messages
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[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{2} \\right)^x\\]on the interval $(0,100 \\pi).$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"100"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"376"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A point $Q$ is chosen in the interior of $\\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\\triangle DEF$ cuts off a segment from $u_3$ with area $9$. Find the area of $\\triangle DEF$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"225"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Bob is trying to decide between two cell phone plans. Plan A has no flat fee, but the user must pay $10$ cents per minute on the phone. Plan B requires a one-time fee of $\\$20$, but only requires a payment of $5$ cents per minute on the phone. What is the minimum whole number of minutes Bob has to use the phone for to make Plan B the cheaper plan? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"401"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let \\(\\mathcal{S}\\) be a set of 16 points in the plane, no three collinear. Let \\(\\chi(\\mathcal{S})\\) denote the number of ways to draw 8 line segments with endpoints in \\(\\mathcal{S}\\), such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of \\(\\chi(\\mathcal{S})\\) across all such \\(\\mathcal{S}\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"1430"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ (in yuan), $y_{2}$ represent the monthly bill for Package $B$ (in yuan), and $x$ represent the monthly call duration in minutes. <br/>$(1)$ Express the functions of $y_{1}$ with respect to $x$ and $y_{2}$ with respect to $x$. <br/>$(2)$ For how long should the monthly call duration be such that the charges for Packages $A$ and $B$ are the same? <br/>$(3)$ In what scenario is Package $A$ more cost-effective? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"300"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \\ge 4. $$ What is the maximum possible size for the set $S?$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"16"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $2\\sin ^{2} \\frac{A}{2}= \\sqrt{3}\\sin A$, $\\sin (B-C)=2\\cos B\\sin C$, find the value of $\\frac{AC}{AB}$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{1+\\sqrt{13}}{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A community organization begins with twenty members, among which five are leaders. The leaders are replaced annually. Each remaining member persuades three new members to join the organization every year. Additionally, five new leaders are elected from outside the community each year. Determine the total number of members in the community five years later. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"15365"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let \\( f(n) \\) be the number of 0's in the decimal representation of the positive integer \\( n \\). For example, \\( f(10001123) = 3 \\) and \\( f(1234567) = 0 \\). Find the value of\n\\[ f(1) + f(2) + f(3) + \\ldots + f(99999) \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"38889"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "If $S$, $H$, and $E$ are all distinct non-zero digits (each less than $6$) and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $6$: \n$$\\begin{array}{c@{}c@{}c@{}c}\n&S&H&E_6\\\\ \n&+&H&E_6\\\\ \n\\cline{2-4} \n&S&E&S_6\\\\ \n\\end{array}$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"11_6"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A cuckoo clock chimes the number of times corresponding to the current hour (e.g., at 19:00, it chimes 7 times). One morning, Maxim approached the clock when it was 9:05 and started moving the minute hand forward until the clock read 7 hours later. How many times did the cuckoo chime during this period? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"43"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ be a function such that (i) For all $x, y \\in \\mathbb{R}$, $f(x)+f(y)+1 \\geq f(x+y) \\geq f(x)+f(y)$ (ii) For all $x \\in[0,1), f(0) \\geq f(x)$, (iii) $-f(-1)=f(1)=1$. Find all such functions $f$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"f(x) = \\lfloor x \\rfloor"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given \\( x, y, z \\in \\mathbf{R} \\) such that \\( x^2 + y^2 + xy = 1 \\), \\( y^2 + z^2 + yz = 2 \\), \\( x^2 + z^2 + xz = 3 \\), find \\( x + y + z \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\sqrt{3 + \\sqrt{6}}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The price of an article was increased $p\\%$. Later the new price was decreased $p\\%$. If the last price was one dollar, the original price was: Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{10000}{10000-p^2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "What is the value of $x$ if the three numbers $2, x$, and 10 have an average of $x$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"6"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let \\(P\\) and \\(P+2\\) be both prime numbers satisfying \\(P(P+2) \\leq 2007\\). If \\(S\\) represents the sum of such possible values of \\(P\\), find the value of \\(S\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"106"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The number $2013$ is expressed in the form $2013 = \\frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,where $a_1 \\ge a_2 \\ge \\cdots \\ge a_m$ and $b_1 \\ge b_2 \\ge \\cdots \\ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"2"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given that $θ$ is an angle in the second quadrant, if $\\tan \\left(θ+ \\frac {π}{4}\\right)= \\frac {1}{2}$, calculate the value of $\\sin θ-\\cos θ$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac {2 \\sqrt {10}}{5}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Jarris the triangle is playing in the \\((x, y)\\) plane. Let his maximum \\(y\\) coordinate be \\(k\\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \\(x\\)-axis, find the minimum possible value of \\(k\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"24/5"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Determine the share of the Japanese yen in the currency structure of the NWF funds as of 01.07.2021 using one of the following methods:\n\nFirst method:\n\na) Find the total amount of NWF funds placed in Japanese yen as of 01.07.2021:\n\n\\[ JPY_{22} = 1213.76 - 3.36 - 38.4 - 4.25 - 226.6 - 340.56 - 0.29 = 600.3 \\text{ (billion rubles)} \\]\n\nb) Determine the share of Japanese yen in the currency structure of NWF funds as of 01.07.2021:\n\n\\[ \\alpha_{07}^{JPY} = \\frac{600.3}{1213.76} \\approx 49.46\\% \\]\n\nc) Calculate by how many percentage points and in what direction the share of Japanese yen in the currency structure of NWF funds has changed over the period considered in the table:\n\n\\[ \\Delta \\alpha^{JPY} = \\alpha_{07}^{JPY} - \\alpha_{06}^{JPY} = 49.46 - 72.98 = -23.52 \\approx -23.5 \\text{ (p.p.)} \\]\n\nSecond method:\n\na) Determine the share of euros in the currency structure of NWF funds as of 01.07.2021:\n\n\\[ \\alpha_{07}^{\\text{EUR}} = \\frac{38.4}{1213.76} \\approx 3.16\\% \\]\n\nb) Determine the share of Japanese yen in the currency structure of NWF funds as of 01.07.2021:\n\n\\[ \\alpha_{07}^{JPY} = 100 - 0.28 - 3.16 - 0.35 - 18.67 - 28.06 - 0.02 = 49.46\\% \\]\n\nc) Calculate by how many percentage points and in what direction the share of Japanese yen in the currency structure of NWF funds has changed over the period considered in the table:\n\n\\[ \\Delta \\alpha^{JPY} = \\alpha_{07}^{JPY} - \\alpha_{06}^{JPY} = 49.46 - 72.98 = -23.52 \\approx -23.5 \\text{ (p.p.)} \\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"-23.5"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $ABC$ be a triangle with $\\angle A = 90^o, \\angle B = 60^o$ and $BC = 1$ cm. Draw outside of $\\vartriangle ABC$ three equilateral triangles $ABD,ACE$ and $BCF$ . Determine the area of $\\vartriangle DEF$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{9\\sqrt{3}}{16}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $ABCD$ be a convex quadrilateral with $\\angle DAB =\\angle B DC = 90^o$ . Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$ , respectively, with $P$ lying in between $B$ and $Q$ . If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"799"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Triangle $ABC$ is isosceles with angle $A$ congruent to angle $B$. The measure of angle $C$ is 30 degrees more than the measure of angle $A$. What is the number of degrees in the measure of angle $C$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"80"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "At a ball, there were 29 boys and 15 girls. Some boys danced with some girls (no more than once with each partner). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"29"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Suppose in a right triangle LMN, where angle M is the right angle, $\\cos N = \\frac{4}{5}$ and length LM is given by 20 units. What is the length of LN? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"25"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "From 5 students, 4 are to be selected to participate in competitions in four subjects: mathematics, physics, chemistry, and biology, with each competition having only 1 participant. If student A does not participate in the biology competition, then the total number of different selection schemes is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"96"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given a real coefficient fourth-degree polynomial with a leading coefficient of 1 that has four imaginary roots, where the product of two of the roots is \\(32+\\mathrm{i}\\) and the sum of the other two roots is \\(7+\\mathrm{i}\\), determine the coefficient of the quadratic term. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"114"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $x$ and $y$ be positive real numbers such that $4x + 9y = 60.$ Find the maximum value of $xy.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"25"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given an ellipse $C:\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and a point $P(1,\\frac{{\\sqrt{2}}}{2})$ on the ellipse, satisfying $|PF_{1}|+|PF_{2}|=2\\sqrt{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ A line $l$ passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\\triangle AOB$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{\\sqrt{2}}{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find all prime numbers $p$ such that $y^{2}=x^{3}+4x$ has exactly $p$ solutions in integers modulo $p$. In other words, determine all prime numbers $p$ with the following property: there exist exactly $p$ ordered pairs of integers $(x, y)$ such that $x, y \\in\\{0,1, \\ldots, p-1\\}$ and $p \\text{ divides } y^{2}-x^{3}-4x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"p=2 \\text{ and } p \\equiv 3(\\bmod 4)"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"2025"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "There are six unmarked envelopes on a table, each containing a letter for a different person. If the mail is randomly distributed among these six people, with each person getting one letter, what is the probability that exactly three people get the right letter? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{1}{18}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "For any positive integers \\( m \\) and \\( n \\), define \\( r(m, n) \\) as the remainder of \\( m \\div n \\) (for example, \\( r(8,3) \\) represents the remainder of \\( 8 \\div 3 \\), so \\( r(8,3)=2 \\)). What is the smallest positive integer solution satisfying the equation \\( r(m, 1) + r(m, 2) + r(m, 3) + \\cdots + r(m, 10) = 4 \\)? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"120"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let the function \\( f:(0,1) \\rightarrow \\mathbf{R} \\) be defined as\n$$\nf(x)=\\left\\{\\begin{array}{l}\nx, \\text{ if } x \\text{ is irrational; } \\\\\n\\frac{p+1}{q}, \\text{ if } x=\\frac{p}{q}, \\text{ where } (p, q)=1 \\text{ and } 0<p<q.\n\\end{array}\\right.\n$$\n\nFind the maximum value of \\( f(x) \\) on the interval \\( \\left(\\frac{7}{8}, \\frac{8}{9}\\right) \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{16}{17}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $T$ be the set of points $(x, y)$ in the Cartesian plane that satisfy\n\\[\\big|\\big| |x|-3\\big|-1\\big|+\\big|\\big| |y|-3\\big|-1\\big|=2.\\]\nWhat is the total length of all the lines that make up $T$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"32\\sqrt{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A sequence of real numbers $\\{a_n\\}_{n = 1}^\\infty (n=1,2,...)$ has the following property:\n\\begin{align*}\n6a_n+5a_{n-2}=20+11a_{n-1}\\ (\\text{for }n\\geq3).\n\\end{align*}\nThe first two elements are $a_1=0, a_2=1$ . Find the integer closest to $a_{2011}$ . Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"40086"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "For how many integers $n$ between 1 and 150 is the greatest common divisor of 18 and $n$ equal to 6? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"17"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"10"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the remaining cube when the freshly-cut face is placed on a table? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{2\\sqrt{3}}{3}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "If a computer executes the following program:\n1. Initial values: \\( x = 3 \\), \\( S = 0 \\).\n2. \\( x = x + 2 \\).\n3. \\( S = S + x \\).\n4. If \\( S \\geq 10000 \\), go to step 5; otherwise, go back to step 2.\n5. Print \\( x \\).\n6. Stop.\n\nThen the value printed at step 5 is: Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"201"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\sqrt{3}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Suppose that the graph of \\[2x^2 + y^2 + 8x - 10y + c = 0\\]consists of a single point. (In this case, we call the graph a degenerate ellipse.) Find $c.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"33"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Parallelogram $ABCD$ has area $1,\\!000,\\!000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there? (A lattice point is any point whose coordinates are both integers.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"784"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Five identical rectangles are arranged to form a larger rectangle $PQRS$, as shown. The area of $PQRS$ is $4000$. What is the length, $x$, rounded off to the nearest integer? [asy]\nreal x = 1; real w = 2/3;\n\n// Draw outer square and labels\npair s = (0, 0); pair r = (2 * x, 0); pair q = (3 * w, x + w); pair p = (0, x + w);\ndraw(s--r--q--p--cycle);\nlabel(\"$S$\", s, SW); label(\"$R$\", r, SE); label(\"$Q$\", q, NE); label(\"$P$\", p, NW);\n\n// Draw other segments\ndraw((x, 0)--(x, w));\ndraw((0, w)--(2 * x, w));\ndraw((w, x + w)--(w, w)); draw((2 * w, x + w)--(2 * w, w));\n\n// length labels\npair upper = (-0.1, x + w);\npair lower = (-0.1, w);\ndraw(lower--upper);\ndraw((-0.1 - 0.03, x + w)--(-0.1 + 0.03, x + w));\ndraw((-0.1 - 0.03, w)--(-0.1 + 0.03, w));\nlabel(\"$x$\", midpoint((-0.1, w)--(-0.1, x + w)), W);\n\npair left = (0, -0.1); pair right = (x, -0.1);\ndraw((0, -0.1 + 0.03)--(0, -0.1 - 0.03));\ndraw((x, -0.1 - 0.03)--(x, -0.1 + 0.03));\ndraw(left--right);\nlabel(\"$x$\", (x/2, -0.1), S);\n\n[/asy] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"35"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^{\\circ}$ and the largest angle is $160^{\\circ}$, then $n$ equals: Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"16"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "To ensure the safety of property during the Spring Festival holiday, an office needs to arrange for one person to be on duty each day for seven days. Given that there are 4 people in the office, and each person needs to work for either one or two days, the number of different duty arrangements is \\_\\_\\_\\_\\_\\_ . (Answer with a number) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"2520"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "What integer \\( n \\) satisfies \\( 0 \\le n < 23 \\) and\n$$\n54126 \\equiv n \\pmod{23}~?\n$$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"13"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given three coplanar vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, and $\\overrightarrow{c}$, where $\\overrightarrow{a}=(\\sqrt{2}, 2)$, $|\\overrightarrow{b}|=2\\sqrt{3}$, $|\\overrightarrow{c}|=2\\sqrt{6}$, and $\\overrightarrow{a}$ is parallel to $\\overrightarrow{c}$.\n1. Find $|\\overrightarrow{c}-\\overrightarrow{a}|$;\n2. If $\\overrightarrow{a}-\\overrightarrow{b}$ is perpendicular to $3\\overrightarrow{a}+2\\overrightarrow{b}$, find the value of $\\overrightarrow{a}\\cdot(\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c})$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"-12"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Consider an arithmetic sequence where the first four terms are $x+2y$, $x-2y$, $2xy$, and $x/y$. Determine the fifth term of the sequence. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"-27.7"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A square with a perimeter of 36 is inscribed in a square with a perimeter of 40. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?\nA) $\\sqrt{101}$\nB) $9\\sqrt{2}$\nC) $8\\sqrt{2}$\nD) $\\sqrt{90}$\nE) $10$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"9\\sqrt{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Fill in the four boxes with the operations \"+\", \"-\", \"*\", and \"$\\div$\" each exactly once in the expression 10 □ 10 □ 10 □ 10 □ 10 to maximize the value. What is the maximum value? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"109"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Is there a real number $a$ such that the function $y=\\sin^2x+a\\cos x+ \\frac{5}{8}a- \\frac{3}{2}$ has a maximum value of $1$ on the closed interval $\\left[0,\\frac{\\pi}{2}\\right]$? If it exists, find the corresponding value of $a$. If not, explain why. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{3}{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given that the dihedral angle $\\alpha-l-\\beta$ is $60^{\\circ}$, points $P$ and $Q$ are on planes $\\alpha$ and $\\beta$ respectively. The distance from $P$ to plane $\\beta$ is $\\sqrt{3}$, and the distance from $Q$ to plane $\\alpha$ is $2 \\sqrt{3}$. What is the minimum distance between points $P$ and $Q$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"2\\sqrt{3}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A dark drawer contains $90$ red socks, $70$ green socks, $50$ blue socks, and $30$ purple socks. Someone randomly selects socks without seeing their color. What is the smallest number of socks that they must pick to guarantee at least $12$ pairs? A pair of socks consists of two socks of the same color. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"27"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$ , and define the $A$ -*ntipodes* to be the points $A_1,\\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \\cdots = A_{N-1}A_N = A_NC$ , and similarly define the $B$ , $C$ -ntipodes. A line $\\ell_A$ through $A$ is called a *qevian* if it passes through an $A$ -ntipode, and similarly we define qevians through $B$ and $C$ . Compute the number of ordered triples $(\\ell_A, \\ell_B, \\ell_C)$ of concurrent qevians through $A$ , $B$ , $C$ , respectively.\n\n*Proposed by Brandon Wang* Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"2017^3 - 2"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Define the sequence \\( b_1, b_2, b_3, \\ldots \\) by \\( b_n = \\sum\\limits_{k=1}^n \\cos{k} \\), where \\( k \\) represents radian measure. Find the index of the 50th term for which \\( b_n < 0 \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"314"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given the parametric equation of line $l$:\n$$\n\\begin{cases}\nx=t+1 \\\\\ny= \\sqrt {3}t\n\\end{cases}\n$$\n(where $t$ is the parameter), and the polar equation of curve $C$ is $\\rho=2\\cos\\theta$, then the polar radius (taking the positive value) of the intersection point of line $l$ and curve $C$ is ______. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\sqrt{3}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "If two distinct members of the set $\\{ 4, 10, 15, 24, 30, 40, 60 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 120? Express your answer as a common fraction. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{3}{7}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "(1) Given $0 < x < \\frac{1}{2}$, find the maximum value of $y= \\frac{1}{2}x(1-2x)$;\n\n(2) Given $x > 0$, find the maximum value of $y=2-x- \\frac{4}{x}$;\n\n(3) Given $x$, $y\\in\\mathbb{R}_{+}$, and $x+y=4$, find the minimum value of $\\frac{1}{x}+ \\frac{3}{y}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"1+ \\frac{ \\sqrt{3}}{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "When $n$ is divided by 6, a remainder of 1 is given. What is the remainder when $n+2010$ is divided by 6? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"1"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "If triangle $PQR$ has sides of length $PQ = 7,$ $PR = 6,$ and $QR = 8,$ then calculate\n\\[\\frac{\\cos \\frac{P - Q}{2}}{\\sin \\frac{R}{2}} - \\frac{\\sin \\frac{P - Q}{2}}{\\cos \\frac{R}{2}}.\\] Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{12}{7}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The average of a set of data 1, 3, 2, 5, $x$ is 3. What is the standard deviation of this set of data? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\sqrt{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The following is Xiao Liang's problem-solving process. Please read carefully and answer the following questions.\n\nCalculate $({-15})÷({\\frac{1}{3}-3-\\frac{3}{2}})×6$.\n\nSolution: Original expression $=({-15})÷({-\\frac{{25}}{6}})×6\\ldots \\ldots $ First step $=\\left(-15\\right)\\div \\left(-25\\right)\\ldots \\ldots $ Second step $=-\\frac{3}{5}\\ldots \\ldots $ Third step\n\n$(1)$ There are two errors in the solution process. The first error is in the ______ step, the mistake is ______. The second error is in the ______ step, the mistake is ______.\n\n$(2)$ Please write down the correct solution process. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{108}{5}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "What is the largest possible area of a quadrilateral with sidelengths $1, 4, 7$ and $8$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"18"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"18"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c)=P(b, c, a)$ and $P(a, a, b)=0$ for all real $a, b$, and $c$. If $P(1,2,3)=1$, compute $P(2,4,8)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"56"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Three cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade, the second card is a 10, and the third card is a queen? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{17}{11050}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Liam needs to add 125 to 96 and then round the result to the nearest hundred. He plans to round the numbers before adding them. He rounds 125 to 100 and 96 to 100, resulting in an answer of $100 + 100 = 200$. Is this answer correct? What is the correct answer? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"200"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find $0.4 \\cdot 0.6$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"0.24"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let $m \\ge 3$ be an integer and let $S = \\{3,4,5,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.\nNote: a partition of $S$ is a pair of sets $A$, $B$ such that $A \\cap B = \\emptyset$, $A \\cup B = S$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"243"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The expression $\\frac{x-3}{4x}$ is equal to zero for what value of $x$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"3"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A rectangular prism has dimensions of 1 by 1 by 2. Calculate the sum of the areas of all triangles whose vertices are also vertices of this rectangular prism, and express the sum in the form $m + \\sqrt{n} + \\sqrt{p}$, where $m, n,$ and $p$ are integers. Find $m + n + p$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"40"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Seven students are standing in a line for a graduation photo. Among them, student A must be in the middle, and students B and C must stand together. How many different arrangements are possible? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"192"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "What is the smallest integer larger than $(\\sqrt{5}+\\sqrt{3})^4$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"248"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given that a certain middle school has 3500 high school students and 1500 junior high school students, if 70 students are drawn from the high school students, calculate the total sample size $n$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"100"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A container holds $47\\frac{2}{3}$ cups of sugar. If one recipe requires $1\\frac{1}{2}$ cups of sugar, how many full recipes can be made with the sugar in the container? Express your answer as a mixed number. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"31\\frac{7}{9}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The sum of two numbers is $S$. Suppose $3$ is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two numbers? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"2S + 12"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Compute $0.18\\div0.003.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"60"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let \\( M = \\{1, 2, \\ldots, 20\\} \\) and \\( A_1, A_2, \\ldots, A_n \\) be distinct non-empty subsets of \\( M \\). When \\( i \\neq j \\), the intersection of \\( A_i \\) and \\( A_j \\) has at most two elements. Find the maximum value of \\( n \\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"1350"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "The water storage volume of two reservoirs, A and B, changes with time. The relationship between the water storage volume of reservoir A, $f(t)$ (in hundreds of tons), and time, $t$ (in hours), is given by $f(t)=2+\\sin t$, where $t\\in[0,12]$. The relationship between the water storage volume of reservoir B, $g(t)$ (in hundreds of tons), and time, $t$ (in hours), is given by $g(t)=5-|t-6|$, where $t\\in[0,12]$. When do the sum of the water storage volumes of reservoirs A and B reach the maximum value? What is the maximum value? (Note: $\\sin 6\\approx-0.279$.) Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"6.721"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\sqrt{55}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \\ldots + a_{2015}x^{2015}$ ($x \\in \\mathbb{R}$), evaluate the expression $\\frac {1}{2}+ \\frac {a_{2}}{2^{2}a_{1}}+ \\frac {a_{3}}{2^{3}a_{1}}+\\ldots+ \\frac {a_{2015}}{2^{2015}a_{1}}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac {1}{4030}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another painter paints every seventh house blue, and so forth, alternating between red and blue, until 50 painters have been by. After this is finished, how many houses will be red? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"52"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find $\\sum_{k=0}^{\\infty}\\left\\lfloor\\frac{1+\\sqrt{\\frac{2000000}{4^{k}}}}{2}\\right\\rfloor$ where $\\lfloor x\\rfloor$ denotes the largest integer less than or equal to $x$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"1414"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given that $α \\in (0, \\frac{π}{2})$ and $\\sin (\\frac{π}{4} - α) = \\frac{\\sqrt{10}}{10}$,\n(1) find the value of $\\tan 2α$;\n(2) find the value of $\\frac{\\sin (α + \\frac{π}{4})}{\\sin 2α + \\cos 2α + 1}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{\\sqrt{10}}{8}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "There exists a positive real number $x$ such that $\\cos (\\tan^{-1}(x))=x$. Find the value of $x^{2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"(-1+\\sqrt{5})/2"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find the minimum of the function\n\\[\\frac{xy}{x^2 + y^2}\\]in the domain $\\frac{2}{5} \\le x \\le \\frac{1}{2}$ and $\\frac{1}{3} \\le y \\le \\frac{3}{8}.$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{6}{13}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "How many positive integers \\( n \\) exist such that both \\(\\frac{n+1}{3}\\) and \\(3n+1\\) are three-digit integers? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"12"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Let\n\\[\\bold{A} = \\begin{pmatrix} 0 & 1 & 2 \\\\ 1 & 0 & 1 \\\\ 2 & 1 & 0 \\end{pmatrix}.\\]There exist constants $p$, $q$, and $r$ such that\n\\[\\bold{A}^3 + p \\bold{A}^2 + q \\bold{A} + r \\bold{I} = \\bold{0},\\]where $\\bold{I}$ and $\\bold{0}$ are the $3 \\times 3$ identity matrix and zero matrix, respectively. Enter the ordered triple $(p,q,r).$ Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"(0,-6,-4)"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Find the minimum value of the sum of the distances from a point in space to the vertices of a regular tetrahedron with edge length 1. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\sqrt{6}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Evaluate the expression \\[ \\frac{a^2 + 2a}{a^2 + a} \\cdot \\frac{b^2 - 4}{b^2 - 6b + 8} \\cdot \\frac{c^2 + 16c + 64}{c^2 + 12c + 36} \\]\ngiven that \\(c = b - 20\\), \\(b = a + 4\\), \\(a = 2\\), and ensuring none of the denominators are zero. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{3}{4}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "$(1)$ Calculate: $(\\frac{1}{2})^{-1}+(\\sqrt{2})^{2}-4\\times |-\\frac{1}{2}|$. \n$(2)$ Simplify first, then find the value: $(1+\\frac{4}{a-1})÷\\frac{{a}^{2}+6a+9}{{a}^{2}-a}$, where $a=2$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{2}{5}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "What is the slope of a line parallel to the line $2x - 4y = 9$? Express your answer as a common fraction. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{1}{2}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Triangle \\(\\triangle P N R\\) has side lengths \\(P N=20, N R=18\\), and \\(P R=19\\). Consider a point \\(A\\) on \\(P N\\). \\(\\triangle N R A\\) is rotated about \\(R\\) to \\(\\triangle N^{\\prime} R A^{\\prime}\\) so that \\(R, N^{\\prime}\\), and \\(P\\) lie on the same line and \\(A A^{\\prime}\\) is perpendicular to \\(P R\\). Find \\(\\frac{P A}{A N}\\). Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{19}{18}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "In parallelogram $ABCD$, angle $B$ measures $110^\\circ$. What is the number of degrees in the measure of angle $C$? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"70^\\circ"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given the parabola $C$: $x^2 = 2py (p > 0)$ and the line $2x-y+2=0$, they intersect at points $A$ and $B$. A vertical line is drawn from the midpoint of the line segment $AB$ to the $x$-axis, intersecting the parabola $C$ at point $Q$. If $\\overrightarrow{QA} \\cdot \\overrightarrow{QB}=0$, calculate the value of $p$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\frac{1}{4}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A hexagon inscribed in a circle has three consecutive sides, each of length 4, and three consecutive sides, each of length 7. The chord of the circle that divides the hexagon into two trapezoids, one with three sides each of length 4, and the other with three sides each of length 7, has length equal to $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"1017"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards are $\\spadesuit$, $\\heartsuit$, and $\\spadesuit$ in that sequence? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"\\dfrac{78}{5100}"
] |
[
"math"
] |
[
{
"content": "Your task is to follow a systematic, thorough reasoning process before providing the final solution. This involves analyzing, summarizing, exploring, reassessing, and refining your thought process through multiple iterations. Structure your response into two sections: Thought and Solution. In the Thought section, present your reasoning using the format: \"<think>\n {thoughts} </think>\n\". Each thought should include detailed analysis, brainstorming, verification, and refinement of ideas. After \"</think>\n,\" in the Solution section, provide the final, logical, and accurate answer, clearly derived from the exploration in the Thought section. If applicable, include the answer in \\boxed{} for closed-form results like multiple choices or mathematical solutions.",
"role": "system"
},
{
"content": "Given the function $f(x)=2\\sin(\\omega x+\\varphi)$, where $(\\omega > 0, |\\varphi| < \\frac{\\pi}{2})$, the graph passes through the point $B(0,-1)$, and is monotonically increasing on the interval $\\left(\\frac{\\pi}{18}, \\frac{\\pi}{3}\\right)$. Additionally, the graph of $f(x)$ coincides with its original graph after being shifted to the left by $\\pi$ units. If $x_{1}, x_{2} \\in \\left(-\\frac{17\\pi}{12}, -\\frac{2\\pi}{3}\\right)$ and $x_{1} \\neq x_{2}$, and $f(x_{1}) = f(x_{2})$, calculate $f(x_{1}+x_{2})$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] |
[
"-1"
] |
[
"math"
] |
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