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Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
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"value": "II. (25 points) If $2 x+y \\geqslant 1$. Try to find the minimum value of the function $u=y^{2}-2 y$ $+x^{2}+4 x$.\n\n"
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this problem here where I need to find the minimum value of the function u... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Task B-4.6.\n\nDetermine the area of the set of all points associated with complex numbers $z$ for which\n\n$$\n|z| \\leqslant\\left|\\frac{1}{z}\\right| \\quad \\text{and} \\quad \\frac{28277 \\pi}{12} \\leqslant \\arg \\left(\\frac{z^{2020}}{1+i}\\right) \\leqslant \\frac{40... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "The letter $A$ has a vertical line of symmetry and the letter $B$ does not. How many of the letters H L O R X D P E have a vertical line of symmetry?\n(A) 1\n(B) 2\n(C) 3\n(D) 4\n(E) 5"
},
{
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"value": "<|begin_of_thought|>\nOkay, so I need to figur... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
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"value": "II. Find the last four digits of $2^{1999}$.\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly."
},
{
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"value": "<|begin_of_thought|>\nOkay, so I need to fin... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "18. Let $a_{k}$ be the coefficient of $x^{k}$ in the expansion of\n$$\n(x+1)+(x+1)^{2}+(x+1)^{3}+(x+1)^{4}+\\cdots+(x+1)^{99} \\text {. }\n$$\n\nDetermine the value of $\\left\\lfloor a_{4} / a_{3}\\right\\rfloor$."
},
{
"from": "assistant",
"value": "<|begin_of_thoug... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Task B-4.1.\n\nIn the set of complex numbers, solve the equation\n\n$$\n(x+5)^{5}=x^{5}+5^{5}\n$$"
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's try to solve this equation: (x + 5)^5 = x^5 + 5^5. Hmm, so we have a complex equation here. Sinc... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "10.083. The sum of the lengths of the diagonals of a rhombus is $m$, and its area is $S$. Find the side of the rhombus."
},
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"value": "<|begin_of_thought|>\nOkay, so I need to find the side of a rhombus when I know the sum of the lengths of its d... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for H... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "\\section*{Problem \\(6-330916=331016\\)}\n\nIt is known that \\(2^{10}=1024\\).\n\nFormulate a computer program that can determine the smallest natural exponent \\(p>10\\) for which the number \\(2^p\\) also ends in the digits ...024! Explain why the program you have formulated ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
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"value": "5. Fyodor starts writing pairs of natural numbers $(a, b)$ on the board, where $a<b$ and each does not exceed 2018. Moreover, if a pair of numbers $(a, b)$ is already written on the board, he cannot write any pair of the form $(c, a)$ or $(b, d)$. What is the maximum number of pa... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
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"value": "2.1 Find $f(2013)$, if for any real $x$ and $y$ the equality holds\n\n$$\nf(x-y)=f(x)+f(y)-2xy\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's try to figure out how to find f(2013) given that for any real numbers x and y, the equation f(x - ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 6.35. Find the confidence interval for estimating the mathematical expectation $a$ of a normal distribution with a reliability of 0.95, knowing the sample mean $\\bar{X}_{\\mathrm{B}}=10.43$, the sample size $n=100$, and the standard deviation $\\sigma=5$."
},
{
"... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
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"value": "13. As shown in the figure, natural numbers starting from 1 are arranged according to a certain rule. What is the number in the 3rd row and 51st column?\n$\\qquad$\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\n\\hline 3 & & 7 & 9 & 12 & & 16 & 18 & 21 & $\\ldots$ \\\\\n\\hline 2 & 4 ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
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"value": "7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The smaller wheel makes 30 revolutions per minute. Determine how many seconds the larger wheel spend... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Problem 4. A segment is divided by three points into four unequal parts. The distance between the midpoints of the inner parts is $6 \\mathrm{~cm}$, and the distance between the midpoints of the outer parts is $16 \\mathrm{~cm}$. What is the length of the given segment?"
},
{... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Find values of the parameter $u$ for which the expression \r\n\\[y = \\frac{ \\tan(x-u) + \\tan(x) + \\tan(x+u)}{ \\tan(x-u)\\tan(x)\\tan(x+u)}\\] \r\ndoes not depend on $x.$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this problem here ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ an... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Task 3. Find the smallest possible value of\n\n$$\nx y+y z+z x+\\frac{1}{x}+\\frac{2}{y}+\\frac{5}{z}\n$$\n\nfor positive real numbers $x, y$ and $z$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have to find the smallest possible value of the... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Determine the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out the value of 2 to the power of 4 multiplied by this sum: 1 plus 1/2... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "16. If a positive integer cannot be written as the difference of two square numbers, then the integer is called a \"cute\" integer. For example, 1,2 and 4 are the first three \"cute\" integers. Find the $2010^{\\text {th }}$ \"cute\" integer.\n(Note: A square number is the square... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "【Question 6】Using 2 unit squares (unit squares) can form a 2-connected square, which is commonly known as a domino. Obviously, dominoes that can coincide after translation, rotation, or symmetry transformation should be considered as the same one, so there is only one domino.\nSi... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "3. Given a convex quadrilateral $ABCD$ with area $P$. We extend side $AB$ beyond $B$ to $A_1$ such that $\\overline{AB}=\\overline{BA_1}$, then $BC$ beyond $C$ to $B_1$ such that $\\overline{BC}=\\overline{CB_1}$, then $CD$ beyond $D$ to $C_1$ so that $\\overline{CD}=\\overline{D... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions."
},
{
"from": "assistant",
"value": "<|begin_of_thoug... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "12. The maximum value of the function $S=\\sqrt{x^{4}-5 x^{2}-8 x+25}-\\sqrt{x^{4}-3 x^{2}+4}$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the maximum value of the function S, which is defined as the square root of (x⁴ - 5x² -... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Problem 6. (Option 1).\n\nIn the right triangle $ABC: \\angle ACB=90^{\\circ}, AC=6, BC=4$. On the line $BC$, a point $D (CD>BD)$ is marked such that $\\angle ADC=45^{\\circ}$. On the line $AD$, a point $E$ is marked such that the perimeter of triangle $CBE$ is the smallest possi... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "7.2. Find the GCD of all numbers obtained by all possible permutations of the digits of the number 202120222023\n\nSolution. By the divisibility rule, all these numbers are divisible by 9 (the sum of the digits is 18). A sufficient condition to prove that there are no other numbe... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Task 1 - 250921\n\nDetermine all triples $(a, b, c)$ of natural numbers $a, b, c$ for which $a \\leq b \\leq c$ and $a \\cdot b \\cdot c=19 \\cdot 85$ hold!"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, let's see. I need to find all triples (a... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Zaslavsky A.A.\n\nTwo ants each crawled along their own closed path on a $7 \\times 7$ board. Each ant only crawled along the sides of the cells and visited each of the 64 vertices of the cells exactly once. What is the minimum possible number of such sides that both the first an... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "5. Let $a_{0}, a_{1}, a_{2}, \\ldots$ be integers such that $a_{0}=19, a_{1}=25$, and for every $n \\geq 0$ we have $a_{n+2}=2 a_{n+1}-a_{n}$. What is the smallest $i>0$ for which $a_{i}$ is a multiple of 19?\n(A) 19\n(B) 25\n(C) 38\n(D) 44\n(E) 50."
},
{
"from": "assista... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "B. As shown in Figure 2, in the square $A B C D$ with side length 1, $E$ and $F$ are points on $B C$ and $C D$ respectively, and $\\triangle A E F$ is an equilateral triangle. Then the area of $\\triangle A E F$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thoug... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "14. Let $k$ be a positive integer, such that for any positive numbers $a, b, c$ satisfying the inequality $k(a b+b c+c a)>5\\left(a^{2}+b^{2}\\right.$ $+c^{2}$), there must exist a triangle with side lengths $a, b, c$. Find the minimum value of $k$.\n\n"
},
{
"from": "ass... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Find consecutive integers bounding the expression \\[\\frac{1}{x_1 + 1}+\\frac{1}{x_2 + 1}+\\frac{1}{x_3 + 1}+...\u0001 +\\frac{1}{x_{2001} + 1}+\\frac{1}{x_{2002} + 1}\\]\nwhere $x_1 = 1/3$ and $x_{n+1} = x_n^2 + x_n.$"
},
{
"from": "assistant",
"value": "<|begin_of_... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Task B-2.1. (20 points) For complex numbers $z, w$ such that $|z|=|w|=|z-w|$, calculate $\\left(\\frac{z}{w}\\right)^{99}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this problem here: For complex numbers z and w where |z| = |w| = |z -... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "1. If $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$.\n(1 mark)\nIf $x-y=12$, find the value of $x^{3}-y^{3}-36 x y$.\n(1 mark)"
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the value of \\( x^3 - y^3 - 36xy \\) given that \\( x ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## 232. Math Puzzle $9 / 84$\n\nA motorcycle battery ( $6 \\mathrm{~V}$ ) is discharged through a lamp $6 \\mathrm{~V} / 0.5 \\mathrm{~W}$. The lamp burns for 48 hours.\n\nHow many Ah was the battery charged if we assume that the discharge current remains constant?"
},
{
... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "1. If $\\log _{4}(x+2 y)+\\log _{4}(x-2 y)=1$, then the minimum value of $|x|-|y|$ is $\\qquad$"
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"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I have this problem here: If log base 4 of (x + 2y) plus log base 4 of (x - 2y) equals 1, then what's... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 1-21 Laying a $1 \\times 7$ module with $1 \\times 1, 1 \\times 2, 1 \\times 3$ blocks, how many patterns are there?"
},
{
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"value": "<|begin_of_thought|>\nOkay, so I need to figure out how many ways there are to lay a 1x7 module using 1x1,... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "45. The sum of the interior angles of a convex $n$-sided polygon is less than $2021^{\\circ}$, then the maximum value of $n$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I need to find the maximum number of sides, n, that a convex polygon ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "5. Given a sphere with a radius of 6. Then the maximum volume of a regular tetrahedron inscribed in the sphere is ( ).\n(A) $32 \\sqrt{3}$\n(B) $54 \\sqrt{3}$\n(C) $64 \\sqrt{3}$\n(D) $72 \\sqrt{3}$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, s... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "2. Solve the equation\n\n$$\n2^{\\sqrt[12]{x}}+2^{\\sqrt[4]{x}}=2 \\cdot 2^{\\sqrt[6]{x}}\n$$"
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"value": "<|begin_of_thought|>\nOkay, so I have this equation to solve: 2^{12th root of x} + 2^{4th root of x} = 2 times 2^{6th root of x}. Hmm, le... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Task 1 - 191231\n\nLet $n$ and $m$ be natural numbers with $n \\geq 1, m \\geq 1$; $N$ be the set of natural numbers from 1 to $n$ and $M$ the set of natural numbers from 1 to $m$.\n\nDetermine the number of all those subsets of $N$ that have common elements with $M$."
},
... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Let $n$ be a nonzero natural number, and $x_1, x_2,..., x_n$ positive real numbers that $ \\frac{1}{x_1}+\\frac{1}{x_2}+...+\\frac{1}{x_n}= n$. Find the minimum value of the expression $x_1 +\\frac{x_2^2}{2}++\\frac{x_3^3}{3}+...++\\frac{x_n^n}{n}$."
},
{
"from": "assista... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all p... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "11. Given $\\frac{x y}{x+y}=2, \\frac{x z}{x+z}=3, \\frac{y z}{y+z}=4$. Find the value of $7 x+5 y-2 z$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to solve this problem where I have three equations involving x, y, and z, and I need to ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "8. Determine the value of the sum\n$$\n\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}} .\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the value... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "8. Let $a>0, b>0$. Then the inequality that does not always hold is ( ).\n(A) $\\frac{2}{\\frac{1}{a}+\\frac{1}{b}} \\geqslant \\sqrt{a b}$\n(B) $\\frac{1}{a}+\\frac{1}{b} \\geqslant \\frac{4}{a+b}$\n(C) $\\sqrt{|a-b|} \\geqslant \\sqrt{a}-\\sqrt{b}$\n(D) $a^{2}+b^{2}+1>a+b$"
}... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 5: On the ground, there are 10 birds pecking, and among any 5 birds, at least 4 birds are on the same circumference. What is the maximum number of birds on the circumference that has the most birds? (6th China Mathematical Olympiad)"
},
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"from": "assistant",
... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Task 5 - 211235\n\n37 cards, each of which is colored red on one side and blue on the other, are laid on a table so that exactly 9 of them show their blue side on top.\n\nNow, in work steps, cards are to be flipped, and in each individual work step, exactly 20 of the 37 cards ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "9. In the diagram, $P Q=P R=Q S$ and $\\angle Q P R=20^{\\circ}$. What is $\\angle R Q S$ ?\nA $50^{\\circ}$\nB $60^{\\circ}$\nC $65^{\\circ}$\nD $70^{\\circ}$\nE $75^{\\circ}$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's try to figure out th... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\\in \\mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\\le j$, then $a_i\\le a_j$.)"
},
{
"from": "assistant",
... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "20.26 In the attached figure, $T P$ and $T^{\\prime} Q$ are two parallel tangent lines of this circle with radius $r$, $T$ and $T^{\\prime}$ are the points of tangency. $P T^{\\prime \\prime} Q$ is a third tangent line with the point of tangency $T^{\\prime \\prime}$.\nIf $T P=4,... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "[ [ equations in integers ]\n\nSolve the equation $12 a+11 b=2002$ in natural numbers.\n\n#"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to solve the equation 12a + 11b = 2002 where a and b are natural numbers. Hmm, natural numbers mean t... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Exercise 8. The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams). Martin and Théodore unearth a treasure composed of precious stones which are eith... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "32. How many five-digit numbers are there that read the same from left to right and right to left?"
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out how many five-digit numbers are palindromes. Hmm, palindromes are numbers that r... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Leibniz Gottfried Wilhelm\n\nConsider the numerical triangle:\n\n$$\n\\begin{array}{cccccc}\n1 & \\frac{1}{2} & \\frac{1}{3} & \\frac{1}{4} & \\ldots & \\frac{1}{1993} \\\\\n\\frac{1}{2} & \\frac{1}{6} & \\frac{1}{12} & \\ldots & \\frac{1}{1992 \\cdot 1993} \\\\\n\\frac{1}{3} & \... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "For some positive integer $n$, a coin will be flipped $n$ times to obtain a sequence of $n$ heads and tails. For each flip of the coin, there is probability $p$ of obtaining a head and probability $1-p$ of obtaining a tail, where $0<p<1$ is a rational number.\nKim writes all $2^n$ ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "24. Find all positive functions defined on $(0,+\\infty)$ that satisfy\n$$\na b \\leqslant \\frac{1}{2}\\left\\{a f(a)+b f^{-1}(b)\\right\\}\n$$\n\nwhere $a, b \\in(0,+\\infty)$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find all po... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Triangle $ ABC$ has fixed vertices $ B$ and $ C$, so that $ BC \\equal{} 2$ and $ A$ is variable. Denote by $ H$ and $ G$ the orthocenter and the centroid, respectively, of triangle $ ABC$. Let $ F\\in(HG)$ so that $ \\frac {HF}{FG} \\equal{} 3$. Find the locus of the point $ A$ ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "8. Divide a circle into a group of $n$ equal parts and color each point either red or blue. Starting from any point, record the colors of $k(k \\leqslant n)$ consecutive points in a counterclockwise direction, which is called a “$k$-order color sequence” of the circle. Two $k$-or... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Tomongho A.K.\n\nFind all increasing arithmetic progressions with a finite number of terms, the sum of which is 1, and each term has the form $1 / k$, where $k$ is a natural number."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find all... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "2. Given that the three vertices $A, B, C$ of the right triangle $\\triangle A B C$ are all on the parabola $y=x^{2}$, and the hypotenuse $A B$ is parallel to the $x$-axis, then the altitude $C D$ from $C$ to $A B$ equals $\\qquad$ ."
},
{
"from": "assistant",
"value"... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Problem Statement\n\nWrite the equation of the plane passing through point $A$ and perpendicular to vector $\\overrightarrow{B C}$.\n\n$A(5 ; 3 ;-1)$\n\n$B(0 ; 0 ;-3)$\n\n$C(5 ;-1 ; 0)$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to f... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "A cuboid has an integer volume. Three of the faces have different areas, namely $7, 27$, and $L$. What is the smallest possible integer value for $L$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, let's tackle this cuboid problem. So, we have a c... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "11. The solution set of the equation $x^{2}|x|+|x|^{2}-x^{2}-|x|=0$ in the complex number set corresponds to a figure in the complex plane which is ( ).\nA. Several points and a line\nB. Unit circle and a line\nC. Several lines\nD. Origin and unit circle"
},
{
"from": "as... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "The lateral sides of the trapezoid are 7 and 11, and the bases are 5 and 15. A line drawn through the vertex of the smaller base parallel to the larger lateral side cuts off a triangle from the trapezoid. Find the sides of this triangle.\n\n#"
},
{
"from": "assistant",
... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "1. In tetrahedron $ABCD$, let $AB=1$, $CD=\\sqrt{3}$, the distance between lines $AB$ and $CD$ is 2, and the angle between them is $\\frac{\\pi}{3}$. Then the volume of tetrahedron $ABCD$ is ( ).\n(2003 National High School League Question)\nA. $\\frac{\\sqrt{3}}{2}$\nB. $\\frac{... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "10. The solution set of the inequality $\\sqrt{a x+b}>x \\quad(a>0, b>0)$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to solve the inequality √(ax + b) > x where a and b are both positive. Hmm, let me start by understanding what this ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Simplify the following expression as much as possible:\n\n$$\n\\sqrt{\\frac{2-\\sqrt{2}}{2+\\sqrt{2}}}+\\sqrt{\\frac{2+\\sqrt{2}}{2-\\sqrt{2}}}\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to simplify this expression: √[(2 - √2)/(2 + ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "A2. Peter wrote down the set of all numbers that are greater than twice the value of their squares in the form of an interval. What did he write?\n(A) $(0,1)$\n(B) $[0,1]$\n(C) $\\left(0, \\frac{1}{2}\\right)$\n(D) $\\left(\\frac{1}{2}, 2\\right)$\n(E) $\\left[\\frac{1}{2}, 1\\ri... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Hammie is in $6^\\text{th}$ grade and weighs 106 pounds. His quadruplet sisters are tiny babies and weigh 5, 5, 6, and 8 pounds. Which is greater, the average (mean) weight of these five children or the median weight, and by how many pounds?\n$\\textbf{(A)}\\ \\text{median, by 60... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "A3. The length of the aquarium is $50 \\mathrm{~cm}$, the width is $20 \\mathrm{~cm}$, and the height is $25 \\mathrm{~cm}$. How many $\\mathrm{cm}$ from the top edge of the aquarium will the water level be if we pour 19 liters of water into it?\n(A) $19 \\mathrm{~cm}$\n(B) $1,9 ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "5. Find all positive integers $m, n$ such that $m^{2}+1$ is a prime number, and\n$$\n10\\left(m^{2}+1\\right)=n^{2}+1 \\text {. }\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, let's try to solve this problem. We need to find all positive inte... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "\\section*{Problem 1 - 101211}\n\nIn a parent-teacher meeting, exactly 18 fathers and exactly 24 mothers were present, with at least one parent of each student in the class attending.\n\nOf exactly 10 boys and exactly 8 girls, both parents were present for each. For exactly 4 boy... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## 133. Math Puzzle $6 / 76$\n\nA catch device that secures the hoist basket in a shaft fails in at most one out of 1000 operational cases. Another independent safety device fails in at most one out of 100 cases where it is called upon.\n\nWhat is the probability that the occupan... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Three, (10 points) Let $a+b+c=0, a^{2}+b^{2}+c^{2}=1$.\n(1) Find the value of $ab+bc+ca$;\n(2) Find the value of $a^{4}+b^{4}+c^{4}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this problem here where a, b, and c are numbers that satisf... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "8. In $\\triangle A B C$, the medians $A E, B F$, and $C D$ intersect at $M$. Given that $E, C, F$, and $M$ are concyclic, and $C D=n$. Find the length of segment $A B$.\n(18th All-Russian Competition Problem)"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "We consider a polyhedron with 20 triangular faces. How many edges and vertices does it have?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the number of edges and vertices of a polyhedron that has 20 triangular faces. Hmm, let me t... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "4. (6 points) A number when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. This number is $\\qquad$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, let's tackle this p... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "11. (15 points) Let real numbers $a, b$ be such that the equation $a x^{3}-x^{2} + b x-1=0$ has three positive real roots. For all real numbers $a, b$ that satisfy the condition, find the minimum value of $P=\\frac{5 a_{2}^{2}-3 a b+2}{a^{2}(b-a)}$.\n\n"
},
{
"from": "ass... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$."
},
{
"from": "assistant",
"value": "<|b... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "I1.1 Let $a$ be a real number and $\\sqrt{a}=\\sqrt{7+\\sqrt{13}}-\\sqrt{7-\\sqrt{13}}$. Find the value of $a$.\nI1.2 In Figure 1, the straight line $\\ell$ passes though the point $(a, 3)$, and makes an angle $45^{\\circ}$ with the $x$-axis. If the equation of $\\ell$ is $x+m y+... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Problem Statement\n\nCalculate the limit of the function:\n\n$\\lim _{x \\rightarrow 2} \\frac{\\operatorname{arctg}\\left(x^{2}-2 x\\right)}{\\sin 3 \\pi x}$"
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. I need to find the limit as x ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 13 Given $a+b+c=0, a^{3}+b^{3}+c^{3}$ $=0$. Find the value of $a^{15}+b^{15}+c^{15}$."
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the value of \\( a^{15} + b^{15} + c^{15} \\) given that \\( a + b + c = 0 \\) and \\( a^3 ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Task B-3.4.\n\nHow many nine-digit numbers divisible by 75 are there, all of whose digits are different, and the hundreds digit is 7?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out how many nine-digit numbers there are that... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "7.1. Pasha and Sasha made three identical toy cars. Sasha made a third of the first car, a fifth of the second car, and a fifteenth of the third car. What part of the entire work did Sasha do?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is:\n$\\textbf{(A)}\\ 0\\qquad \\textbf{(B)}\\ 1\\qquad \\textbf{(C)}\\ 2\\qquad \\textbf{(D)}\\ \\text{finite but more than 2}\\qquad \\textbf{(E)}\\ \\in... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed\nby the curve $C: y=\\frac{x}{x+k}\\ (x\\geq 0)$, the line $x=a$ and the $x$-axis around the $x$-axis, and denote by $V_2$ that of\nthe solid by a rotati... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "\nProblem 8.2. Given a $\\triangle A B C$. Let $M$ be the midpoint of $A B$, $\\angle C A B=15^{\\circ}$ and $\\angle A B C=30^{\\circ}$.\n\na) Find $\\angle A C M$.\nb) Prove that $C M=\\frac{A B \\cdot B C}{2 A C}$.\n"
},
{
"from": "assistant",
"value": "<|begin_of_... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations\n\\begin{align*}\nabcd &= 2007,\\\\\na &= \\sqrt{55 + \\sqrt{k+a}},\\\\\nb &= \\sqrt{55 - \\sqrt{k+b}},\\\\\nc &= \\sqrt{55 + \\sqrt{k-c}},\\\\\nd &= \\sqrt{55 - \\sqrt... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Problem 3. Decrypt the numerical riddle $\\overline{A B B}+\\overline{B B}=\\overline{B B A}$, where different letters represent different digits, and the same letters represent the same digits."
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's tr... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "3. In the positive geometric sequence $\\left\\{a_{n}\\right\\}$,\n$$\na_{5}=\\frac{1}{2}, a_{6}+a_{7}=3 \\text {. }\n$$\n\nThen the maximum positive integer $n$ that satisfies $a_{1}+a_{2}+\\cdots+a_{n}>a_{1} a_{2} \\cdots a_{n}$ is $\\qquad$"
},
{
"from": "assistant",
... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 2 Let $x \\in \\mathbf{R}$, try to find the minimum value of the function $f(x)=\\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$."
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I have this problem here: find the minim... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "6. The set of all real number pairs $(x, y)$ that satisfy $\\left\\{\\begin{array}{l}4^{-x}+27^{-y}=\\frac{5}{6} \\\\ \\log _{27} y-\\log _{4} x \\geqslant \\frac{1}{6} \\text { is } \\\\ 27^{y}-4^{x} \\leqslant 1\\end{array}\\right.$ $\\qquad$ . $\\qquad$"
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"from": "... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "(7) Let $z \\in \\mathbf{C}$, and satisfy $|z-\\mathrm{i}| \\leqslant 1, A=\\operatorname{Re}(z)\\left(|z-\\mathrm{i}|^{2}-1\\right)$. Find $\\max A$ and $\\min A$."
},
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"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this complex number problem ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "(3) For a positive integer $n$, if $n=p \\cdot q\\left(p \\geqslant q, p, q \\in \\mathbf{N}^{*}\\right)$, when $p-q$ is the smallest, we call $p \\cdot q$ the \"best factorization\" of $n$, and define $f(n)=\\frac{q}{p}$. For example, the factorizations of 12 are $12 \\times 1, ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "3. Before the trial run of one of the units of the hydropower station under construction, it was found that a fishing net is located $S$ km upstream from the dam. The river current speed is $v$ km/h. The hydropower station workers decided to go there by boat. Removing the net wil... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Four. (30 points) Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies\n$$\na_{1}=\\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\\left(n \\in \\mathbf{N}_{+}\\right) \\text {. }\n$$\n(1) Find the general term formula of the sequence $\\left\\{a_{n}\\right\\}$;\n(2) If the seque... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "In a unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, if we use the planes $A B_{1} C$, $B C_{1} D$, $C D_{1} A$, $D A_{1} B$, $A_{1} B C_{1}$, $B_{1} C D_{1}$, $C_{1} D A_{1}$, and $D_{1} A B_{1}$ to cut this unit cube, then the volume of the part containing the center of the cube i... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 24 (Problem from the 48th Slovenian Mathematical Olympiad) Find all five-digit numbers $\\overline{a b c d e}$, such that the number is divisible by 9, and $\\overline{a c e}-\\overline{b d a}=760$."
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"from": "assistant",
"value": "<|begin_of_thought|>\nAl... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 6 Let real numbers $a$, $b$, $c$ satisfy\n$$\n\\left\\{\\begin{array}{l}\na^{2}-b c-8 a+7=0, \\\\\nb^{2}+c^{2}+b c-6 a+6=0 .\n\\end{array}\\right.\n$$\n\nFind the range of values for $a$.\n(1995, Jilin Province Junior High School Mathematics Competition)"
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"fr... |