messages
listlengths 2
2
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stringlengths 1
155
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stringclasses 7
values | dataset
stringclasses 1
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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": "G1.3 A six-digit number 1234xy is divisible by both 8 and 9 . Given that $x+y=c$, find the value of $c$.",
"role": "user"
}
] |
8
|
olympiads
|
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. Let the right vertex of the ellipse $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$ be $A$, the upper vertex be $B$, and the left focus be $F$. If $\\angle A B F=90^{\\circ}$, then the eccentricity of the ellipse is $\\qquad$.",
"role": "user"
}
] |
\frac{\sqrt{5}-1}{2}
|
olympiads
|
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": "$9.133 \\frac{|x+2|-|x|}{\\sqrt{4-x^{3}}}>0$.\n\nTranslate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.\n\n$9.133 \\frac{|x+2|-|x|}{\\sqrt{4-x^{3}}}>0$.",
"role": "user"
}
] |
x\in(-1;\sqrt[3]{4})
|
olympiads
|
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": "7.197. $5^{\\frac{x}{\\sqrt{x}+2}} \\cdot 0.2^{\\frac{4}{\\sqrt{x}+2}}=125^{x-4} \\cdot 0.04^{x-2}$.",
"role": "user"
}
] |
9
|
olympiads
|
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": "7・72 If $f(x)$ is a non-constant function defined on all real numbers, then\n$$\nF(x)=f(x)-f(-x)\n$$\n(A) is an odd function.\n(B) is an even function.\n(C) is both an odd and an even function.\n(D) is neither an odd nor an even function.\n(China Beijing High School Mathematics Competition, 1990)",
"role": "user"
}
] |
A
|
olympiads
|
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 $A B C D$ be a trapezoid such that $(A B)$ is parallel to $(C D), A B=3 C D=3 D A$ and $\\widehat{A D C}=120^{\\circ}$. Determine the angle $\\widehat{C B A}$ in degrees.",
"role": "user"
}
] |
30
|
olympiads
|
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 $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
"role": "user"
}
] |
51
|
amc_aime
|
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": "Two circles with centers $A$ and $B$ intersect at points $X$ and $Y$. The minor arc $\\angle{XY}=120$ degrees with respect to circle $A$, and $\\angle{XY}=60$ degrees with respect to circle $B$. If $XY=2$, \ffind the area shared by the two circles.",
"role": "user"
}
] |
\frac{10\pi - 12\sqrt{3}}{9}
|
aops_forum
|
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. [2] How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute values of the numbers in the sequence are nonincreasing?",
"role": "user"
}
] |
128
|
olympiads
|
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": "$\\left[\\begin{array}{c}\\text { Quadrilateral Pyramid }\\end{array}\\right]$\n\nIn the pyramid $S A B C D$, the base is a quadrilateral $A B C D$, where sides $A D$ and $B C$ are parallel, side $A B$ is 4, side $B C$ is 8, and angle $A B C$ is $60^{\\circ}$. The edge $S B$ is $8 \\sqrt{2}$. Find the volume of the pyramid, given that two planes can be drawn through the lines $A D$ and $B C$, not coinciding with the base of the pyramid, and intersecting the pyramid in equal quadrilaterals.",
"role": "user"
}
] |
\frac{160\sqrt{3}}{3}
|
olympiads
|
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": "3. Let $a^{2}+b^{2}=1, c^{2}+d^{2}=1$ and $a c+b d=0$. Determine the value of the expression $a b+c d$?",
"role": "user"
}
] |
0
|
olympiads
|
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": "Example 13 Find the maximum value of the function $f(x)=\\sqrt{10 x-9-x^{2}}+\\sqrt{68 x-256-x^{2}}$.",
"role": "user"
}
] |
3\sqrt{35}
|
olympiads
|
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": "9. There is a fast clock and a slow clock. The fast clock is 2 minutes fast every hour, and the slow clock is 2 minutes slow every hour. Now, after adjusting their time, the fast clock will show a time that is 1 hour ahead of the time shown by the slow clock after $\\qquad$ hours.",
"role": "user"
}
] |
15
|
olympiads
|
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": "20. Given $a, b, c \\in \\mathbf{Z}_{+}, [a, b]=12, [b, c]$ $=15$. Then the minimum possible value of $[a, c]$ is ( ).\n( A) 20\n(B) 30\n(C) 60\n(D) 120\n(E) 180",
"role": "user"
}
] |
A
|
cn_contest
|
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": "## Problem Statement\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow 0}\\left(\\frac{1+8 x}{2+11 x}\\right)^{\\frac{1}{x^{2}+1}}$",
"role": "user"
}
] |
\frac{1}{2}
|
olympiads
|
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": "9. Place 3 identical white balls, 4 identical red balls, and 5 identical yellow balls into three different boxes, allowing some boxes to contain balls of different colors. The total number of different ways to do this is (answer in numbers).\n\n允许有的盒子中球的颜色不全的不同放法共有种 (要求用数字做答).\nAllowing some boxes to contain balls of different colors, the total number of different ways to do this is (answer in numbers).",
"role": "user"
}
] |
3150
|
cn_contest
|
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": "7.166. $\\sqrt{\\log _{0.04} x+1}+\\sqrt{\\log _{0.2} x+3}=1$.",
"role": "user"
}
] |
25
|
olympiads
|
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": "6. If $x_{1} 、 x_{2}$ are the roots of the equation $x^{2}+x-3=0$, then the value of $x_{1}^{3}-4 x_{2}^{2}+19$ is ( ).\n(A)6\n(B) 4\n(C) 2\n(D) 0",
"role": "user"
}
] |
D
|
cn_contest
|
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": "10. The requirements for deciphering codes in modern society are getting higher and higher. In cryptography, the content that can be seen directly is called plaintext, and the content obtained after processing the plaintext in a certain way is called ciphertext. There is a kind of ciphertext that maps the 26 English letters $a, b, \\cdots, z$ (regardless of case) to the 26 natural numbers $1,2, \\cdots, 26$, as shown in Table 1.\nTable 1\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline$a$ & $b$ & $c$ & $d$ & $e$ & $f$ & $g$ & $h$ & $i$ & $j$ & $k$ & $l$ & $m$ \\\\\n\\hline 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\\\\n\\hline$n$ & $o$ & $p$ & $q$ & $r$ & $s$ & $t$ & $u$ & $v$ & $w$ & $x$ & $y$ & $z$ \\\\\n\\hline 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 \\\\\n\\hline\n\\end{tabular}\n\nThe transformation formula for the sequence number $x$ of the plaintext and the sequence number $y$ of the ciphertext is:\n$$\ny=\\left\\{\\begin{array}{ll}\n\\frac{x+1}{2}, & x \\text { is odd, and } 1 \\leqslant x \\leqslant 26 ; \\\\\n\\frac{x}{2}+13, & x \\text { is even, and } 1 \\leqslant x \\leqslant 26 .\n\\end{array}\\right.\n$$\n\nUsing this formula, plaintext can be converted into ciphertext, such as $5 \\rightarrow \\frac{5+1}{2}=3$, i.e., $e$ becomes $c$; $8 \\rightarrow \\frac{8}{2}+13=17$, i.e., $h$ becomes $q$.\n\nAccording to the above formula, if the ciphertext obtained from a certain plaintext is $s h x c$, then the original plaintext is",
"role": "user"
}
] |
love
|
cn_contest
|
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": "## Problem Statement\n\nCalculate approximately using the differential.\n\n$$\ny=\\frac{1}{\\sqrt{2 x^{2}+x+1}}, x=1.016\n$$",
"role": "user"
}
] |
0.495
|
olympiads
|
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": "16. (3 points) As shown in the figure, the area of rectangle $A D E F$ is 16, the area of triangle $A D B$ is 3, and the area of triangle $A C F$ is 4. Therefore, the area of triangle $A B C$ is $\\qquad$",
"role": "user"
}
] |
6.5
|
olympiads
|
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": "176. Among 150 schoolchildren, only boys collect stamps. 67 people collect stamps of the USSR, 48 people - of Africa, and 32 people - of America, 11 people only collect stamps of the USSR, 7 people - only of Africa, 4 people - only of America, and only Ivanov collected stamps of the USSR, Africa, and America. Find the maximum number of girls.",
"role": "user"
}
] |
66
|
olympiads
|
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": "815. Find the differentials of the functions:\n1) $y=x^{3}$ at the point $x=0$, if $\\left.\\Delta x=0.3 ; 2\\right) y=x^{3}-x^{2}+$ $+3 x-1$; 3) $r=\\varphi^{4}+2^{\\sin 3 \\varphi}$; 4) $y=\\ln \\left(x^{2}+1\\right)+\\operatorname{arctg} \\sqrt{x}$ at the point $x=1$, if $\\Delta x=0.1$.",
"role": "user"
}
] |
0.125
|
olympiads
|
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": "5.5. The hedgehogs collected 65 mushrooms and divided them so that each hedgehog got at least one mushroom, but no two hedgehogs had the same number of mushrooms. What is the maximum number of hedgehogs that could be",
"role": "user"
}
] |
10
|
olympiads
|
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\n\n$\\frac{((3!)!)!}{3!} = k \\cdot n!,$\nwhere $k$ and $n$ are [positive integers](https://artofproblemsolving.com/wiki/index.php/Positive_integer) and $n$ is as large as possible, find $k + n.$",
"role": "user"
}
] |
839
|
amc_aime
|
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 $\\triangle{ABC}$ with $AB = 12$, $BC = 13$, and $AC = 15$, let $M$ be a point on $\\overline{AC}$ such that the [incircles](https://artofproblemsolving.com/wiki/index.php/Incircle) of $\\triangle{ABM}$ and $\\triangle{BCM}$ have equal [radii](https://artofproblemsolving.com/wiki/index.php/Inradius). Then $\\frac{AM}{CM} = \\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.",
"role": "user"
}
] |
45
|
amc_aime
|
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 prime $p>100$ for which there exists an integer $a>1$ such that $p$ divides $\\frac{a^{89}-1}{a-1}$.",
"role": "user"
}
] |
179
|
olympiads
|
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": "3. In a circle with radius $R$, determine the central angle corresponding to a circular segment with the following property: the chord of the segment is equal to the perimeter of the circle with the largest radius that is inscribed in it.",
"role": "user"
}
] |
\alpha=2\arcsin\frac{4\pi}{4+\pi^{2}}
|
olympiads
|
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": "Toner Drum and Celery Hilton are both running for president. A total of $129$ million people cast their vote in a random order, with exactly $63$ million and $66$ million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page?\n\n[i]2017 CCA Math Bonanza Individual Round #13[/i]",
"role": "user"
}
] |
\frac{1}{43}
|
aops_forum
|
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's determine the maximum possible value of the expression $x^{2} y-y^{2} x$, where $x$ and $y$ independently run through the interval $[0,1]$. $(\\mathbf{H})$",
"role": "user"
}
] |
\frac{1}{4}
|
olympiads
|
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": "$4 \\cdot 248$ To make the equation\n$$\\left[\\frac{10^{n}}{x}\\right]=1989$$\n\nhave integer solutions, what is the smallest value of the positive integer $n$?",
"role": "user"
}
] |
7
|
inequalities
|
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": "2. A school organized three extracurricular activity groups in mathematics, Chinese, and foreign language. Each group meets twice a week, with no overlapping schedules. Each student can freely join one group, or two groups, or all three groups simultaneously. A total of 1200 students participate in the extracurricular groups, with 550 students joining the mathematics group, 460 students joining the Chinese group, and 350 students joining the foreign language group. Among them, 100 students participate in both the mathematics and foreign language groups, 120 students participate in both the mathematics and Chinese groups, and 140 students participate in all three groups. How many students participate in both the Chinese and foreign language groups?",
"role": "user"
}
] |
80
|
number_theory
|
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": "22. In $\\triangle A B C$, $A D \\perp B C$ at $D$, points $E, F, G$ are the midpoints of $A B, B D, A C$ respectively. If $E G=\\frac{3}{2} E F, A D+E F=12 \\mathrm{~cm}$, then the area of $\\triangle A B C$ is $\\qquad$ $\\mathrm{cm}^{2}$.",
"role": "user"
}
] |
48
|
olympiads
|
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 the interior of triangle $ABC$, we have chosen point $P$ such that the lines drawn through $P$ parallel to the sides of the triangle determine 3 triangles and 3 parallelograms. The areas of the resulting triangles are 4, 9, and 49 square units. What is the area of the original triangle?",
"role": "user"
}
] |
144
|
olympiads
|
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": "64. Rearrange the digits of a three-digit number, the difference between the largest and smallest three-digit numbers obtained is exactly equal to the original three-digit number. This three-digit number is $\\qquad$ .",
"role": "user"
}
] |
495
|
olympiads
|
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 $f(x)=\\operatorname{tg}\\left(\\frac{\\pi}{4}-\\operatorname{arctg} x\\right)$. Then $f\\left(\\frac{1-\\sqrt[3]{3}-\\sqrt{2}}{1+\\sqrt[3]{3}+\\sqrt{2}}\\right)=$ $\\qquad$ 2.",
"role": "user"
}
] |
\sqrt[3]{3}+\sqrt{2}
|
cn_contest
|
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": "Example 7. Find $\\lim _{x \\rightarrow+\\infty} \\frac{7 x^{2}+6 x-3}{9 x^{3}+8 x^{2}-2}$.",
"role": "user"
}
] |
0
|
olympiads
|
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": "2. Let the solution set of $x^{2}-5 x+6 \\leqslant 0$ be $P$, and the solution set of $x^{2}-2 a x+a^{2}-1 \\leqslant 0$ be $Q$. Given that $P \\subset Q$. Then the range of real number $a$ is $\\qquad$",
"role": "user"
}
] |
2 \leqslant a \leqslant 3
|
cn_contest
|
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 odd integers $n \\geq 1$ such that $n$ divides $3^{n}+1$.\n\n",
"role": "user"
}
] |
1
|
olympiads
|
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": "11.3. How many points are there on the hyperbola $y=\\frac{2013}{x}$ such that the tangent at these points intersects both coordinate axes at points with integer coordinates",
"role": "user"
}
] |
48
|
olympiads
|
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": "Problem 1. The sum of the first thirteen terms of a certain arithmetic progression is $50 \\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression except the first three is to the sum of all terms except the last three as $4: 3$. Find the number of terms in this progression.",
"role": "user"
}
] |
20
|
olympiads
|
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": "3. In the number $2 * 0 * 1 * 6 * 07 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,2,4,5,6,7$ (digits can repeat) so that the resulting 11-digit number is divisible by 75. In how many ways can this be done?",
"role": "user"
}
] |
432
|
olympiads
|
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": "8. Solution. Let's find the probability of the opposite event “the groups will not be able to contact each other.” Consider pairs of tourists where the first tourist is from the first group and the second is from the second group. There are a total of $5 \\cdot 8=40$ such pairs. Therefore, the probability that none of the tourists have a phone number of anyone from the other group is $(1-p)^{40}$.\n\nThus, the desired probability is $1-(1-p)^{40}$.",
"role": "user"
}
] |
1-(1-p)^{40}
|
olympiads
|
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": "G2.1 If the tens digit of $5^{5^{5}}$ is $a$, find the value of $a$.",
"role": "user"
}
] |
2
|
olympiads
|
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": "10.3.1. (12 points) The curve given by the equation $y=2^{p} x^{2}+5 p x-2^{p^{2}}$ intersects the $O x$ axis at points $A$ and $B$, and the $O y$ axis at point $C$. Find the sum of all values of the parameter $p$ for which the center of the circle circumscribed around triangle $A B C$ lies on the $O x$ axis.",
"role": "user"
}
] |
-1
|
olympiads
|
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": "8. 确定 $\\sqrt{25^{\\frac{1}{\\log _{6} 5}}+49^{\\frac{1}{\\log _{8} 7}}}$ 的值.",
"role": "user"
}
] |
10
|
olympiads
|
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": "45th Putnam 1984 Problem A2 Evaluate 6/( (9 - 4)(3 - 2) ) + 36/( (27 - 8)(9 - 4) ) + ... + 6 n /( (3 n+1 - 2 n+1 )(3 n - 2 n ) ) + ... . Solution",
"role": "user"
}
] |
2
|
olympiads
|
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": "On a $7 \\times 7$ chessboard, place 1 red, 1 black, and 1 white chess piece each. If the three pieces of different colors cannot be placed in the same row or column, how many different ways are there to place them?",
"role": "user"
}
] |
44100
|
olympiads
|
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": "4. (1996 National College Entrance Examination) Given the functions $f(x)=a x^{2}+b x+c, g(x)=a x+b$, where $a, b, c$ are real numbers, and $a>0$. When $|x| \\leqslant 1$, $|f(x)| \\leqslant 1$ and the maximum value of $g(x)$ is 2. Find $f(x)$.",
"role": "user"
}
] |
f(x)=2x^2-1
|
olympiads
|
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": "Example 6. Find the integral $\\int \\frac{\\ln x}{x} d x$.",
"role": "user"
}
] |
\frac{1}{2}\ln^{2}x+C
|
olympiads
|
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": "11.3. The perimeter of triangle $ABC$ is 24 cm, and the segment connecting the point of intersection of its medians with the point of intersection of its angle bisectors is parallel to side $AC$. Find the length of $AC$.",
"role": "user"
}
] |
8
|
olympiads
|
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": "Problem condition\n\nFind the derivative.\n\n$y=\\frac{1}{\\sqrt{2}} \\cdot \\operatorname{arctg} \\frac{3 x-1}{\\sqrt{2}}+\\frac{1}{3} \\cdot \\frac{3 x-1}{3 x^{2}-2 x+1}$",
"role": "user"
}
] |
\frac{4}{3(3x^{2}-2x+1)^{2}}
|
olympiads
|
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": "One, (20 points) Given real numbers $a, b, c, d$ satisfy $2a^2 + 3c^2 = 2b^2 + 3d^2 = (ad - bc)^2 = 6$. Find the value of $\\left(a^2 + \\dot{b}^2\\right)\\left(c^2 + d^2\\right)$.",
"role": "user"
}
] |
6
|
cn_contest
|
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": "10.4. Find the maximum value of the expression $a+b+c+d-ab-bc-cd-da$, if each of the numbers $a, b, c$ and $d$ belongs to the interval $[0 ; 1]$.",
"role": "user"
}
] |
2
|
olympiads
|
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": "4. If $4n+1$ and $6n+1$ are both perfect squares, then the smallest positive integer $n$ is $\\qquad$",
"role": "user"
}
] |
20
|
cn_contest
|
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": "4. Find the number of pairs of integers $(x ; y)$ that satisfy the equation $x^{2}+x y=30000000$.",
"role": "user"
}
] |
256
|
olympiads
|
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": "(4) If $\\cos ^{5} \\theta-\\sin ^{5} \\theta<7\\left(\\sin ^{3} \\theta-\\cos ^{3} \\theta\\right), \\theta \\in[0,2 \\pi)$, then the range of $\\theta$ is $\\qquad$ .",
"role": "user"
}
] |
(\frac{\pi}{4},\frac{5\pi}{4})
|
olympiads
|
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": "4. Given the sequence $\\left\\{a_{n}\\right\\}(n \\geqslant 1)$ satisfies $a_{n+2}=$ $a_{n+1}-a_{n}$ and $a_{2}=1$. If the sum of the first 1999 terms of this sequence is 2000, then the sum of the first 2000 terms equals ( ).\n(A) 1999\n(B) 2000\n(C) 2001\n(D) 2002",
"role": "user"
}
] |
C
|
cn_contest
|
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": "2. In $\\triangle A B C$, $A B \\leqslant A C \\leqslant B C$, and the smallest interior angle is not less than $59^{\\circ}$. Then the maximum value of the largest interior angle is $\\qquad$ degrees.",
"role": "user"
}
] |
62
|
cn_contest
|
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": "2. In $\\triangle A B C$, $|A B|=13,|B C|=14$, $|C A|=15$, and an internal point $P$ satisfies\n$$\n\\overrightarrow{B P} \\cdot \\overrightarrow{C A}=18, \\overrightarrow{C P} \\cdot \\overrightarrow{B A}=32 \\text {. }\n$$\n\nThen $\\overrightarrow{A P} \\cdot \\overrightarrow{B C}=$ $\\qquad$",
"role": "user"
}
] |
14
|
olympiads
|
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": "3. In a square $ABCD$ with side length 2, $E$ is the midpoint of $AB$. Now, fold the figure along segments $EC$ and $ED$ so that segments $EA$ and $EB$ coincide (point $B$ overlaps with $A$), forming a tetrahedron $CDEA$. Then the volume of this tetrahedron is $\\qquad$",
"role": "user"
}
] |
\frac{\sqrt{3}}{3}
|
olympiads
|
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": "\\section*{Problem 1 - 141011}\n\nSomeone chooses a natural number \\(n\\), adds the natural numbers from 1 to \\(n\\) together, and obtains the sum \\(1+2+\\ldots+n\\) as a three-digit number, which (like 777) consists entirely of the same digits.\n\nDetermine all possibilities for choosing a number \\(n\\) for which this is true!",
"role": "user"
}
] |
36
|
olympiads
|
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": "11. What is the greatest value that the expression\n\n$$\na e k-a f h+b f g-b d k+c d h-c e g\n$$\n\ncan take if each of the numbers $a, \\ldots, k$ is equal to 1 or -1?",
"role": "user"
}
] |
4
|
olympiads
|
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": "2. (10 points) There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five packaging boxes priced at 3 yuan, 5 yuan, 7 yuan, 9 yuan, and 11 yuan. One gift is paired with one packaging box, resulting in $\\qquad$ different prices.",
"role": "user"
}
] |
19
|
olympiads
|
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": "9.3. On the board, there are $N$ prime numbers (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. For what largest $N$ is this possible",
"role": "user"
}
] |
4
|
olympiads
|
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 summer camp gathers fifteen participants. Each day, three of them are chosen to do the dishes. At the end of the stay, it is observed that no matter how two participants are chosen, there is exactly one day where they were both on duty. How many days did the summer camp last?",
"role": "user"
}
] |
35
|
olympiads
|
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": "## Task Condition\n\nFind the derivative.\n\n$$\ny=3 x-\\ln \\left(1+\\sqrt{1-e^{6 x}}\\right)-e^{-3 x} \\cdot \\arcsin \\left(e^{3 x}\\right)\n$$",
"role": "user"
}
] |
3e^{-3x}\cdot\arcsin(e^{3x})
|
olympiads
|
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": "6. Suppose that $\\log _{a} 125=\\log _{5} 3$ and $\\log _{b} 16=\\log _{4} 7$. Find the value of $a^{\\left(\\log _{5} 3\\right)^{2}}-b^{\\left(\\log _{4} 7\\right)^{2}}$.",
"role": "user"
}
] |
-22
|
olympiads
|
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": "3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. For example, $[3]=3,[\\sqrt{2}]=1,[-\\pi]=-4$. Let $x$ be a positive real number. If $\\left[\\log _{2} x\\right]$ is even, then $x$ is called a lucky number. Then the probability that a number randomly selected from the interval $(0,1)$ is a lucky number is",
"role": "user"
}
] |
\frac{1}{3}
|
cn_contest
|
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": "6. Given $a+b+c \\neq 0$, and\n$$\n\\frac{a+b}{c}=\\frac{b+c}{a}=\\frac{a+c}{b}=p \\text {. }\n$$\n\nThen the line $y=p x+p$ does not pass through the ( ) quadrant.\n(A) first\n(B) second\n(C) third\n(D) fourth",
"role": "user"
}
] |
D
|
cn_contest
|
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": "Solve in the set of real numbers, the system:\n$$x(3y^2+1)=y(y^2+3)$$\n$$y(3z^2+1)=z(z^2+3)$$\n$$z(3x^2+1)=x(x^2+3)$$",
"role": "user"
}
] |
(1, 1, 1), (-1, -1, -1), (0, 0, 0)
|
aops_forum
|
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": "(3) Given $f(x)=x^{2}-2 x+3, g(x)=k x-1$, then “ $|k| \\leqslant 2$ ” is “ $f(x) \\geqslant g(x)$ holds for all $x \\in \\mathbf{R}$ ” ( ).\n(A) A sufficient but not necessary condition\n(B) A necessary but not sufficient condition\n(C) A necessary and sufficient condition\n(D) Neither a sufficient nor a necessary condition",
"role": "user"
}
] |
A
|
olympiads
|
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": "## Task Condition\n\nCalculate the area of the figure bounded by the graphs of the functions:\n\n$$\ny=x^{2} \\cdot \\cos x, y=0,\\left(0 \\leq x \\leq \\frac{\\pi}{2}\\right)\n$$",
"role": "user"
}
] |
\frac{\pi^{2}}{4}-2
|
olympiads
|
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": "## Task A-2.2.\n\nTwo circles with radii 1 and 3 touch each other externally at point $A$, and their external common tangent touches them at points $B$ and $C$. Determine the sum of the squares of the lengths of the sides of triangle $A B C$.",
"role": "user"
}
] |
24
|
olympiads
|
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's calculate as simply as possible the value of\n\n$$\na^{3}+b^{3}+3\\left(a^{3} b+a b^{3}\\right)+6\\left(a^{3} b^{2}+a^{2} b^{3}\\right)\n$$\n\nif $a+b=1$.",
"role": "user"
}
] |
1
|
olympiads
|
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 nonzero numbers $x{}$ and $y{}$ satisfy the inequalities $x^{2n}-y^{2n}>x$ and $y^{2n}-x^{2n}>y$ for some natural number $n{}$. Can the product $xy$ be a negative number?\n\n[i]Proposed by N. Agakhanov[/i]",
"role": "user"
}
] |
\text{No}
|
aops_forum
|
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": "3. Determine the largest prime number whose all digits are distinct such that any permutation of its digits yields another prime number.",
"role": "user"
}
] |
97
|
olympiads
|
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": "6. Let $a, b$ be positive integers, and $a+b \\sqrt{2}$ $=(1+\\sqrt{2})^{100}$. Then the units digit of $a b$ is $\\qquad$",
"role": "user"
}
] |
4
|
cn_contest
|
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": "2. (12 points) Find a natural number $n$ such that the numbers $n+15$ and $n-14$ are squares of other numbers.",
"role": "user"
}
] |
210
|
olympiads
|
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": "2.248. $\\frac{\\sqrt{11+\\sqrt{3}}}{\\sqrt{59}} \\cdot \\sqrt{4+\\sqrt{5+\\sqrt{3}}} \\cdot \\sqrt{3+\\sqrt{5+\\sqrt{5+\\sqrt{3}}}} \\cdot \\sqrt{3-\\sqrt{5+\\sqrt{5+\\sqrt{3}}}}$.",
"role": "user"
}
] |
\sqrt{2}
|
olympiads
|
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 $N_0=\\{0, 1, 2 \\cdots \\}$. Find all functions: $N_0 \\to N_0$ such that: \r\n\r\n(1) $f(n) < f(n+1)$, all $n \\in N_0$;\r\n\r\n(2) $f(2)=2$;\r\n\r\n(3) $f(mn)=f(m)f(n)$, all $m, n \\in N_0$.",
"role": "user"
}
] |
f(n) = n
|
aops_forum
|
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": "29.8. Compute $\\int \\frac{d x}{1+\\sqrt{x}}$.",
"role": "user"
}
] |
2\sqrt{x}-2\ln(1+\sqrt{x})+C
|
olympiads
|
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": "4. Grandpa Stamen gathered 258 kilograms of pears from his orchard and set off to the market. That day, he sold a part of them. If he had sold 15 kg more, he would have had one-sixth of the total amount of pears left. In addition, $\\frac{3}{8}$ of the sold pears and another 5 kg were sold in the morning at a price of 30 den/kг. In the afternoon, he increased the price and earned $1 \\frac{5}{8}$ times more money from the pears he sold in the afternoon compared to those he sold in the morning. At what price did Grandpa Stamen sell the pears in the afternoon?",
"role": "user"
}
] |
32.5
|
olympiads
|
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": "[Coordinate method on the plane $]$ $[\\underline{\\text { Orthocenter and orthotriangle }]}$\n\nGiven points $A(5, -1), B(4, -8), C(-4, -4)$. Find the coordinates of the intersection point of the altitudes of triangle $ABC$.\n\n#",
"role": "user"
}
] |
(3,-5)
|
olympiads
|
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": "6. In the Cartesian coordinate plane, the number of integer points (i.e., points with both coordinates as integers) on the circumference of a circle centered at $(199,0)$ with a radius of 199 is $\\qquad$ .",
"role": "user"
}
] |
4
|
cn_contest
|
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": "## Task 4 - 110514\n\nThe product $21 \\cdot 12 \\cdot 25$ is to be calculated.\n\nManfred wants to solve this problem in writing.\n\nAnnerose says: \"With the help of a computational advantage, I can also solve the problem in my head.\"\n\nIndicate which computational advantage Annerose could have used!",
"role": "user"
}
] |
6300
|
olympiads
|
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": "11. Observe the array: $(1),(3,5),(7,9,11),(13,15,17$,\n19), $\\cdots \\cdots$. Then 2003 is in the group.",
"role": "user"
}
] |
45
|
cn_contest
|
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": "## Problem Statement\n\nWrite the equation of the plane passing through point $A$ and perpendicular to vector $\\overrightarrow{B C}$.\n\n$A(2 ; 1 ; 7)$\n\n$B(9 ; 0 ; 2)$\n\n$C(9 ; 2 ; 3)$",
"role": "user"
}
] |
2y+z-9=0
|
olympiads
|
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": "Task B-1.2. How many integer solutions does the inequality $\\frac{x^{2}-2015 x+2014}{2013 x^{2}-2013 x} \\leq 0$ have?",
"role": "user"
}
] |
2013
|
olympiads
|
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": "2. Given $x=\\frac{1}{\\sqrt{3}-\\sqrt{2}}$. Then $x^{6}-2 \\sqrt{2} x^{5}-x^{4}$ $+x^{3}-2 \\sqrt{3} x^{2}+2 x-\\sqrt{2}$ is $\\qquad$.",
"role": "user"
}
] |
\sqrt{3}
|
cn_contest
|
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": "## Task Condition\n\nDerive the equation of the tangent line to the given curve at the point with abscissa \\( x_{0} \\).\n\n$$\ny=\\frac{-2\\left(x^{8}+2\\right)}{3\\left(x^{4}+1\\right)}, x_{0}=1\n$$",
"role": "user"
}
] |
-\frac{2}{3}\cdotx-\frac{1}{3}
|
olympiads
|
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": "## Task B-3.3.\n\nFor which natural numbers $n$ is the value of the expression $\\frac{n^{2}-4 n+4}{n+1}$ an integer?",
"role": "user"
}
] |
n\in{2,8}
|
olympiads
|
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": "Task 4\n\nCalculate $y$.\na) $d=576+386, \\quad y=d-463$\nb) $e=3784-489, \\quad y=5680-e$",
"role": "user"
}
] |
2385
|
olympiads
|
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": "10. As shown in the figure, two squares $\\mathrm{ABEG}, \\mathrm{GECD}$, point $\\mathrm{H}$ is the midpoint of $\\mathrm{GE}$, $\\frac{D F}{D C}=\\frac{1}{3}$. Connect $\\mathrm{DH}$, $\\mathrm{CH}$, $\\mathrm{AF}$, $\\mathrm{BF}$. The area of square $\\mathrm{ABEG}$ is $\\mathrm{m}$ square centimeters, and the area of the shaded part is $\\mathrm{n}$ square centimeters. Given that $\\mathrm{m}$ and $\\mathrm{n}$ are positive integers, and $\\mathrm{m}$ has 9 divisors, then the side length of square $\\mathrm{ABEG}$ is $\\qquad$ centimeters.",
"role": "user"
}
] |
10
|
olympiads
|
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": "## Task 1\n\nOut of 33 pioneers in a pioneer group, 8 pioneers were able to win medals at the pioneer sports festival because they took first, second, or third place. All other pioneers received certificates for participation.\n\nHow many pioneers in this group received certificates?",
"role": "user"
}
] |
25
|
olympiads
|
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": "7.240. $\\lg ^{4}(x-1)^{2}+\\lg ^{2}(x-1)^{3}=25$.",
"role": "user"
}
] |
1.1;11
|
olympiads
|
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": "Example 1. Find $\\lim _{x \\rightarrow+\\infty}\\left(\\sqrt{x^{2}+6 x+5}-x\\right)$.",
"role": "user"
}
] |
3
|
olympiads
|
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": "GS. 2 If $R$ is the remainder of $1^{6}+2^{6}+3^{6}+4^{6}+5^{6}+6^{6}$ divided by 7 , find the value of $R$.",
"role": "user"
}
] |
6
|
olympiads
|
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": "A2. The center of a circle with radius 1 coincides with the intersection of the diagonals of a square.\n\nThe circle and the square have the same\n\n\narea. The circle intersects a side of the square at points $A$ and $B$.\n\nHow long is the line segment $A B$?",
"role": "user"
}
] |
\sqrt{4-\pi}
|
olympiads
|
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 integer pairs $(x, y)$ for which\n\n$$\n(x+2)^{4}-x^{4}=y^{3}\n$$",
"role": "user"
}
] |
(x,y)=(-1,0)
|
olympiads
|
math
|
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