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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 cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal to the nearest tenth. [asy]\n\nimport markers;\nsize(150... | 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 projection of the vector $\\bar{a}=3 \\bar{i}+4 \\bar{j}-\\bar{k}$ in the direction of the vector $\\bar{b}=\\bar{i}+\\bar{j}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find 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{}. Let \\( P \\in \\mathbb{R}[X] \\) be a monic polynomial of degree 2020 such that \\( P(n) = n \\) for all \\( n \\in \\{0, 1, \\ldots, 2019\\} \\). Calculate \\( P(2020) \\)."
},
{
"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": "Return your final response within \\boxed{}. Find all subsequences \\( \\{a_{1}, a_{2}, \\cdots, a_{n}\\} \\) of the sequence \\(\\{1, 2, \\cdots, n\\}\\) that satisfy the property \\( i + 1 \\mid 2(a_{1} + a_{2} + \\cdots + a_{j}) \\) for \\(1 \\leq i \\leq 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{}. During a trip, person A first drives 50 miles on the highway and then 10 miles on the coastal road. The speed at which person A drives on the highway is 3 times the speed on the coastal road. If person A spent 30 minutes driving on the... | 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.You are given two images img1 and img2 both of size n x n, represented as binary, square matrices of the same size. (A binary matrix has only... | 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 an acute triangle \\(ABC\\). The altitude \\(AA_1\\) is extended through vertex \\(A\\) by a segment \\(AA_2 = BC\\). The altitude \\(CC_1\\) is extended through vertex \\(C\\) by a segment \\(CC_2 = AB\\). Find the angles of tri... | 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 [Russian] and [Bengali].\n\nThere's a zebra crossing appearing in the mi... | 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 the function\n$$\ny(x) = (\\sqrt{1+x} + \\sqrt{1-x} + 2) \\left(\\sqrt{1-x^2} + 1 \\right),\n$$\n\nwhere \\(x \\in [0,1]\\). Determine the minimum value of \\(y(x)\\)."
},
{
"from": "assistant",
"value": "<|begin_of_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{}. Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at... | 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 sequence of \\( n \\) distinct numbers, it is required to simultaneously find the largest and the second largest numbers.\n\na) Prove that this can be done by comparing \\( n+m-2 \\) pairs of numbers, where the integer \\( m \\) 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{}. A hexahedron \\(ABCD E\\) is made of two regular congruent tetrahedra \\(ABCD\\) and \\(ABCE\\). Prove that there exists only one isometry \\(\\mathcal{Z}\\) that maps points \\(A, B, C, D, E\\) onto \\(B, C, A, E, D\\), respectively. ... | 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 smallest positive integer \\( p \\) for which \\(\\sqrt{2^{3} \\times 5 \\times p}\\) is an integer?\n(A) 2\n(B) 5\n(C) 10\n(D) 1\n(E) 20"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let'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{}. (F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \\angle A \\equal{} 50^{\\circ}$ . Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \\angle KAC \\equal{} \\angle LAB \\equal{} 10^{\\circ}$ . Det... | 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": "\"What is the expected temperature range for dark atoms or dark molecules to remain stable in a dark matter halo based on current theoretical models?\""
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so the user is asking about the expected temperat... | 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 line \\( l \\) and a point \\( A \\). An arbitrary line passing through \\( A \\) and skew to \\( l \\) is constructed. Let \\( M \\) and \\( N \\) be the endpoints of the common perpendicular to this arbitrary line and \\( l \... | 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 2013 cards, each with a unique number written on them, all face down. In one move, you can point to ten cards, and you will be told one of the numbers written on those ten cards (but not which card the number is on). What is ... | 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 $P(x), Q(x), $ and $R(x)$ be three monic quadratic polynomials with only real roots, satisfying $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$ $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$ for all real numbers $x.$ What is $P(0)+Q(0)+R(0... | 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 we connect the point of intersection $A$ of two tangents to an ellipse with one of its foci $G$, the line $AG$ bisects the angle $KGH$, where $K$ and $H$ are the points where the tangents touch the ellipse. Prove this theorem if the... | 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{}. \nKing Arthur has two equally wise advisors – Merlin and Percival. Each of them finds the correct answer to any question with a probability \\( p \\), or an incorrect answer with a probability \\( q = 1-p \\).\n\nIf both advisors give ... | 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 natural numbers \\( n \\) such that the sum of the digits of the number \\( 2^n \\) in decimal notation equals 5."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find all natural num... | 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}{2-x} \\), for \\( n \\in \\mathbf{N} \\), define \\( f_{1}(x)=f(x) \\) and \\( f_{n+1}(x)=f\\left[f_{n}(x)\\right] \\). If \\( f_{13}(x)=f_{31}(x) \\), what is the expression for \\( f_{16}(x) \\)?\nA. \\( ... | 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.Takahashi has many red balls and blue balls. Now, he will place them in a row.\n\nInitially, there 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{}. The side $AC$ of triangle $ABC$ is extended at segment $CD = AB = 1$ . It is known that $\\angle ABC = 90^o$ , $\\angle CBD = 30^o$ . Calculate $AC$ ."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOka... | 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 $A, B, C, D, E$ be integers, $B \\neq 0$ and $F=AD^{2}-BCD+B^{2}E \\neq 0$ . Prove that the number $N$ of pairs of integers $(x, y)$ such that \\[Ax^{2}+Bxy+Cx+Dy+E=0,\\] satisfies $N \\le 2 d( \\vert F \\vert )$ , where ... | 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{}. Buying 2 bottles of soda and 1 bottle of mineral water costs 7 yuan in total. Buying 4 bottles of soda and 3 bottles of mineral water costs 16 yuan in total. How much will it cost to buy 10 bottles of soda and 10 bottles of mineral wat... | 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 budget of the Petrov family consists of the following sources of income:\n\n- Parents' salary after income tax deduction: 56000 rubles;\n- Grandmother's pension: 14300 rubles;\n- Son's scholarship: 2500 rubles.\n\nThe average month... | 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 ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \\Big(x+\\frac{1}{3}\\Big) + f \\Big(x+\\frac{1}{2}\\Big)=f(x)+ f \\Big(x+\\frac{5}{6}\\Big)}$ . Show... | 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 $ABC$, the altitude $AH$ passes through the midpoint of the median $BM$. Prove that in triangle $BMC$, one of the altitudes also passes through the midpoint of one of the medians."
},
{
"from": "assistant",
"val... | 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 $(4n + 2)\\times (4n + 2)$ square grid, a turtle can move between squares sharing a side.The turtle begins in a corner square of the grid and enters each square exactly once, ending in the square where she started. In terms 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": "Return your final response within \\boxed{}. Define $[a,b,c]$ to mean $\\frac {a+b}c$, where $c \\neq 0$. What is the value of \n\n$\\left[[60,30,90],[2,1,3],[10,5,15]\\right]?$\n$\\mathrm{(A) \\ }0 \\qquad \\mathrm{(B) \\ }0.5 \\qquad \\mathrm{(C) \\ }1 \\qquad \\mathrm{(D) \\ }... | 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 volume of the tetrahedron \\(ABCD\\) is 5. A plane passes through the midpoints of edges \\(AD\\) and \\(BC\\) and intersects edge \\(CD\\) at point \\(M\\). The ratio of the length of segment \\(DM\\) to the length of segment \\(C... | 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.Little Shino is interested in the fighting tournaments. Once she went to watch one of the tournament... | 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 the area of a triangle, whose sides are equal to the medians of a triangle with area $S$, is equal to $3S / 4$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to prove that the area ... | 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_n \\) be the least common multiple (LCM) of the numbers \\( 1, 2, \\ldots, n \\). For which positive integers \\( n \\) does \\( M_{n-1} = M_n \\) hold? Prove your conclusion."
},
{
"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{}. In an isosceles triangle with a side length of $b$, the angle bisectors of the base angles are drawn. The segment of the line between the points where these bisectors intersect the sides is equal to $m$. Determine the base of the trian... | 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 the product of three consecutive natural numbers is divisible by 504, given that the middle number is a perfect cube."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see. I need to prove ... | 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{}. \nA large conference has 500 delegates attending. If each delegate knows 400 other delegates, is it always possible to select 6 delegates such that every two of them know each other?"
},
{
"from": "assistant",
"value": "<|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": "Return your final response within \\boxed{}. Given $\\frac{x}{y}=\\frac{10}{4}.$ If $y = 18$, what is the value of $x$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the value of x when y is 18, given that the ratio of x to y is ... | 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 convex pentagon \\(ABCDE\\), where \\(DE \\parallel AC\\), \\(\\angle ADC = 90^\\circ\\), \\(AC = 12\\), \\(CD = 6\\), \\(AC\\) bisects \\(\\angle DAB\\), and \\(\\angle BCE = 60^\\circ\\), find the minimum area of \\(\\triangle ... | 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.Berland traffic is very different from traffic in other countries. The capital of Berland consists 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 focus of the hyperbola \\(\\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}=1 (a>0, b>0)\\) is \\(2c\\). The line \\(l\\) passes through points \\((a, 0)\\) and \\((0, b)\\), and the sum of the distances from point \\((1,0)\\) to line \\(l... | 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 \\( a + \\log_{3} 2017 \\), \\( a + \\log_{9} 2017 \\), \\( a + \\log_{27} 2017 \\) (\\( a \\in \\mathbb{R} \\)) form a geometric sequence, what is the common ratio?"
},
{
"from": "assistant",
"value": "<|begin_of_though... | 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 range of the function \\( f(x) = \\sqrt{x-3} + \\sqrt{12-3x} \\) is ( ).\n\n(A) \\( [1, \\sqrt{2}] \\)\n\n(B) \\( \\left[1, \\frac{3}{2}\\right] \\)\n\n(C) \\( [1, \\sqrt{3}] \\)\n\n(D) \\( [1, 2] \\)"
},
{
"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": "Return your final response within \\boxed{}. \nConsider the following two sets of numbers:\n\n$$\nA=\\left\\{\\frac{3 n-4}{5 n-3}: n \\in \\mathbb{Z}\\right\\} \\quad \\text{and} \\quad B=\\left\\{\\frac{4 k-3}{7 k-6}: k \\in \\mathbb{Z}\\right\\} \\text{.}\n$$\n\nHow many elemen... | 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{}. Fold two opposite vertices of a rectangular sheet of paper inward so that the resulting shape is a regular hexagon. When is this task possible?"
},
{
"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": "Return your final response within \\boxed{}. Inscribe a rectangle of base $b$ and height $h$ in a circle of radius one, and inscribe an isosceles triangle in the region of the circle cut off by one base of the rectangle (with that side as the base of the triangle). For what v... | 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, russian and vietnamese as well. \n\nAda has an array 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": "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.Stepan has a set of n strings. Also, he has a favorite string s. \n\nStepan wants to do the followin... | 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 the set $A = \\{1, 2, 3, \\cdots, n\\}$ and its every non-empty subset, define a unique \"alternating sum\" as follows: rearrange the subset in decreasing order, and then starting from the largest number, alternately subtract or ad... | 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 \\(\\left(a_{1}, a_{2}, a_{3}, a_{4}\\right)\\) is a 4-term sequence of real numbers satisfying the following two conditions:\n- \\(a_{3} = a_{2} + a_{1}\\) and \\(a_{4} = a_{3} + a_{2}\\);\n- there exist real numbers \\(a, 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": "Return your final response within \\boxed{}. In a family of 4 people, if Masha's scholarship is doubled, the total income of the family will increase by 5%. If instead, the mother's salary is doubled, the total income will increase by 15%. If the father's salary is doubled, the 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{}. To calculate the product of all natural numbers from 1 to 50 inclusive, it is recommended to use a computer. However, you can easily determine the number of zeros at the end of this product without the help of a computer. How many zero... | 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{}. Tea and a cake cost £4.50. Tea and an éclair cost £4. A cake and an éclair cost £6.50. What is the cost of tea, a cake, and an éclair?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's try to solve 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{}. Let \\( M \\) be a finite set of points in the plane. Prove that there exists a point in \\( M \\) that has at most 3 nearest points within \\( M \\)."
},
{
"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{}. All divisors of a certain natural number, except for one and the number itself, are listed. Some two numbers from this list differ by six times. By how much do the two largest numbers in this list differ?"
},
{
"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": "Return your final response within \\boxed{}. When the integer \\( 300^8 \\) is written out, it has \\( d \\) digits. What is the value of \\( d \\) ?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find out how many digits the number 30... | 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{}. Using the result of problem $\\underline{61403}$, prove the inequalities:\n\na) $\\sqrt[n]{a_{1} \\cdots a_{n}} \\leq \\frac{a_{1} + \\cdots + a_{n}}{n}$ (Cauchy's inequality).\n\nb) $\\left(\\frac{b_{1} + \\cdots + b_{n}}{n}\\right)^{... | 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 $BE$ and $CF$ be altitudes in triangle $ABC$ . If $AE = 24$ , $EC = 60$ , and $BF = 31$ , determine $AF$ ."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to solve this geometry 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{}. For \\( p = 1, 2, \\cdots, 10 \\), let \\( S_{p} \\) be the sum of the first 40 terms of an arithmetic sequence with the first term \\( p \\) and the common difference \\( 2p - 1 \\). What is \\( S_{1} + S_{2} + \\cdots + S_{10} \\)?\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 capacities of three containers $A$, $B$, and $C$, each of which is shaped like a cube, are in the ratio $1: 8: 27$, and the volumes of water initially filled in them are in the ratio $1:2:3$. After transferring some water from cont... | 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 has $K$ chocolates and he wants to distribute them to $N$ people (numbered $1$ through $N$). 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{}. 10 different positive numbers are given. In what order should they be denoted as \\(a_{1}, a_{2}, \\ldots, a_{10}\\) so that the sum \\(a_{1} + 2a_{2} + 3a_{3} + \\ldots + 10a_{10}\\) is maximized?"
},
{
"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{}. Express the number 80 as the sum of two prime numbers. In how many ways can this be done? Note that the number one is not a prime number."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figur... | 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 the prime numbers \\( p \\) and \\( q \\) such that \\( q^5 - 2p^2 = 1 \\), find the value of \\( p + q \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this problem here: given prime ... | 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 that the integer part of the real number \\(\\frac{2+\\sqrt{2}}{2-\\sqrt{2}}\\) is \\(a\\), and the decimal part is \\(1 - b\\), find the value of \\(\\frac{(b-1) \\cdot (5-b)}{\\sqrt{a^{2}-3^{2}}}\\)."
},
{
"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{}. Suppose the prosecutor from the previous problem made the following two statements in court:\n\n1) Either $\\mathrm{X}$ is guilty, or $\\mathrm{Y}$ is guilty.\n2) $\\mathrm{X}$ is not guilty.\n\nWhat conclusion would you reach based on... | 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 rectangle has an area of $16$ and a perimeter of $18$ ; determine the length of the diagonal of the rectangle.\n\n*2015 CCA Math Bonanza Individual Round #8*"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\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{}. We have that $2a + 1 = 1$ and $b - a = 1.$ What is the value of $b$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to figure out the value of b here. So, we have two equations: 2a + 1 = 1 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{}. Let \\( a, b, c, d \\) be real numbers such that \\( a^{2}+b^{2}+c^{2}+d^{2}=1 \\). Determine the minimum value of \\( (a-b)(b-c)(c-d)(d-a) \\) and determine all values of \\( (a, b, c, d) \\) such that the minimum value is achieved."
... | 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{}. Two springs with stiffnesses of 3 kN/m and 6 kN/m are connected in series. How much work is required to stretch this system by 5 cm?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see. The problem ... | 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{}. $2n+1$ students are positioned such that the distances between any two of them are all different. At a given moment, each of them sprays the closest student with a water gun.\n\na) Show that there exists a pair of students who spray ea... | 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 polyhedron has faces that are all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?"
},
{... | 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{}. Solve the following equation for real numbers: $\\log _{a} x=x$, where $a=x^{\\log _{4} x}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to solve this equation step by step. The problem is ... | 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 $\\Gamma$ be a circle with center $O$ and $P$ a point outside the circle. Let $A \\in \\Gamma$ such that the line segment $PA$ is tangent to $\\Gamma$, and let $B$ be the point diametrically opposite to $A$. Let $\\Delta$ be a line... | 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.Last summer Peter was at his granny's in the country, when a wolf attacked sheep in the nearby fores... | 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{}. \nLet \\( f(x) \\) be a function satisfying the following conditions:\n1. If \\( x > y \\) and \\( f(x) + x \\geq w \\geq f(y) + y \\), then there exists a real number \\( z \\in [y, x] \\) such that \\( f(z) = w - z \\);\n2. The equat... | 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 real numbers \\( a_{1}, a_{2}, \\cdots, a_{2016} \\) satisfying \\( 9 a_{i} > 11 a_{i+1}^{2} \\) for \\( i = 1, 2, \\cdots, 2015 \\). Find the maximum value of \\( \\left( a_{1} - a_{2}^{2} \\right) \\left( a_{2} - a_{3}^{2} \\ri... | 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{}. We have three piles of 51, 49, and 5 pebbles. At each step, we can either merge two piles or divide a pile into two equal parts. Is it possible that after a certain number of steps, we end up with 105 piles of one pebble each?"
},
... | 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 $10 \\times 10$ grid, place 1 white piece in any 91 squares. Each time, take a white piece from the grid and place a black piece in an empty square. Then again, take a white piece and place a black piece. Continue this process unt... | 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 positive integers $n$ satisfy the following condition:\n$(130n)^{50} > n^{100} > 2^{200}\\ ?$\n$\\textbf{(A) } 0\\qquad \\textbf{(B) } 7\\qquad \\textbf{(C) } 12\\qquad \\textbf{(D) } 65\\qquad \\textbf{(E) } 125$"
},
{
... | 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's phone shows a Battery Low notification if the battery level is 15 \\% or less.\n\nGiven that ... | 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 Cartesian coordinate system $xOy$, given points $M(-1,2)$ and $N(1,4)$, point $P$ moves along the x-axis. When the angle $\\angle MPN$ reaches its maximum value, the x-coordinate of point $P$ is ______."
},
{
"from": "as... | 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 the ellipse \\(\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1\\) with \\(a > b > 0\\), let points \\(A\\) and \\(B\\) be the left and right vertices of the ellipse, respectively. Point \\(P\\) is another point on the ellipse dif... | 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{}. Define \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$?\n$\\textbf{(A) } 4900 \\qquad \\textbf{(B) } 4950\\qquad \\textbf{(C) } 5000\\qquad \\textbf{(D) } 5050 \\qquad \\textbf{(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": "Introduction\nYou are given a random integer generator with the following implementation\n\nThe first invocation always returns 1.\nThe second invocation returns a random integer between 1 and 2.\nThe third invocation returns a random integer between 1 and 3.\nThe nth invocation ... | Introduction
You are given a random integer generator with the following implementation
The first invocation always returns 1.
The second invocation returns a random integer between 1 and 2.
The third invocation returns a random integer between 1 and 3.
The nth invocation returns a random integer between 1 and n, incl... | Okay, I need to create a dice generator that uses this ascending random number generator. The goal is to get perfectly random numbers between 1 and 6, each with equal probability. Hmm, the challenge is that the random generator's range increases each time it's called. So the first call gives 1, the second gives 1-2, th... | To solve this problem, we need to generate a perfectly random dice roll (values between 1 and 6) using an ascending random number generator. The generator's range increases with each call, starting from 1 and incrementing by 1 each time. The challenge is to ensure uniform probability for each dice roll using this gener... | 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 plane passing through the vertex of a cone intersects the base along a chord, the length of which is equal to the radius of the base. Determine the ratio of the volumes of the resulting parts of the cone."
},
{
"from": "assis... | 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 \\( K(n) \\) be the number of triples in a given system. Prove that:\na) \\( K(n) > \\frac{n}{6} - 1 \\);\n\nb) \\( K(n) < \\frac{2n}{9} \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need ... | 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{}. Is it possible to wrap a unit cube in a cloth of size $3 \\times 3$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out if a 3x3 cloth can wrap a unit cube. Let me start by understand... | 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{}. Consider the sequence \\( a_1, a_2, a_3, \\cdots a_n, \\cdots \\) which satisfies \\( a_1 = a_2 = 1, a_3 = 2 \\), and for any natural number \\( n \\), we have \\( a_n \\cdot a_{n+1} \\cdot a_{n+2} \\neq 1 \\). Additionally, \\( 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": "Return your final response within \\boxed{}. Find the envelope of a family of straight lines that form a triangle with a right angle and have an area of $a^{2} / 2$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the envelope of 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 volumes of solids generated by rotating the region bounded by the graphs of the functions about the x-axis.\n\n$$\ny = \\sin^2(x), x = \\frac{\\pi}{2}, y = 0\n$$\n"
},
{
"from": "assistant",
"value": "<|begin_... | 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 the numbers on the opposite faces of a die always equals 7."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to figure out why the sum of the numbers on opposite faces of a die always... | 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 $\\xi$ and $\\eta$ be two random variables with $\\mathrm{E}[\\xi] = \\mathrm{E}[\\eta] = 0$ and $\\mathrm{D}[\\xi] = \\mathrm{D}[\\eta] = 1$, and the correlation coefficient $\\rho = \\rho(\\xi, \\eta)$. Show that\n\n$$\n\\mathrm{... | 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 smallest positive integer $N$ such that $N / 3$, $N / 4$, $N / 5$, $N / 6$, and $N / 7$ are all integers?\n(a) 420\n(b) 350\n(c) 210\n(d) 300\n(e) 280"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\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{}. In an isosceles trapezoid that is inscribed around a circle, the ratio of the parallel sides is $k$. Find the angle at the base."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this problem abou... | 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 trapezoid \\(ABCD\\), the base \\(AD\\) is equal to 16, the sum of the diagonals \\(AC\\) and \\(BD\\) is 36, and the angle \\(\\angle CAD\\) is \\(60^\\circ\\). The ratio of the areas of triangles \\(AOD\\) and \\(BOC\\), where \\(... | 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": "The Catalan numbers (OEIS) are a sequence of natural numbers often appearing in combinatorics.\nThe nth Catalan number is the number of Dyck words (balanced strings of parenthesis or brackets such as [[][]]; formally defined as a string using two characters a and b such that any ... | The Catalan numbers (OEIS) are a sequence of natural numbers often appearing in combinatorics.
The nth Catalan number is the number of Dyck words (balanced strings of parenthesis or brackets such as [[][]]; formally defined as a string using two characters a and b such that any substring starting from the beginning has... | Okay, I need to write a program that takes a non-negative integer n and outputs the nth Catalan number. The Catalan numbers are given by the formula C_n = 1/(n+1) * binomial(2n, n). Hmm, how do I compute that efficiently?
First, let's think about calculating the binomial coefficient. The binomial coefficient for (2n c... | To solve this problem, we need to compute the nth Catalan number efficiently. The Catalan numbers are a sequence of natural numbers with significant applications in combinatorics. The nth Catalan number can be calculated using the formula:
\[ C_n = \frac{1}{n+1} \binom{2n}{n} \]
### Approach
The approach involves lev... | 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{}. \nThere are 40 identical gas cylinders with unknown and potentially different gas pressures. It is permitted to connect any number of cylinders together, not exceeding a given natural number $k$, and then disconnect them; during the 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{}. Solve the equations \\(2x^3 - 5x^2 + 6x - 2 = 0\\) and \\(6x^3 - 3x^2 - 2x + 1 = 0\\), given that they have a common root."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to solve these two cubi... | null | null | null | null | null | null | null | null | null | null |
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