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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": "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 denote correct match equation (we will denote it as CME) an equation a + b = c there all 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{}. In an equilateral triangle $ABC$, $K$ is the midpoint of side $AB$, point $L$ lies one-third of the way along side $BC$ closer to point $C$, and point $M$ lies one-third of the way along side $AC$ closer to point $A$.\n\nDetermine what... | 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 number \\( A \\) consisting of eight non-zero digits is added to a seven-digit number consisting of identical digits, resulting in an eight-digit number \\( B \\). It turns out that \\( B \\) can be obtained by permuting some of 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{}. Given $k > 0$ and $k \\neq 1$, the line $l: y = kx + 1$ and the line $l_{1}: y = k_{1}x + 1$ are symmetric with respect to the line $y = x + 1$. The lines $l$ and $l_{1}$ intersect the ellipse $E : \\frac{x^{2}}{4} + y^{2} = 1$ at poin... | 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 $a$ white balls and $b$ black balls $(a \\neq b)$ in a jar. All balls are drawn out one by one without replacement. Which of the following events is more likely to happen:\nEvent $A$: At some point, the number of drawn white ... | 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 picture shows several circles connected by segments. Tanya chooses a natural number \\( n \\) and places different natural numbers in the circles so that the following property holds for all these numbers:\n\nIf numbers \\( a \\) 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{}. We select $16$ cells on an $8\\times 8$ chessboard. What is the minimal number of pairs of selected cells in the same row or column?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so 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{}. In a $15 \\times 15$ grid, some centers of adjacent squares are connected by segments to form a closed line that does not intersect itself and is symmetric with respect to one of the main diagonals. Show that the length of the line 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{}. Give a projective proof of the following lemma. Let $ABCD$ be a trapezoid such that $(AB)$ and $(CD)$ are parallel. Let $X$ be the intersection of the diagonals of the trapezoid and $Y$ be the intersection of the lines $(AD)$ and $(BC)... | 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 diagonals $AC$ and $BD$ of quadrilateral $ABCD$ are extended beyond vertices $C$ and $D$ and doubled. Prove that the ratio of the area of the original quadrilateral $ABCD$ to the area of the new quadrilateral $ABC_{1}D_{1}$ (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{}. A line passing through the incenter $I$ of the triangle $ABC$ intersect its incircle at $D$ and $E$ and its circumcircle at $F$ and $G$ , in such a way that the point $D$ lies between $I$ and $F$ . Prove that:\r $DF \\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": "Return your final response within \\boxed{}. Given 1980 non-collinear vectors on a plane, any 1979 of these vectors sum to be collinear with another vector outside this set. Prove that the sum of these 1980 vectors is the zero vector."
},
{
"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": "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 prime number is number x which has only divisors as 1 and x itself.\nHarsh is playing a game with ... | 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 = 0.48181\\ldots \\) be an infinite repeating decimal, where the digits 8 and 1 repeat. When \\( F \\) is written as a simplified fraction, the denominator exceeds the numerator by:\n(A) 13\n(B) 14\n(C) 29\n(D) 57\n(E) 126"
... | 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{}. Inside a triangle \\(ABC\\), an arbitrary point \\(O\\) is chosen. Prove that the following equality holds:\n\\[\nS_A \\cdot \\overrightarrow{OA} + S_B \\cdot \\overrightarrow{OB} + S_C \\cdot \\overrightarrow{OC} = 0\n\\]\nwhere \\(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{}. In triangle \\( ABC \\), the medians \\( BD \\) and \\( CE \\) intersect at point \\( M \\). The circle constructed using segment \\( BM \\) as a diameter passes through vertex \\( C \\) and is tangent to line \\( DE \\). It is known 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{}. Find the greatest integer less than or equal to \\((2+\\sqrt{3})^{3}\\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the greatest integer less than or equal to (2 + √3)³. Hmm, let me... | 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 integer $n$ , between 10000 and 99999, is $abcde$ when written in decimal notation. The digit $a$ is the remainder when $n$ is divided by 2, the digit $b$ is the remainder when $n$ is divided by 3, the digit $c$ is 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": "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.You have to restore the wall. The wall consists of $N$ pillars of bricks, the height of the $i$-th p... | 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 \\( n \\in \\mathbf{N} \\), is there a non-zero \\( n \\)-variable polynomial \\( P \\) and \\( Q \\) with integer coefficients, such that \\((x_{1} + x_{2} + \\cdots + x_{n}) \\cdot P(x_{1}, x_{2}, \\cdots, x_{n}) \\equiv Q(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{}. In triangle \\( \\triangle ABC \\), where \\( AB \\) is the longest side, given that \\( \\sin A \\sin B = \\frac{2 - \\sqrt{3}}{4} \\), find the maximum value of \\( \\cos A \\cos B \\)."
},
{
"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{}. The sequence $f_{n}$ for $n \\geq 0$ is the Fibonacci sequence, defined as follows: $f_{0}=1, f_{1}=1$, and $f_{n+1}=f_{n}+f_{n-1}$ for $n = 1, 2, \\cdots$. Solve the equation:\n$$\nn f_{n} f_{n-1}=\\left(f_{n-2}-1\\right)^{2}\n$$"
}... | 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{}. Given \\( x_{1} = 1 \\), \\( x_{2} = 2 \\), and \\( x_{n+2} = \\frac{x_{n+1}^{2} - 7}{x_{n}} \\left( n \\in \\mathbf{N}^{*} \\right) \\), find the smallest positive period of the sequence \\( \\left\\{ x_{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": "It is the year 87,539,319 and solitary space-walking is now commonplace, many people travel into space by themselves, propelled by nothing but a jetpack on their backs, programming their course with a personal computer and keyboard as they go.\nYou are one such person; you were o... | It is the year 87,539,319 and solitary space-walking is now commonplace, many people travel into space by themselves, propelled by nothing but a jetpack on their backs, programming their course with a personal computer and keyboard as they go.
You are one such person; you were out on a lovely, peaceful spacewalk, when,... | Okay, I need to solve this programming problem where the goal is to write a program that prints "HELP!" using as few keystrokes as possible. The catch is that after each keypress, the leftmost column of the keyboard is ripped off. So each subsequent key can't be from the columns that were already removed. Let's break t... | To solve this problem, we need to write a program that prints "HELP!" to stdout using the fewest possible keypresses while adhering to the constraint that each subsequent keypress must be from a column further to the right than the previous one. The keyboard layout is QWERTY, and after each keypress, the leftmost colum... | 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{}. It is known that \\(a x^{3}+b x^{2}+c x+d\\), where \\(a, b, c, d\\) are given integers, is divisible by 5 for any integer \\(x\\). Prove that all numbers \\(a, b, c, d\\) are divisible by 5."
},
{
"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{}. Given a triangle \\( \\triangle ABC \\) with all three sides unequal, the incircle touches the sides \\( BC, CA, \\) and \\( AB \\) at points \\( D, E, \\) and \\( F \\) respectively. Let \\( L \\) be the reflection of point \\( 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{}. Determine whether or not there are any positive integral solutions of the simultaneous equations \n\\begin{align*} x_1^2 +x_2^2 +\\cdots +x_{1985}^2 & = y^3,\\\\ x_1^3 +x_2^3 +\\cdots +x_{1985}^3 & = z^2 \\end{align*}\nwith distinct in... | 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 regular hexagonal prism \\(ABCDEF - A'B'C'D'E'F'\\), \\(M\\) is the midpoint of \\(DE\\). Determine the shape of the cross-section passing through the points \\(A'\\), \\(C\\), and \\(M\\).\n\nA. Triangle\n\nB. Quadrilateral\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{}. Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads?\n$\\textbf{... | 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 the sides $BC$ and $CD$ of the square $ABCD$, points $E$ and $F$ are chosen in such a way that the angle $EAF$ is $45^\\circ$. The side length of the square is 1. Find the perimeter of the triangle $CEF$. Justify your solution."
}... | 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 \\), \\(AB = AC = a\\). Construct an equilateral triangle \\(BCD\\) externally on the side \\(BC\\). Find the condition when the area of the quadrilateral \\(ABCD\\) is maximized."
},
{
"from": "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{}. Determine the point on an arc $AB$ such that the sum of its distances from points $A$ and $B$ is maximized."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the point on an arc AB where 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 \\(ABC\\), point \\(D\\) is taken on side \\(AC\\). It is given that \\(BC > AB = 5\\), \\(BD = 4\\), and triangles \\(ABD\\) and \\(BCD\\) are isosceles. Find \\(AC\\). If there are multiple answers, list them in any order... | 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 derivative of the specified order.\n\n$$\ny=x \\ln (1-3 x), y^{(IV)}=?\n$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the fourth derivative of y = x ln(1 - 3x). Hmm, deriva... | 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 \\(ABCD\\) be a convex quadrilateral with diagonals intersecting at point \\(E\\). Consider the line connecting the orthocenters of triangles \\(EAB\\) and \\(ECD\\), and the line connecting the centroids of triangles \\(EBC\\) 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": "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.In a class of N students, a class test was held. The i^{th} student scored A_{i} marks. It is also 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": "Return your final response within \\boxed{}. For which values of the real parameters \\( a \\) and \\( b \\) will the maximum of the function \\( \\left|x^{2}+a x+b\\right| \\) on the interval \\([-1, 1]\\) be the smallest?"
},
{
"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{}. If $y = -x^2 + 5$ and $x$ is a real number, then what is the maximum value possible for $y$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the maximum value of y given the equation y =... | 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 this and the next two problems, we meet three inhabitants of the forest: A, B, and C, each of whom is either a knight or a liar. Statements are only made by two of them: A and B. In their statements, the word \"us\" refers to all ... | 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 $P^{*}$ be the set of primes less than $10000$ . Find all possible primes $p\\in P^{*}$ such that for each subset $S=\\{p_{1},p_{2},...,p_{k}\\}$ of $P^{*}$ with $k\\geq 2$ and each $p\\not\\in S$ , there is a $q\\in P... | 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 maximum and minimum values of the function\n\n$$\nz = 4x^2 + y^2 - 16x - 4y + 20\n$$\n\nin the closed region \\( D \\), defined by the inequalities:\n\n$$\nx \\geq 0, \\quad x - 2y \\leq 0, \\quad x + y - 6 \\leq 0.\n$$"
},
... | 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{}. Given the expression $\\frac{\\cos \\alpha \\cos \\frac{\\beta}{2}}{\\cos \\left(a-\\frac{\\beta}{2}\\right)} + \\frac{\\cos \\beta \\cos \\frac{\\alpha}{2}}{\\cos \\left(\\beta-\\frac{\\alpha}{2}\\right)} - 1$, find the value of $\\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": "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 Little Elephant from the Zoo of Lviv likes lucky strings, i.e., the strings that consist only 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": "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.Recently, Chef visited his doctor. The doctor advised Chef to drink at least 2000 ml of water each 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": "What can you pill outside, eat and throw away the inner part?\nA: medicine chest\nB: preserve\nC: mealie cob\nD: medicine bottle\nE: feast"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see. The question is asking what you can \"pill outside,... | 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 integers $a,b,c,x,y,z$ such that \\[a+b+c=xyz, \\; x+y+z=abc, \\; a \\ge b \\ge c \\ge 1, \\; x \\ge y \\ge z \\ge 1.\\]"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, so I need to find all... | 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 hotel has 100 standard rooms, with a room rate of 400 yuan/day, but the occupancy rate is only 50%. If the room rate is reduced by 20 yuan, the occupancy increases by 5 rooms. Find the appropriate room rate so that the hotel's reve... | 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 state with $n$ flags of ON or OFF can be represented by a sequence of bits where $0, 1, ..., n-1$ ... | 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{}. The graph of the quadratic function \\( y = ax^2 + c \\) intersects the coordinate axes at the vertices of an equilateral triangle. What is the value of \\( ac \\)?"
},
{
"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": "Return your final response within \\boxed{}. Solve the system\n$$\\left\\{\\begin{array}{l}x^{\\prime}=3x-y+z, \\\\ y^{\\prime}=-x+5y-z, \\\\ z^{\\prime}=x-y+3z\\end{array}\\right.$$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to solve... | 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 positive integers \\( n \\) for which there exists \\( k \\in \\mathbf{N}^{*}, k \\geqslant 2 \\) and positive rational numbers \\( a_{1}, a_{2}, \\cdots, a_{k} \\) such that\n\\[ \na_{1}+a_{2}+\\cdots+a_{k}=a_{1} \\cdots a_{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": "Return your final response within \\boxed{}. In the convex quadrilateral $ABCD$ inscribed in a circle, $\\odot\\left(O_{1}, R_{1}\\right), \\odot\\left(O_{2}, R_{2}\\right), \\odot\\left(O_{3}, R_{3}\\right)$, and $\\odot\\left(O_{4}, R_{4}\\right)$ are the incircles of $\\triang... | 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{}. Vitya has five math lessons per week, one on each day from Monday to Friday. He knows that there is a probability of \\( \\frac{1}{2} \\) that the teacher will not check his homework at all during the week, and a probability of \\( \\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{}. Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $P$ be a point inside the $ABD$ satisfying $\\angle PAD=90^\\circ - \\angle PBD=\\angle CAD$ . Prove that $\\angle PQB=\\angle BAC$ , where $Q$ is the inter... | 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}<\\ldots<a_{n} \\) be a sequence of positive integers with the property that for \\( i<j \\), the decimal representation of \\( a_{j} \\) does not start with that of \\( a_{i} \\). (For example, the numbers 137 and 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{}. Give the exact proof that if \\(a\\) and \\(b\\) are relatively prime numbers, then \\(\\varphi(a \\cdot b) = \\varphi(a) \\cdot \\varphi(b)\\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, so I nee... | 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 if the equation \\( x^{4} - a x^{3} + 2 x^{2} - b x + 1 = 0 \\) has a real root, then \\( a^{2} + b^{2} \\geqslant 8 \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this equation... | 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 monkey and a weight are suspended on opposite ends of a rope thrown over a block. Both ends of the rope are of equal length, and the system is in equilibrium. Each foot of the rope weighs 4 ounces. The combined age of the monkey 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{}. The intersection of a plane with a regular tetrahedron with edge \\( a \\) is a quadrilateral with perimeter \\( P \\). Prove that \\( 2a \\leq P \\leq 3a \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOk... | 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 is a safe with \\( n \\) cells numbered from 1 to \\( n \\). Initially, each cell contained a card with its own number. Vasya rearranged the cards in some order such that the card in the \\( i \\)-th cell has a number \\( a_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) Two circles with radii $\\mathrm{R}$ and $\\mathrm{r}$ are externally tangent. Demonstrate that the segment determined by the common external tangent $\\ell$ measures $d = 2 \\sqrt{\\mathrm{Rr}}$.\n\n(b) Consider, as illustrated, 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{}. Two native inhabitants A and B of this island stated:\n\nA: We are both liars, and this island is called Maya.\n\nB: That's the truth.\n\nCan it be asserted that the second island is indeed called Maya?"
},
{
"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{}. How many integers $n$ with $10 \\le n \\le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different?\n\n*Proposed by Evan Chen*"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\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{}. We seek the true statement. There are one hundred statements written in a notebook:\n\n1) There is exactly one false statement in this notebook.\n2) There are exactly two false statements in this notebook.\n...\n100) There are exactly ... | 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 group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$ , and four full-solution problems $F_1, F_2, F_3, F_4$ . Each person in the group correctly solved exactly 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{}. In the isosceles triangle \\(ABC\\) (with \\(AB = BC\\)), an incircle with center \\(O\\) is inscribed and touches side \\(AB\\) at point \\(E\\). Point \\(D\\) is chosen on the extension of side \\(AC\\) beyond point \\(A\\) such 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{}. A function \\( f(x) \\) defined on the interval \\([1,2017]\\) satisfies \\( f(1)=f(2017) \\), and for any \\( x, y \\in [1,2017] \\), \\( |f(x) - f(y)| \\leqslant 2|x - y| \\). If the real number \\( m \\) satisfies \\( |f(x) - f(y)| ... | 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 $r$ is positive and the line whose equation is $x + y = r$ is tangent to the circle whose equation is $x^2 + y ^2 = r$, then $r$ equals\n$\\textbf{(A) }\\frac{1}{2}\\qquad \\textbf{(B) }1\\qquad \\textbf{(C) }2\\qquad \\textbf{(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{}. Let the complex coefficient polynomial \n$$\nf(z) = a z^{2} + b z + c\n$$\nsatisfy: For any $|z| \\leqslant 1$, $|f(z)| \\leqslant 1$. Find the maximum value of $|a| + |b|$."
},
{
"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": "Introduction\nNumber theory is full of wonders, in the form of unexpected connections. Here's one of them.\nTwo integers are co-prime if they have no factors in common other than 1. Given a number \\$N\\$, consider all integers from 1 to \\$N\\$. Draw two such integers at random ... | Introduction
Number theory is full of wonders, in the form of unexpected connections. Here's one of them.
Two integers are co-prime if they have no factors in common other than 1. Given a number \$N\$, consider all integers from 1 to \$N\$. Draw two such integers at random (all integers have the same probability of bei... | Okay, I need to solve this problem where given a number N, I have to compute the probability that two randomly selected numbers from 1 to N are coprime. The challenge is to find an efficient way to compute this without enumerating all pairs, especially since N can be as large as 1000, which would have a million pairs, ... | To solve this problem, we need to compute the probability that two randomly selected integers from 1 to N are coprime. The solution involves using the Möbius function and the inclusion-exclusion principle to efficiently determine the count of coprime pairs without enumerating all possible pairs.
### Approach
1. **Siev... | 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.On the planet Mars a year lasts exactly n days (there are no leap years on Mars). But Martians have ... | 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{}. Determine all functions $f: \\mathbb{R} \\mapsto \\mathbb{R}$ satisfying\r\n\r\n\\[f\\left(\\sqrt{2} \\cdot x\\right) + f\\left(4 + 3 \\cdot \\sqrt{2} \\cdot x \\right) = 2 \\cdot f\\left(\\left(2 + \\sqrt{2}\\right) \\cdot x\\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{}. If $2^a+2^b=3^c+3^d$, the number of integers $a,b,c,d$ which can possibly be negative, is, at most:\n$\\textbf{(A)}\\ 4 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 2 \\qquad \\textbf{(D)}\\ 1 \\qquad \\textbf{(E)}\\ 0$"
},
{
... | 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": "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.You are given a permutation p of integers from 1 to n, where n is an even number. \n\nYour goal is 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{}. **p1.** An ant starts at the top vertex of a triangular pyramid (tetrahedron). Each day, the ant randomly chooses an adjacent vertex to move to. What is the probability that it is back at the top vertex after three days?**p2.** A squar... | 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 \\( x_{1}, x_{2}, \\cdots, x_{n} \\in \\mathbf{R}_{+} \\) and \\( \\sum_{i=1}^{n} \\frac{1}{1+x_{i}} = \\frac{n}{2} \\), prove:\n\\[ \\sum_{1 \\leqslant i, j \\leqslant n} \\frac{1}{x_{i}+x_{j}} \\geqslant \\frac{n^{2}}{2}. \\]"
... | 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 sides of triangle \\(ABC\\) satisfy the condition:\n\\[ b^2 + c^2 \\geq 5a^2 \\]\nProve that side \\(BC\\) is the smallest side of the triangle."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I ha... | 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 $P_{0}(x)=0,P_{1}(x)=x$ and \\[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \\quad n\\geq 2\\] For every natural number $n\\geq 1$ , find all real numbers $x$ satisfying the equation $P_{n}(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{}. On a cubic planet, cubic mice live only on the faces of the cube and not on the edges or vertices. The number of mice on different faces is different and on any two neighboring faces, this number differs by at least 2. What is the mini... | 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 \\(ABCD\\), it is known that \\(\\angle A + \\angle D = 120^\\circ\\) and \\(AB = BC = CD\\).\n\nProve that the point of intersection of the diagonals is equidistant from the vertices \\(A\\) and \\(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{}. Prove or disprove the inequality involving the angles of a triangle:\n\n$$\n\\sin 2 \\alpha + \\sin 2 \\beta + \\sin 2 \\gamma \\leq \\sin (\\alpha + \\beta) + \\sin (\\beta + \\gamma) + \\sin (\\gamma + \\alpha)\n$$"
},
{
"fro... | 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 convex quadrilateral is inscribed in a circle of radius 1. Show that its perimeter minus the sum of its two diagonals lies between 0 and 2."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to 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{}. The vertices of quadrilateral \\(ABCD\\) lie on a circle. Diagonals \\(AC\\) and \\(BD\\) are drawn. Identify four pairs of equal angles whose vertices lie on the circle."
},
{
"from": "assistant",
"value": "<|begin_of_thou... | 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 integer \\( n \\geq 2 \\), let \\( M_{0}\\left(x_{0}, y_{0}\\right) \\) be a point of intersection between the parabola \\( y^{2} = n x - 1 \\) and the line \\( y = x \\). Prove that for any positive integer \\( m \\), there 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": "Return your final response within \\boxed{}. The medians $A A_{1}$ and $B B_{1}$ of triangle $A B C$ intersect at point $M$. Prove that if the quadrilateral $A_{1} M B_{1} C$ is circumscribed, then $A C = B C$."
},
{
"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": "Return your final response within \\boxed{}. Let $a,b,c,d,e,f$ be positive numbers such that $a,b,c,d$ is an arithmetic progression, and $a,e,f,d$ is a geometric progression. Prove that $bc\\ge ef$ ."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\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{}. During a secret meeting, 20 trainees elect their favorite supervisor. Each trainee votes for two supervisors. It is known that for any two trainees, there is always at least one supervisor for whom both have voted. What is the minimum ... | 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{}. Crestwood Elementary School has an active four-square league, consisting of ten players, including Justin and Tim. Each day at recess, the ten players split into two four-square games, each with five players in no relevant order. Over ... | 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{}. Four children were discussing the answer to a problem.\n\nKolia said: \"The number is 9.\"\n\nRoman said: \"The number is a prime number.\"\n\nKatya said: \"The number is even.\"\n\nNatasha said: \"The number is divisible by 15.\"\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{}. Suppose \\( m \\) is an integer with \\( |m| \\geqslant 2 \\). Let the sequence of integers \\( a_{1}, a_{2}, \\cdots \\) satisfy the condition that \\( a_{1} \\) and \\( a_{2} \\) are not both zero, and for any positive integer \\( 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 altitudes \\(A A^{\\prime}\\) and \\(C C^{\\prime}\\) of the acute triangle \\(A B C\\) intersect at point \\(H\\). Point \\(B_{0}\\) is the midpoint of side \\(A C\\). Prove that the intersection point of the lines symmetrical 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{}. What are the integers \\( p \\) such that both \\( p \\) and \\( p+1 \\) are prime?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find all integers p where both p and p+1 are prime numbers.... | 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 multiplication by recurrence in the same way. Complete: \"multiplication is to $[\\ldots]$ what exponentiation is to multiplication.\""
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's 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{}. Given \\( n \\) points on a plane, with no three points collinear (\\( n > 4 \\)), any two points can be connected to form \\( \\mathrm{C}_{n}^{4} \\) quadrilaterals. Prove that at least 20% of these quadrilaterals are convex."
},
... | 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 the following equation, the letters $A, B, C, D, E, F, G, H, I$ each represent different digits from $1$ to $9$:\n\n$$\n\\overline{\\mathrm{ABCD}}+\\overline{\\mathrm{EF}} \\times \\overline{\\mathrm{GH}}-I=X\n$$\n\nFind the minim... | 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 can we design a system for the production of hydrogen fuel using the water-gas shift reaction and what are the optimal operating conditions required for maximum yield of hydrogen gas?\""
},
{
"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{}. Given a regular triangular prism $P-ABC$ with a base equilateral triangle having a side length of $2\\sqrt{3}$ and an inscribed sphere radius of $\\sqrt{2}-1$, find the volume of the prism."
},
{
"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{}. \nProve that for any positive integer $n$, the sum\n\n$$\n\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{2n}\n$$\n\nis greater than $\\frac{1}{2}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I 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{}. Define the sequence $(a_n)_{n=1}^\\infty$ of positive integers by $a_1=1$ and the condition that $a_{n+1}$ is the least integer such that \\[\\mathrm{lcm}(a_1, a_2, \\ldots, a_{n+1})>\\mathrm{lcm}(a_1, a_2, \\ldots, a_n)\\mbox{.}... | 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 tree is a connected undirected graph with n - 1 edges, where n denotes the number of vertices. Ver... | null | null | null | null | null | null | null | null | null | null |
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