \\star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{(...TRUNCATED) | "Given $44<\\sqrt{2005}<45,25<\\sqrt{\\frac{2005}{3}}<26$, so\n$$\n26 \\leqslant c \\leqslant 44 .\n(...TRUNCATED) | (23,24,30),(12,30,31),(9,30,32),(4,30,33),(15,22,36),(9,18,40),(4,15,42) | Number Theory | math-word-problem | olympiads | 1e57f229-b82e-5c03-ab5e-e465aef6a830 | [
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"7. Given positive integers $a, b$ satisfy\n$$\n\\sqrt{\\frac{a b}{2 b^{2}-a}}=\\frac{a+2 b}{4 b} \\(...TRUNCATED) | "7.2012.\n\nNotice,\n$$\n\\begin{array}{l}\n\\sqrt{\\frac{a b}{2 b^{2}-a}}=\\frac{a+2 b}{4 b} \\\\\n(...TRUNCATED) | 2012 | Algebra | math-word-problem | cn_contest | 2e8a9edd-dfb2-5c52-a7c3-ce50ae674f3a | [
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"16. If a positive integer cannot be written as the difference of two square numbers, then the integ(...TRUNCATED) | "16. Answer: 8030\nAny odd number greater than 1 can be written as $2 k+1$, where $2 k+1=(k+1)^{2}-k(...TRUNCATED) | 8030 | Number Theory | math-word-problem | olympiads | 56235d59-8371-5924-88cd-e24ac98bd30e | [
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"Let $\\triangle ABC$ be a triangle with $AB = 7$, $AC = 8$, and $BC = 3$. Let $P_1$ and $P_2$ be tw(...TRUNCATED) | "To find the distance between the circumcenters of $\\triangle BP_1P_2$ and $\\triangle CQ_1Q_2$, we(...TRUNCATED) | 3 | Geometry | math-word-problem | aops_forum | ca30b970-fe84-5b67-94f3-b26737b563b2 | [
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"To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many(...TRUNCATED) | "1. **Identify the problem and the goal:**\n We need to determine all integers \\( n \\) for which(...TRUNCATED) | n = 4 | Geometry | math-word-problem | aops_forum | 84b967ad-0515-5261-b6ef-e7ad07e90929 | [
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] | 1 | [{"content":"To [i]dissect [/i] a polygon means to divide it into several regions by cutting along f(...TRUNCATED) | "<think>\nOkay, so I need to figure out for which integers n it's possible to dissect a regular n-si(...TRUNCATED) | 42,509 | 309 | 81 | 0.516286 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"9. (15 points) As shown in Figure 2, circle $\\odot M$ is externally tangent to circle $\\odot O$ a(...TRUNCATED) | "II. 9. As shown in Figure 6, extend $PC$ and $QC$ to intersect $\\odot M$ at points $D$ and $E$ res(...TRUNCATED) | \sqrt{\frac{R}{R+r}} | Geometry | math-word-problem | olympiads | 81c2df6d-6639-56b1-94b2-90c4644d236b | [
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"7. The function\n$$\nf(x)=\\frac{\\sin x-1}{\\sqrt{3-2 \\cos x-2 \\sin x}}(0 \\leqslant x \\leqslan(...TRUNCATED) | "7. $[-1,0]$.\n\nLet $1-\\sin x=a, 1-\\cos x=b$. Then\n$$\ny=f(x)=\\frac{-a}{\\sqrt{a^{2}+b^{2}}}(0 (...TRUNCATED) | [-1,0] | Calculus | math-word-problem | olympiads | 183158a5-72af-5e09-86c3-29c255051d88 | [
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"Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system:\n\\[ ax+by=((...TRUNCATED) | "\\( (0, 0, c) \\).\n\nThe final answer is \\( \\boxed{ (0, 0, 0), (0, 0, c), (a, 0, 0), (0, b, 0) }(...TRUNCATED) | (0, 0, 0), (0, 0, c), (a, 0, 0), (0, b, 0) | Algebra | math-word-problem | aops_forum | 6d9ffb53-4602-5285-8faa-8ac19cc999f0 | [
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"22. Let $P M O$ be a triangle with $P M=2$ and $\\angle P M O=120^{\\circ}$. Let $B$ be a point on (...TRUNCATED) | "22. 28\nExtend $P M$ to a point $C$ such that $P C \\perp O C$. Since $\\angle P M O=120^{\\circ}, (...TRUNCATED) | 28 | Geometry | math-word-problem | olympiads | 6951f08f-04d0-5142-acae-6a705e0caa52 | [
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] | 2 | [{"content":"22. Let $P M O$ be a triangle with $P M=2$ and $\\angle P M O=120^{\\circ}$. Let $B$ be(...TRUNCATED) | "<think>\nOkay, let's try to solve this geometry problem step by step. First, I need to understand t(...TRUNCATED) | 35,744 | 516 | 65 | 0.521363 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
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