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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... | [
{
"from": "user",
"value": "## Task B-1.3.\n\nA ship traveling along a river has covered $24 \\mathrm{~km}$ upstream and $28 \\mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \\mathrm{~km}$ upstream and $21 \\mathrm{~km}$ downstream, or half an hour more than for ... |
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. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As 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": "Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?"
},
{
"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": "4. Given the three sides of an obtuse triangle are 3, $4, x$, then the range of values for $x$ is ( ).\n(A) $1<x<7$.\n(B) $5 \\ll x<7$.\n(C) $1<x<\\sqrt{7}$.\n(D) $5<x<7$ or $1<x<\\sqrt{7}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have 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": "1. Solve the equation: $\\frac{8 x+13}{3}=\\frac{19-12 x}{2}-\\left(16 x-\\frac{7-4 x}{6}\\right)$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to solve this equation: (8x + 13)/3 = (19 - 12x)/2 - (16x - (7 - 4x)/6). Hmm, that looks a bi... |
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 right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out the possible... |
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 $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt\\ (x\\geq 0)$ is maximal. Find $\\lim_{n\\to\\infty} \\ln a_n.$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the limit as n approaches infinity of the natural ... |
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 that the internal angles $A, B, C$ of $\\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\\sqrt{3} b \\cos \\frac{A+B}{2}=c \\sin B$.\n(1) Find the size of $\\angle C$;\n(2) If $a+b=\\sqrt{3} c$, find $\\sin A$."
},
{
"from": "assistant",
"val... |
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-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\\mathbb{C}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to sol... |
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. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. 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": "8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation\n$$\nf(x)+f(y)=f(x+y)-x y-1 .\n$$\n\nIf $f(1)=1$, then the number of integers $n$ (where $n \\neq 1$) that satisfy $f(n)=n$ is\n(A) 0.\n(B) 1.\n(C) 2.\n(D) 3.\n(E) infinitely many.\n(30t... |
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": "9.6. Find the minimum value of the expression $(\\sqrt{2(1+\\cos 2 x)}-\\sqrt{36-4 \\sqrt{5}} \\sin x+2) \\cdot(3+2 \\sqrt{10-\\sqrt{5}} \\cos y-\\cos 2 y) \\cdot$ If the answer is not an integer, round it to the nearest integer."
},
{
"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": "Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I have this equation to solve: \n\n\\[\n\\frac{x}{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": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{3 / 2} \\frac{x^{2} \\cdot d x}{\\sqrt{9-x^{2}}}\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to calculate the definite integral of x squared over the square 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": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person... |
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. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOk... |
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. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed ... |
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. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that 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": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 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": "Example 5 Given that $x_{1}, x_{2}, \\cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\\cdots+x_{10}^{2}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so ... |
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, (50 points) In an $n \\times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.\n\n---\n\nThe translation ... |
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 the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$ is a subset of $A$, and $C \\cap B \\neq \\varnothing$, then the number of such subsets $C$ is $(\\quad)$.\n(A) 256\n(B) 959\n(C) 960\n(D) 961"
},
{
"from": "assistant",
"value": "<|begin_of_though... |
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": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag... |
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": "29. Choose any three numbers from $a, b, c, d, e$ and find their sum, exactly obtaining the ten different numbers $7,11,13,14,19,21,22,25,26,28$. Then $a+b+c+d+e=$ ( i). MATHEMATICE YoUTH CLUE A. 25\nB. 31\nC. 37\nD. 43"
},
{
"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": "Problem 3. In the school, there are 50 teachers, of whom 29 drink coffee, 28 drink tea, and 16 do not drink either coffee or tea. How many teachers drink only coffee, and how many drink only tea?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's 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": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let'... |
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. Let $AB$ be a chord of the unit circle $\\odot O$. If the area of $\\odot O$ is exactly equal to the area of the square with side $AB$, then $\\angle AOB=$ $\\qquad$ (to 0.001 degree)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so the 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": "83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\\qquad$. $(a+b$ and $b+a$ are considered the same equation)"
},
{
"from": "assi... |
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": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and... |
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. The number of real solutions to the equation $\\left|x^{2}-3 x+2\\right|+\\left|x^{2}+2 x-3\\right|=11$ is ( ).\n(A) 0\n(B) 1\n(C) 2\n(D) 4"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the number of real solutions to the equati... |
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 numerical sequence:\n\n$\\lim _{n \\rightarrow \\infty} \\frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the limit as n approaches 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": "Condition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{1}{24}\\left(x^{2}+8\\right) \\sqrt{x^{2}-4}+\\frac{x^{2}}{16} \\arcsin \\frac{2}{x}, x>0\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the derivative of this fun... |
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 5 - 071225}\n\nAll ordered pairs of real numbers \\((x, y)\\) are to be determined for which the system of equations\n\n\\[\n\\begin{aligned}\nx \\cdot\\left(a x^{2}+b y^{2}-a\\right) & =0 \\\\\ny \\cdot\\left(a x^{2}+b y^{2}-b\\right) & =0\n\\end{aligned}\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": "6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,\n$$\n\\begin{array}{c}\nS=\\left[\\frac{1}{1}\\right]+\\left[\\frac{2}{1}\\right]+\\left[\\frac{1}{2}\\right]+\\left[\\frac{2}{2}\\right]+\\left[\\frac{3}{2}\\right]+ \\\\\n{\\left[\\frac{4}{2}\\right]+\\... |
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": "40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find three consecutive odd numbers whose sum equals 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": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so 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": "19. Given $m \\in\\{11,13,15,17,19\\}$, $n \\in\\{1999,2000, \\cdots, 2018\\}$.\nThen the probability that the unit digit of $m^{n}$ is 1 is ( ).\n(A) $\\frac{1}{5}$\n(B) $\\frac{1}{4}$\n(C) $\\frac{3}{10}$\n(D) $\\frac{7}{20}$\n(E) $\\frac{2}{5}$"
},
{
"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. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to calculate ... |
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": "Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?\n"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the ratio of the area inside the incircle of a regular hexag... |
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 $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\\cdot}AG + BP{\\cdot}BG + CP{\\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$."
},
{
"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": "1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need 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": "The function $f$ maps the set of positive integers into itself, and satisfies the equation\n\n$$\nf(f(n))+f(n)=2 n+6\n$$\n\nWhat could this function be?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out what the function f could ... |
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": "30. Find the remainder when the 2018-digit number $\\underbrace{\\overline{55 \\cdots}}_{2018 \\text { 555 }}$ is divided by 13."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the remainder when this huge 2018-digit number made up e... |
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. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).\nA. 1\nB. 2\nC. 3\nD. 4"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out the maximum number of points where a line can intersect 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": "1. Let the universal set be the set of real numbers. If $A=\\{x \\mid \\sqrt{x-2} \\leqslant 0\\}, B=\\left\\{x \\mid 10^{x^{2}-2}=10^{x}\\right\\}$, then $A \\cap \\bar{B}$ is\nA. $\\{2\\}$\nB. $\\{-1\\}$\nC. $\\{x \\mid x \\leqslant 2\\}$\nD. $\\varnothing$"
},
{
"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": "4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \\perp A M$, then $\\angle B A C$ equals ( ).\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": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. I need to solve the equation: \n\n\\[16 \\cdot 5^{2x-1} - 2 \\cdot 5^{x-1} - 0.048 = 0\\]\... |
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 In $\\triangle ABC$, it is known that $x \\sin A + y \\sin B + z \\sin C = 0$. Find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlrig... |
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 order for Mateen to walk a kilometer (1000m) in his rectangular backyard, he must walk the length 25 times or walk its perimeter 10 times. What is the area of Mateen's backyard in square meters?\n$\\text{(A)}\\ 40 \\qquad \\text{(B)}\\ 200 \\qquad \\text{(C)}\\ 400 \\qquad \\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": "11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. I need to find the ra... |
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 knights in a certain kingdom come in two colors. $\\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What 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": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to 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": "Example 11 Let $x>0, y>0, \\sqrt{x}(\\sqrt{x}+2 \\sqrt{y})$ $=\\sqrt{y}(6 \\sqrt{x}+5 \\sqrt{y})$. Find the value of $\\frac{x+\\sqrt{x y}-y}{2 x+\\sqrt{x y}+3 y}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to solve this problem where ... |
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. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. I have a regula... |
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's determine all the triples of numbers $(x, y, m)$ for which\n\n$$\n-2 x + 3 y = 2 m, \\quad x - 5 y = -11\n$$\n\nand $x$ is a negative integer, $y$ is a positive integer, and $m$ is a real number."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "## Zadatak B-1.2.\n\nNa slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.\n\n\n\n"
},
{
"from... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "The equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of ... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Let's determine $m$ such that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$.\n\n---\n\nDetermine $m$ so that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$."
},
{
"from": "assistant",
"value": "<... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. The problem is asking h... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "Example 1 The range of the function $y=-x^{2}-2 x+3(-5 \\leqslant x \\leqslant 0)$ is $(\\quad)$.\n(A) $(-\\infty, 4]$\n(B) $[3,12]$\n(C) $[-12,4]$\n(D) $[4,12]$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. I need to find the range of th... |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process
before providing the final precise and accurate solutions.
This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to devel... | [
{
"from": "user",
"value": "4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30... |
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. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|... |
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": "All three vertices of $\\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? \n$\\textbf{(A)}\\ 4\\qquad\\textbf{(B)}\\ 6\\qquad\\textbf{(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": "1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this 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": "2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).\n(A) $\\frac{\\sqrt{5}+1}{2}$\n(B) $\\frac{\\sqrt{5}-1}{2}$\n(C) 1\n(D) 2"
},
{
"from": "assistant",
"v... |
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": "V-2 If one side of the rectangle is reduced by $3 \\mathrm{~cm}$, and the other side is reduced by $2 \\mathrm{~cm}$, we get a square whose area is $21 \\mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.\n\n![](https://cdn.mathpix.co... |
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. Given $a, b>0, a \\neq 1$, and $a^{b}=\\log _{a} b$, then the value of $a^{a^{b}}-\\log _{a} \\log _{a} b^{a}$ is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this problem here: Given that a and b are both positive numbers, a isn't equ... |
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. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{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": "In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?"
},
{
"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": "18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues tow... |
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": "53. How many four-digit numbers contain at least one even digit?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to figure out how many four-digit numbers contain at least one even digit. Hmm, let's see. First, let me make sure I understand ... |
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. The range of the function $f(x)=\\sin x+\\cos x+\\tan x+$ $\\arcsin x+\\arccos x+\\arctan x$ is $\\qquad$ ."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the range of the function f(x) = sin x + cos x + tan x + arcsin x + arccos... |
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 $a_1,a_2,\\ldots,a_n$ be a permutation of the numbers $1,2,\\ldots,n$, with $n\\geq 2$. Determine the largest possible value of the sum \\[ S(n)=|a_2-a_1|+ |a_3-a_2| + \\cdots + |a_n-a_{n-1}| . \\]\r\n[i]Romania[/i]"
},
{
"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": "There is a regular $17$-gon $\\mathcal{P}$ and its circumcircle $\\mathcal{Y}$ on the plane. \nThe vertices of $\\mathcal{P}$ are coloured in such a way that $A,B \\in \\mathcal{P}$ are of diff\u000berent colour, if the shorter arc connecting $A$ and $B$ on $\\mathcal{Y}$ has $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": "10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's try to figure out this problem. So, we have an isosc... |
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. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, 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": "1. Let $S=\\{1,2, \\cdots, n\\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find 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": "Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\\theta$. The ma... |
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": "9.171. $0.6^{\\lg ^{2}(-x)+3} \\leq\\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to solve this inequality: 0.6^(lgΒ²(-x) + 3) β€ (5/3)^(2 lg xΒ²). Hmm, let me start by breaking it down. \n\nFirst, ... |
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. Find all integer solutions of the inequality\n\n$$\nx^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \\leq 0\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I have this inequality to solve for integer solutions:\n\nxΒ²yΒ² + ... |
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": "Solve the triangle whose area $t=357.18 \\mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.\n\n"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's see. I have to solve a triangle where the area is 357.18 cmΒ², and the sides are 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": "Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.\n\\[ 2^m p^2 + 1 = q^5 \\]"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I need to find all triples (m, p, q) where m is a positive integer and p, q are pri... |
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": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two... |
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": "25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna 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": "Given the complex number $z$ has a modulus of 1. Find\n$$\nu=\\frac{(z+4)^{2}-(\\bar{z}+4)^{2}}{4 i} \\text {. }\n$$\n\nthe maximum value."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, let's try to figure out how to find the maximum value of this ex... |
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 1\n\nPerform the calculations:\n\na) $7 \\cdot 147 - 7 \\cdot 47$ (1p)\n\nb) $(2+4+6+8+\\cdots+50)-(1+3+5+7+\\cdots+49)$ (2p)\n\nc) $10 \\cdot 9^{2} : 3^{2} - 3^{4} \\quad(2 \\text{p})$\n\nd) $(\\overline{a b} + \\overline{b c} + \\overline{c a}) : (a + b + c) \\quad$ ... |
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. The curve $(x+2 y+a)\\left(x^{2}-y^{2}\\right)=0$ represents three straight lines intersecting at one point on the plane if and only if\nA. $a=0$\nB. $a=1$\nC. $a=-1$\nD. $a \\in \\mathbf{R}$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I hav... |
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": "9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so the problem is: At the 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": "2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).\n(A) the I quadrant\n(B) the II quadrant\n(C) the III quadrant\n(D) the IV quadra... |
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": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\tex... |
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": "Father played chess with uncle. For a won game, the winner received 8 crowns from the opponent, and for a draw, nobody got anything. Uncle won four times, there were five draws, and in the end, father earned 24 crowns.\n\nHow many games did father play with uncle?\n\n(M. VolfovΓ‘)... |
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\n\nGiven the numbers $1,2,3, \\ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.\n\nSelected problems by Prof. 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": "Problem 5.3. In five of the nine circles in the picture, the numbers 1, 2, 3, 4, 5 are written. Replace the digits $6, 7, 8, 9$ in the remaining circles $A, B, C, D$ so that the sums of the four numbers along each of the three sides of the triangle are the same.\n\n![](https://cd... |
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 function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $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": "18.3.19 $\\star \\star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \\leqslant b \\leqslant c$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I need to find all positive integer triples (a, b, 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": "5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $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": "4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \\neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$."
... |
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. Given that $z$ is a complex number, and $|z|=1$. When $\\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\\qquad$ or . $\\qquad$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nAlright, so I have this problem here wher... |
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: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \\sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\\prime}$ such that $B^{\\prime} C: C B$ $=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": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\nOkay, so I need to find the integer part of M, which is defined as the sum of two squar... |
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