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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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. A circle passing through the vertices \\(A, B, D\\) of trapezoid \\(ABCD\\) intersects its lateral side \\(CD\\) at point \\(K\\). Prove that the circle circumscribed around triangle \\(BCK\\) is tangent to line \\(AB\\)."
},
{
... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Problem statement\n\nThere is a positive integer sequence $ a_1, a_2, \\ ldots, a_N $ of length $ N ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Petya has a string of length *n* consisting of small and large English letters and digits.\n\nHe per... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a convex quadrilateral, two adjacent sides, \\(a = 28.29 \\mathrm{~cm}\\) and \\(b = 11.71 \\mathrm{~cm}\\), are perpendicular to each other. The other two sides are equal and also form a right angle. Calculate the area of the quadr... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given \\( f(x) = \\frac{1 + x}{1 - 3x} \\), \\( f_{1}(x) = f[f(x)] \\), \\( f_{2}(x) = f\\left[f_{1}(x)\\right] \\), \\(\\cdots\\), \\( f_{n+1}(x) = f\\left[f_{n}(x)\\right] \\), find \\( f_{2017}(-2) \\).\n\nA) \\( -\\frac{1}{7} \\)\n... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Cut out a $2 \\times 2$ square from one corner of an $8 \\times 8$ grid. Can the remaining 60 squares be divided into 15 pieces, each shaped like a $2 \\times 1$ domino?"
},
{
"from": "assistant",
"value": "<|begin_of_thoug... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( A, B, C, D, E, F \\) be 6 points on a circle in that order. Let \\( X \\) be the intersection of \\( AD \\) and \\( BE \\), \\( Y \\) is the intersection of \\( AD \\) and \\( CF \\), and \\( Z \\) is the intersection of \\( CF... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. For which natural numbers \\( n \\) is the sum \\( 5^n + n^5 \\) divisible by 13? What is the smallest \\( n \\) that satisfies this condition?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "How do non-abelian gauge fields affect particle physics in the context of string theory? Specifically, what are the implications for the behavior and interactions of particles at the quantum level?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "How do environmental toxins affect the development and functioning of the brain, and what measures can be taken to prevent or mitigate their negative effects?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out how environmental ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. Return the function body without invoking it at the final solution.## Sum Even Fibonacci Numbers\n\n* Write a func named SumEvenFibonacci that takes a parameter of type int and returns a value of type int\n\n... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The students were fishing. It is known that \\( a_{1} \\) fishermen caught at least one fish, \\( a_{2} \\) fishermen caught more than one fish, \\( a_{3} \\) - more than two fish, and so on, up to \\( a_{10} \\) - more than nine fish.... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a right triangle, inscribe a rectangle with a common right angle, such that the diagonal of the rectangle is minimal."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out how to i... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a triangle \\(ABC\\). An incircle is inscribed in it, touching the sides \\(AB\\), \\(AC\\), and \\(BC\\) at the points \\(C_1\\), \\(B_1\\), \\(A_1\\) respectively. Find the radius of the excircle \\(w\\), which touches the side... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. What is the value of $1+3+5+\\cdots+2017+2019-2-4-6-\\cdots-2016-2018$?\n$\\textbf{(A) }-1010\\qquad\\textbf{(B) }-1009\\qquad\\textbf{(C) }1008\\qquad\\textbf{(D) }1009\\qquad \\textbf{(E) }1010$"
},
{
"from": "assistant",
... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nConvert the temperature to the Kelvin scale: \\( T = 273 + 95 = 368 \\) K. To calculate \\( \\Delta G^{0}{ }_{368} \\), use the equation:\n\\[ \\Delta G^{\\mathrm{o}}{ }_{368} = \\Delta H^{o} - T \\Delta S^{o} \\]\n\nThe entropy chan... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In triangle \\( \\triangle ABC \\), prove:\n(1) \\( \\cos \\frac{A}{2} < \\cos \\frac{B}{2} + \\cos \\frac{C}{2} \\);\n(2) \\( \\cos \\frac{A}{2} < \\sin \\frac{B}{2} + \\sin \\frac{C}{2} \\)."
},
{
"from": "assistant",
"va... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find all functions $ f:R \\implies R $ , such for all $x,y,z$ $f(xy)+f(xz)\\geq f(x)f(yz) + 1$ "
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, let's tackle this functional equation problem. We ne... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The segment connecting the centers of two intersecting circles is divided by their common chord into segments equal to 5 and 2. Find the common chord, given that the radius of one circle is twice the radius of the other."
},
{
... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. After marking 12 balls with the numbers $1, 2, 3, \\cdots, 12$ and placing them in a box, three people (A, B, and C) each take out 4 balls. It is known that the sum of the numbers on the balls each person took is equal. A has two balls... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. (1) Does there exist positive integers \\( a \\) and \\( b \\) such that for any positive integer \\( n \\), the number \\( 2^{n}a + 5^{n}b \\) is a perfect square?\n\n(2) Does there exist positive integers \\( a \\), \\( b \\), and \\... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Prove that for all natural numbers \\( n \\), the smallest integer greater than \\( (\\sqrt{3}+1)^{2n} \\) is divisible by \\( 2^{n+1} \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to pro... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Calculate the dihedral angle at the apex of a regular square pyramid, given that the centers of the inscribed and circumscribed spheres coincide."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's try ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.The MEX (minimum excluded) of an array is the smallest non-negative integer that does not belong to ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The sum of $n$ terms of a geometric sequence $\\left\\{b_{n}\\right\\}$ is 6 times less than the sum of the reciprocals of these terms. Find the product $b_{1} b_{n}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Chef is asked to write a program that takes an array A of length N and two integers X, Y as input an... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \\( f(1)=1 \\) and \\( f(x y + f(x)) = x f(y) + f(x) \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's try to tackle t... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nIn an acute-angled triangle \\(SAP\\), the altitude \\(AK\\) is drawn. A point \\(L\\) is selected on the side \\(PA\\), and a point \\(M\\) is chosen on the extension of the side \\(SA\\) beyond point \\(A\\) such that \\(\\angle LS... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. \\( f \\) is a one-to-one function from \\( \\mathbf{R} \\) to \\( \\mathbf{R} \\), and \\( y = f(x) \\) is strictly increasing. The solution set of the equation \\( x = f(x) \\) is \\( P \\), and the solution set of the equation \\( x... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given point \\( A \\) is a point inside a unit circle centered at \\( O \\), satisfying \\( |\\overrightarrow{OA}| = \\frac{1}{2} \\). Points \\( B \\) and \\( C \\) are any two points on the unit circle \\( O \\). Determine the range ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $(a_n)_{n\\ge0}$ be a sequence of positive real numbers such that\n\\[\\sum_{k=0}^nC_n^ka_ka_{n-k}=a_n^2,\\ \\text{for any }n\\ge 0.\\]\nProve that $(a_n)_{n\\ge0}$ is a geometric sequence.\n\n*Lucian Dragomir*"
},
{
"f... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find all real numbers \\(x, y\\) satisfying \\(x^{5}+y^{5}=33\\) and \\(x+y=3\\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, let me try to figure out how to solve this system of equations: \\(x^5 ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. What is the degree measure of an angle whose measure is double the measure of its complement?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see. The problem is asking for the degree measure of an ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "I was annoyed the other day when I got a point off in math class because I didn't distribute the factors of my function. \nThat is, I put (x+2)(x-3) when I should have put x^2 - x - 6.\nYour job is to write a program that takes a string with the expression and convert it to the e... | I was annoyed the other day when I got a point off in math class because I didn't distribute the factors of my function.
That is, I put (x+2)(x-3) when I should have put x^2 - x - 6.
Your job is to write a program that takes a string with the expression and convert it to the expanded form so I don't miss future points... | Okay, so I need to write a program that takes a string representing a product of factors in parentheses, like (x+2)(x-3), and expands it into the standard polynomial form, like x^2 -x -6. The challenge is to do this without using any built-in expansion libraries, and there's a code-golf aspect so I need to make the cod... | To solve this problem, we need to expand a product of polynomial factors into their standard polynomial form. The solution involves parsing the input, expanding each factor, multiplying the polynomials, combining like terms, sorting the terms by degree, and formatting the output correctly.
### Approach
1. **Parse Inpu... | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. On semicircle, with diameter $|AB|=d$ , are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$ "
},
{
"from": "assistan... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a row, 23 natural numbers (not necessarily distinct) are written. Prove that it is possible to place parentheses, addition, and multiplication signs between them such that the value of the resulting expression is divisible by 2000 e... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Read problems statements in Mandarin chinese and Vietnamese as well. \n\nYou are given a string $S$.... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the solution set of the equation \\( x^{\\lg x} = \\frac{x^{3}}{100} \\). \n\nOptions:\n(A) Only \\( \\frac{1}{10} \\).\n(B) Only 10.\n(C) Only 100.\n(D) Only 10 or 100.\n(E) More than two real numbers.\n\n(Note: From the 13th Ann... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. On a plane, several regular $n$-gons are given. Prove that the convex hull of their vertices has at least $n$ vertices."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to prove that the convex h... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a rectangle \\(ABCD\\) where \\(AB = 5\\) and \\(BC = 6\\). Point \\(K\\) lies on side \\(CD\\), point \\(L\\) on side \\(AB\\), point \\(M\\) on side \\(BC\\), and point \\(N\\) on side \\(CD\\). Prove that the length of the clo... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. Return the function body without invoking it at the final solution.In this Kata, two players, Alice and Bob, are playing a palindrome game. Alice starts with `string1`, Bob starts with `string2`, and the boar... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In the convex quadrilateral \\(ABCD\\), \\(AD\\) and \\(BC\\) are not parallel, and there exists a circle \\(\\Gamma\\) that is tangent to sides \\(BC\\) and \\(AD\\) at points \\(C\\) and \\(D\\) respectively. Furthermore, let \\(AC\\... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Call a number $T$ *persistent* if the following holds: Whenever $a,b,c,d$ are real numbers different from $0$ and $1$ such that $$ a+b+c+d = T $$ and $$ \\frac{1}{a}+\\frac{1}{b} +\\frac{1}{c}+\\frac{1}{d} = T, $$ we also h... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. There are two regular pentagons with one common vertex. The vertices of each pentagon are numbered clockwise from 1 to 5, with the common vertex labeled as 1. Vertices with the same numbers are connected by straight lines. Prove that t... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In triangle $CAT$ , we have $\\angle ACT = \\angle ATC$ and $\\angle CAT = 36^\\circ$ . If $\\overline{TR}$ bisects $\\angle ATC$ , then $\\angle CRT =$ [asy]\npair A,C,T,R;\nC = (0,0); T = (2,0); A = (1,sqrt(5+sqrt(20))); R = ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.A manufacturing project consists of exactly $K$ tasks. The board overviewing the project wants to hi... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given \\( a \\) and \\( b \\) in the interval \\([1, \\sqrt{3}]\\), find the maximum and minimum values of \\(\\frac{a^{2}+b^{2}-1}{a b}\\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, so I have th... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Suppose that $a, b, c$ , and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$ ."
},
{
"from": "assistant",
"value":... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. On a hot day, Megan likes to eat a Popsicle every 15 minutes. Assuming she keeps up that rate of consumption, how many Popsicles can Megan finish in 4 hours and 30 minutes?"
},
{
"from": "assistant",
"value": "<|begin_of_th... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given positive integers \\( x \\) and \\( y \\) such that \\( x < y \\), let \\( p = \\frac{x^3 - y}{1 + xy} \\). Find all possible integer values of \\( p \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nO... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "She celebrated her success by giving a speech, where was she?\nA: graveyard\nB: awards ceremony\nC: various situations\nD: vocation\nE: working hard"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's try to figure out this question. The question ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( a_{1}, a_{2}, a_{3}, \\ldots \\) be an infinite increasing sequence of natural numbers, and let \\( p_{1}, p_{2}, p_{3}, \\ldots \\) be a sequence of prime numbers such that for each natural \\( n \\), the number \\( a_{n} \\) ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Create a program that computes the hamming weight of a string. Winner is the program with the lowest hamming weight.\nRules:\n\nHamming weight for an ASCII character is defined as the total number of bits set to 1 in its binary representation.\nAssume input encoding is 7-bit ASCI... | Create a program that computes the hamming weight of a string. Winner is the program with the lowest hamming weight.
Rules:
Hamming weight for an ASCII character is defined as the total number of bits set to 1 in its binary representation.
Assume input encoding is 7-bit ASCII, passed through whatever input mechanism i... | Okay, I need to create a program that calculates the Hamming weight of a string. The Hamming weight here is the total number of 1 bits in all the characters of the input string. The twist is that the program's own code needs to have the lowest possible Hamming weight. Hmm, that's an interesting challenge.
First, let m... | To solve this problem, we need to compute the Hamming weight of a given string, which is defined as the total number of 1 bits in the binary representation of each character in the string. The challenge is to write a program whose own code has the lowest possible Hamming weight.
### Approach
The optimal solution invo... | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. For which odd values of $n$ is it possible to cover all the black squares of an $n \\times n$ checkerboard, where the corners are black, using L-shaped pieces that each cover 3 squares?"
},
{
"from": "assistant",
"value": "... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( r_0 \\) be the radius of the excircle of a tetrahedron touching the face with area \\( S_0 \\), and \\( S_1, S_2, \\) and \\( S_3 \\) be the areas of the other faces of the tetrahedron. Prove that the volume of the tetrahedron ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. If you cut down all the trees on one hectare of forest, you can produce 100 cubic meters of planks. Assuming all trees in the forest are identical, evenly distributed, and each tree can yield 0.4 cubic meters of planks, determine the a... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the real numbers \\(a\\) and \\(b\\) such that the equation \\(\\frac{a x^{2}-24 x+b}{x^{2}-1}=x\\) has two roots, and the sum of these roots is equal to 12, counting repeated roots as a single root."
},
{
"from": "assista... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Eudokimus, a system administrator is in trouble again. As a result of an error in some script, a lis... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( f: \\mathbf{N}^{+} \\rightarrow \\mathbf{N}^{+} \\) be a function such that for any \\( m, n \\in \\mathbf{N}^{+} \\), \\( f(m+n) \\geqslant f(m) + f(n) \\). Additionally, it is given that \\( f(2) = 0 \\), \\( f(3) > 0 \\), an... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The polynomials \\( p_n(x) \\) are defined by \\( p_0(x) = 0 \\), \\( p_1(x) = x \\), \\( p_{n+2}(x) = x p_{n+1}(x) + (1 - x) p_n(x) \\). Find the real roots of each \\( p_n(x) \\)."
},
{
"from": "assistant",
"value": "<|be... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2882 is a four-digit palindrome and 49194 is a five-digit palindrome. There are pairs of four-digit palindromes whose sum is a f... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find all solutions to the equation \\( x^2 - 8[x] + 7 = 0 \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this equation to solve: \\( x^2 - 8[x] + 7 = 0 \\). Hmm, I need to find all real so... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $A$ and $B$ be two assertions. Show the following equivalences:\n\n1. $\\operatorname{non}(A$ and $B) \\Longleftrightarrow($ not $A)$ or $($ not $B)$\n2. $\\operatorname{non}(A$ or $B) \\Longleftrightarrow($ not $A)$ and $($ not $B... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.We have N logs of lengths A_1,A_2,\\cdots A_N.\n\nWe can cut these logs at most K times in total. Wh... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Investigate the convergence of the series\n\n$$\n\\frac{\\sin 1}{1^{2}}+\\frac{\\sin 2}{2^{2}}+\\frac{\\sin 3}{3^{2}}+\\ldots+\\frac{\\sin n}{n^{2}}+\\ldots\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nO... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nOne vertex of an equilateral triangle lies on a circle, and the other two vertices divide a certain chord into three equal parts.\n\nAt what angle is the chord seen from the center of the circle?"
},
{
"from": "assistant",
... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let the function \\( f: \\{1, 2, \\cdots\\} \\rightarrow \\{2, 3, \\cdots\\} \\) satisfy, for all positive integers \\( m \\) and \\( n \\),\n\\[ f(m+n) \\mid (f(m) + f(n)). \\]\nProve that there exists a positive integer \\( c > 1 \\)... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Ottó decided to assign a number to each pair \\((x, y)\\) and denote it as \\((x \\circ y)\\). He wants the following relationships to hold:\na) \\(x \\circ y = y \\circ x\\)\nb) \\((x \\circ y) \\circ z = (x \\circ z) \\circ (y \\circ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. There exists an integer $n$ such that $\\sqrt{p+n}+\\sqrt{n}$ is an integer prime $p$.\n(A) There does not exist such an integer;\n(B) There exists only one such integer;\n(C) There exist more than one but a finite number of such integ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=a, a_{2}=b, 2a_{n+2}=a_{n+1}+a_{n}$. If $\\lim _{n \\rightarrow \\infty}\\left(a_{1}+a_{2}+\\cdots+a_{n}\\right)=4$, find the values of $a$ and $b$."
},
{
"from": "assi... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Davi goes to a store that sells a bottle of orange juice for $R \\$ 2.80$ and a box with six of these bottles for $R \\$ 15.00$. He needs to buy 22 bottles for his birthday. How much will he spend at the minimum?"
},
{
"from": ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$. Point $X$ is chosen on edge $A_{1} D_{1}$, and point $Y$ is chosen on edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects r... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Prove that for \\( 0 \\leq m \\leq n \\):\n\n$$\n\\sum_{k=0}^{n}\\binom{n}{k}\\binom{k}{m}=\\binom{n}{m} 2^{n-m}\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to prove that the sum from k=... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Farmer John is obsessed with making Bessie exercise more!\n\nBessie is out grazing on the farm, whic... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Suppose $G$ is a graph with chromatic number $n$. Suppose there exist $k$ graphs $G_{1}, G_{2}, \\ldots, G_{k}$ having the same vertex set as $G$ such that $G_{1} \\cup G_{2} \\cup \\cdots \\cup G_{k}=G$ and each $G_{i}$ has chromatic ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a peculiar animal farm, there are cats and dogs. There are 180 more dogs than cats. 20% of the dogs mistake themselves for ( ) animals.\nA. 240 \nB. 248 \nC. 420 \nD. 842"
},
{
"from": "assistant",
"value": "<|begin_o... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The random variable \\( X \\) is given by the probability density function\n\n$$\np(x)=\\frac{1}{\\sigma \\sqrt{2 \\pi}} e^{-x^{2} / 2 \\sigma^{2}}\n$$\n\nFind the probability density function of the random variable \\( Y=X^{3} \\)."
... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the smallest natural number \\( n \\) for which the number \\( A = n^3 + 12n^2 + 15n + 180 \\) is divisible by 23."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the smallest natur... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Write 2004 numbers on the blackboard: $1, 2, \\cdots, 2004$. In each step, erase some numbers and write their sum modulo 167. After several steps, among the two numbers remaining on the blackboard, one of them is 999. What is the other... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Cut a $6 \\times 6$ square grid into \"L-shaped\" tiles consisting of three cells each, such that no two tiles form a $2 \\times 3$ rectangular block."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let m be a positive integer. Prove that there exist infinitely many prime numbers p such that m+p^3 is composite."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, so I need to prove that for any positi... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In a certain math competition, there are 6 questions, each worth 7 points (each question's score is an integer between 0 and 7). The total score of each participant is the product of the scores of the 6 questions. If two participants h... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The function $f$ from the set $\\mathbb{N}$ of positive integers into itself is defined by the equality \\[f(n)=\\sum_{k=1}^{n} \\gcd(k,n),\\qquad n\\in \\mathbb{N}.\\]\na) Prove that $f(mn)=f(m)f(n)$ for every two relatively pri... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "\"How does urbanization affect the diversity of insect species in a given area and what are the specific environmental factors that contribute to this change?\""
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's try to figure out how urbanization... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a regular hexagon \\( A_6 \\) where each of its 6 vertices is colored with either red or blue, determine the number of type II colorings of the vertices of the hexagon."
},
{
"from": "assistant",
"value": "<|begin_of_... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Let $s$ be a string of lowercase Latin letters. Its price is the sum of the indices of letters (an i... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.One of Arkady's friends works at a huge radio telescope. A few decades ago the telescope has sent a ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Angela and Barry share a piece of land. The ratio of the area of Angela's portion to the area of Barry's portion is $3: 2$. They each grow corn and peas on their piece of land. The entire piece of land is covered by corn and peas in th... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a segment \\( AB \\), a line \\( l \\) that is not parallel to it, and a point \\( M \\) on this line, using only a two-sided ruler, construct the points of intersection of the line \\( l \\) with the circle of radius \\( AB \\) ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Xiaojun went to the stationery store for pens. It is known that watercolor pens cost 1 yuan and 7 jiao each, and ballpoint pens cost 1 yuan and 2 jiao each. She spent exactly 15 yuan. How many watercolor pens and ballpoint pens did Xia... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The last digit of a natural number is 2016 times smaller than the number itself. Find all such numbers."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find all natural numbers where the last... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. 5n is a positive integer, and its representation in base b is 777. Find the smallest positive integer b such that n is a perfect fourth power."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's tackle ... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. For each of the numbers 1, 19, 199, 1999, etc., a separate card is made and the number is written on it.\n\na) Is it possible to select at least three cards such that the sum of the numbers on them equals a number that has all digits a... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Calculate the product of all positive integers less than $100$ and having exactly three positive divisors. Show that this product is a square."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need t... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Three parallelograms are constructed with correspondingly parallel sides, having the sides of a given triangle as their diagonals. Prove that the other diagonals of these parallelograms intersect at a single point."
},
{
"from"... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. From the set \\( M = \\{1, 2, \\cdots, 36\\} \\), remove \\( n \\) numbers such that the sum of any two remaining elements is not a factor of 2015. Find the minimum value of \\( n \\)."
},
{
"from": "assistant",
"value": "<... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. How many equilateral hexagons of side length \\(\\sqrt{13}\\) have one vertex at \\((0,0)\\) and the other five vertices at lattice points?\n(A lattice point is a point whose Cartesian coordinates are both integers. A hexagon may be co... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\(\\alpha\\) and \\(\\beta\\) be a pair of conjugate complex numbers. If \\(|\\alpha - \\beta| = 2 \\sqrt{3}\\) and \\(\\frac{\\alpha}{\\beta^{2}}\\) is a real number, then \\(|\\alpha| = \\) \\(\\qquad\\)."
},
{
"from": "... | null | null | null | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Write a Manufactoria program that will accept the empty input tape. But don't do it quickly! I mean, write the program quickly, but don't let it run quickly. The slower the program, the better, as long as it terminates eventually. The example program below takes 3:51 (the \"t... | Write a Manufactoria program that will accept the empty input tape. But don't do it quickly! I mean, write the program quickly, but don't let it run quickly. The slower the program, the better, as long as it terminates eventually. The example program below takes 3:51 (the "total time" reported by the simulator).
h... | Okay, so I need to create a Manufactoria program that accepts the empty input tape but does so as slowly as possible. The example provided takes about 3:51, and I need to make one that's even slower. Let me think about how to approach this.
First, I remember that in Manufactoria, the robot moves the tape through vario... | To create a slow-accepting Manufactoria program for the empty tape, we maximize steps with nested loops and tape manipulations. Here's a solution:
1. **Entry & Initial Write**: Start at (9,8) facing right. Use pushers and conveyors to write multiple colors in a loop.
2. **Extended Processing Loop**: After each write, ... | null | null | null | null | null | null | null |
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 develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Determine the best estimate for $\\lg 2$ using the given table:\n\n\\(10^3=1000, 10^4=10000, 2^{10}=1024, 2^{11}=2048, 2^{12}=4096, 2^{13}=8192\\)\n\nChoose the best range for $\\lg 2$ from the following options:\n(A) \\(\\frac{3}{10}\... | null | null | null | null | null | null | null | null | null | null |
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