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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": "Most of you have probably heard of the famous Turtle Graphics program, initially made popular by Logo. This challenge is to implement a Turtle Graphics program in your language of choice.\nRules:\n\nThe floor must be a diagonal with dimensions input by the user through std-in in... | Most of you have probably heard of the famous Turtle Graphics program, initially made popular by Logo. This challenge is to implement a Turtle Graphics program in your language of choice.
Rules:
The floor must be a diagonal with dimensions input by the user through std-in in the form x/y, representing the x and y len... | Okay, I need to implement a Turtle Graphics program based on the given rules. Let me start by breaking down the requirements.
First, the user inputs the floor dimensions as x/y. So I have to parse that. The x can't exceed 20 and y 45. So I need to validate that. Then, the initial position and direction are given as x,... | To solve this problem, we need to implement a Turtle Graphics program that allows the user to control a turtle's movements on a grid. The turtle can draw or erase based on commands, and the grid dimensions and initial position are specified by the user. The turtle must handle various commands, including moving in diffe... | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Bolyai Farkas provides a method for constructing the n-th part of a segment \\( AB \\) that does not require the construction of parallel lines:\n\nOn a line passing through \\( A \\), but different from \\( AB \\), measure a segment \... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 $P(x)=x^n+a_{n-1}x^{n-1}+\\cdots+a_0$ be a polynomial of degree $n\\geq 3.$ Knowing that $a_{n-1}=-\\binom{n}{1}$ and $a_{n-2}=\\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 values of the positive integer \\( n \\) for which the following system of equations has a solution in positive integers \\( x_{1}, x_{2}, \\ldots, x_{n} \\). Find all solutions for each such \\( n \\).\n\n\\[\n\\begin{al... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 $n \\geq 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$ -gon with the numbers $1, 2,\\ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divid... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 insert the number 48 between the two digits of the number 49. Then we insert the number 48 again between the 4 and 8, and repeat this process several times (e.g., $44 \\ldots 48 \\ldots 89$). Is it true that the resulting number wil... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.Snuke can change a string t of length N into a string t' of length N - 1 under the following rule:\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{}. $2^{n-1}$ different sequences of zeros and ones of length $n$ are written one below the other. It is known that for any three of the written sequences, there exists a position $p$ such that in the $p$-th place, all three have the digit... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 \\) that contains \\( m \\) points inside it, along with points \\( A \\), \\( B \\), and \\( C \\), there are a total of \\( m+3 \\) points. Using these points as vertices, the triangle \\( \\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{}. Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as thems... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Represent the number 180 as the sum of four addends so that they form a geometric progression, in which the third term is 36 more than the first term."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, le... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 snail crawls around the clock face counterclockwise at a constant speed. It started at 12:00 from the 12 o'clock mark and completed a full circle exactly at 14:00. What time was it when the snail met the minute hand during its moveme... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 blank $n \\times n$ ($n \\geqslant 4$) grid, where each cell is arbitrarily filled with either +1 or -1. The product of the numbers in $n$ cells, no two of which share the same row or column, is called a basic term.\n\nProve... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 a positive integer $n$ , an *$n$ -branch* $B$ is an ordered tuple $(S_1, S_2, \\dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \\subset S_2 \\subset \\dots \\subset S_m \\subseteq \\{1,2,\\... | 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{}. Show that the sum of the cubes of three consecutive natural numbers is divisible by 9 times the middle number, provided that the middle number is not divisible by 3."
},
{
"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{}. Find the cosine of the acute angle of a rhombus if the line through one of its vertices divides the angle in the ratio of $1:3$ and the opposite side in the ratio of $3:5$."
},
{
"from": "assistant",
"value": "<|begin_of_th... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. A 2022-digit natural number is given, where any two consecutive digits (in the same order) form a two-digit number that is divisible either by 19 or by 23. The given number starts with the digit 4. What digit does it end 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 the midpoint of side $BC$ of triangle $ABC$ be $D$, and let the midpoint of $AD$ be $E$. Draw the line $BE$, which intersects $AC$ at $F$, and show that $AF = \\frac{1}{3} AC$."
},
{
"from": "assistant",
"value": "<|beg... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 the real number \\(x\\) satisfies \\(\\arcsin x > \\arccos x\\), then what is the range of the function \\( f(x) = \\sqrt{2 x^{2} - x + 3} + 2^{\\sqrt{x^{2} - x}} \\)?"
},
{
"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": "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.There are 2000001 stones placed on a number line. The coordinates of these stones are -1000000, -999... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Segment $AB$ has midpoint $C$, and segment $BC$ has midpoint $D$. Semi-circles are constructed with diameters $\\overline{AB}$ and $\\overline{BC}$ to form the entire region shown. Segment $CP$ splits the region into two sections of eq... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 coefficients \\( a, b, c \\) of the quadratic trinomial \\( a x^{2} + b x + c \\) are natural numbers not greater than 2019. Can such a trinomial have two distinct roots that differ by less than 0.01?"
},
{
"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{}. Let $\\mathbf{R}^{+}$ be the set of all positive real numbers. Find all functions \\( f: \\mathbf{R}^{+} \\rightarrow \\mathbf{R}^{+} \\) such that \\( f(x) + f(y) = f(x y f(x+y)) \\) for all \\( x, y \\in \\mathbf{R}^{+} \\) 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{}. We distribute 2018 coins into $b$ boxes, and we distribute these $b$ boxes into $m$ houses, such that the number of coins in each house is strictly less than $b$. Is it possible for each box to contain at least $m$ coins?"
},
{
... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 regular tetrahedron \\( P-ABCD \\), where each face is an equilateral triangle with side length 1, points \\( M \\) and \\( N \\) are the midpoints of edges \\( AB \\) and \\( BC \\), respectively. Find the distance between the sk... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 set of positive integers less than 20, choose three different numbers such that their sum is divisible by 3. How many different ways are there to choose these numbers?"
},
{
"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{}. An integer is reduced by 9 times by striking out one of its digits; the resulting number is also divisible by 9.\n\na) Prove that in order to divide the resulting number by 9, it is also sufficient to strike out one of its digits.\n\nb... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Petya bought one cake, two muffins, and three bagels; Anya bought three cakes and one bagel; and Kolya bought six muffins. They all paid the same amounts of money. Lena bought two cakes and two bagels. How many muffins could she have 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{}. Let \\(ABC\\) be a triangle, and let \\(P, Q, X, Y\\) be points on the same side as \\(A\\) of line segment \\(BC\\) such that \\(ABYX\\) and \\(ACQP\\) are squares. Let \\(M\\) be the midpoint of \\([BC]\\). Show that \\(PX = 2AM\\)."... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 of equations:\n\n\\[\n\\left\\{\\begin{aligned}\nx_{1}+2 x_{2}+2 x_{3}+\\cdots+2 x_{100} & =1 \\\\\nx_{1}+3 x_{2}+4 x_{3}+\\cdots+4 x_{100} & =2 \\\\\nx_{1}+3 x_{2}+5 x_{3}+\\cdots+6 x_{100} & =3 \\\\\n\\cdots & \\cdot... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 ten weights of different weights, each weighing an integer number of grams. It is known that the weight of the lightest weight and the heaviest weight differ by 9 grams. One weight is lost. Determine its weight if the total w... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 quadrilateral, whose diagonals are not perpendicular, is $\\operatorname{tg} \\varphi \\cdot \\mid a^{2}+c^{2}-b^{2}-d^{2} \\mid / 4$, where $a, b, c,$ and $d$ are the lengths of consecutive sides, and $\\varph... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 picture, there are several circles connected by line segments. Tanya chooses a natural number \\( n \\) and places different natural numbers in the circles such that for all these numbers the following property holds:\n\nIf numb... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 \\( n \\) be a positive integer and let \\( S \\) be the set \\( \\{1,2, \\ldots, n\\} \\). Define a function \\( f: S \\rightarrow S \\) by\n\\[ \nf(x)= \\begin{cases} \n2 x & \\text{if } 2 x \\leq n, \\\\ \n2 n - 2 x + 1 & \\text... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 the first half of the year, lazy Pasha forced himself to solve math problems. Every day he solved no more than 10 problems, and if on any given day he solved more than 7 problems, then for the next two days he solved no more tha... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. $\\triangle P Q R$ has \\(PQ = 150\\) and \\(PR = QR = 125\\). Three line segments are drawn parallel to \\(QR\\), dividing $\\triangle P Q R$ into four sections of equal area. The height, \\(h\\), of the bottom section is closest 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{}. Among any ten out of sixty students, there will be three classmates. Prove that among all of them, there will be 15 classmates."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to prove that amon... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. 12 soccer teams participate in a round-robin tournament. Each pair of teams plays a match, with the winning team earning 2 points and the losing team earning 0 points. If the match results in a draw, each team earns 1 point. Three refe... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 complex numbers \\( z = \\cos \\alpha + i \\sin \\alpha \\) and \\( u = \\cos \\beta + i \\sin \\beta \\), and that \\( z + u = \\frac{4}{5} + \\frac{3}{5} i \\), find \\( \\tan(\\alpha + \\beta) \\)."
},
{
"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{}. On the coordinate plane, the graphs of the functions \\( y = x^{2} + bx + c \\) and \\( y = x^{2} + cx + b \\) are shown.\n\nFind the values of \\( b \\) and \\( c \\). In the answer, write down the equations of each function."
},
... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Xiaoqing is preparing a dish called \"Xiang Qing Fried Eggs,\" which requires 7 steps with the following times:\n- Wash scallions, chop scallions: 1 minute\n- Beat eggs: half a minute\n- Stir egg mixture and scallions: 1 minute\n- Wash... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 sum of the distances from any point inside an equilateral triangle to its three sides is constant."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to prove that if you take any po... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Each cell of a \\(10 \\times 10\\) board is painted either blue or white. We define a cell as happy if exactly two of its neighboring cells are blue. Color the board such that all cells are happy. (Two cells are considered neighbors if... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 $A = \\left\\{a_{1}, a_{2}, \\cdots, a_{n}\\right\\}$ be a set of numbers, and let the arithmetic mean of all elements in $A$ be denoted by $P(A)\\left(P(A)=\\frac{a_{1}+a_{2}+\\cdots+a_{n}}{n}\\right)$. If $B$ is a non-empty subse... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.Fibonacci numbers have the following form F1=1, F2=1, F3=2 ... \nFn=Fn-1 +Fn-2 .\n\nWe have an array... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Chef is good at making pancakes. Generally he gets requests to serve N pancakes at once.\nHe serves ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 line, there are $n+1$ distinct segments marked, and one point on the line belongs to all these segments. Prove that among the marked segments, it is possible to choose distinct segments $I$ and $J$ that intersect over a segment 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": "Where does dust usually end up in the corner?\nA: windowsill\nB: carpet\nC: frogs\nD: ground\nE: corner"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to figure out where dust usually ends up in the corner. The options are A: windowsill,... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Prove that for all non-negative real numbers $x,y,z$ , not all equal to $0$ , the following inequality holds $\\displaystyle \\dfrac{2x^2-x+y+z}{x+y^2+z^2}+\\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\\dfrac{2z^2+x+y-z}{x^2+y^2+z}\\geq 3.$ Determ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 tetrahedron $ABCD$, edge $AB$ has length 3 cm. The area of face $ABC$ is $15\\mbox{cm}^2$ and the area of face $ABD$ is $12 \\mbox { cm}^2$. These two faces meet each other at a $30^\\circ$ angle. Find the volume of the tetrahedron ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find all values of \\( m \\) for which any solution of the equation\n\n\\[ 2018 \\cdot \\sqrt[5]{6.2x - 5.2} + 2019 \\cdot \\log_{5}(4x + 1) + m = 2020 \\]\n\nbelongs to the interval \\([1; 6]\\)."
},
{
"from": "assistant",
... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $A B C$ be a triangle. Let $S$ be the point of intersection of the angle bisector from $A$ with the circumcircle of $A B C$. \n\nShow that the triangle $B C S$ is isosceles."
},
{
"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{}. While working on fruit harvesting, class 5A collected 560 kg of apples, class 5B collected 595 kg, and class 5C collected 735 kg. All collected apples were packed into boxes, with each box containing the same maximum possible number 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{}. The sum of three numbers \\(a, b,\\) and \\(c\\) is divisible by 30. Prove that \\(a^{5} + b^{5} + c^{5}\\) is also divisible by 30."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, let me try to figure... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 point of intersection of the line and the plane.\n\n$\\frac{x+2}{-1}=\\frac{y-1}{1}=\\frac{z+4}{-1}$\n\n$2 x - y + 3 z + 23 = 0$"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to figu... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 \\( i = \\sqrt{-1} \\). Consider a sequence of complex numbers \\( z_{1} = 0 \\), and for \\( n \\geq 1 \\), \\( z_{n+1} = z_{n}^{2} + i \\). On the complex plane, what is the modulus of \\( z_{111} \\)?\n\n(A) 1 \n(B) \\( \\sqrt{... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 values of the following trigonometric functions:\n1. \\(\\sin 35^\\circ\\)\n2. \\(\\cos 167^\\circ\\)\n3. \\(\\tan 3\\)\n4. \\(\\cot(-1.5)\\)"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.The number 105 is quite special - it is odd but still it has eight divisors. Now, your task is this:... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 segment of unit length is cut into eleven smaller segments, each with length of no more than $a$ . For what values of $a$ , can one guarantee that any three segments form a triangle?\n\n*(4 points)*"
},
{
"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{}. The internal and external angle bisectors drawn from point \\( B \\) intersect at points \\( K \\) and \\( L \\); Similarly, the internal and external angle bisectors drawn from point \\( C \\) intersect with those drawn from point \\(... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Given an integer array. You have to check whether it is possible to divide the array in two parts such that the sum of elements of one part is equal to the sum of the element of the other part.If it is possible print \"YES\" otherwise \"NO\".\nSample Input\n4 3 5 5 3\n\nSample Ou... | Given an integer array. You have to check whether it is possible to divide the array in two parts such that the sum of elements of one part is equal to the sum of the element of the other part.If it is possible print "YES" otherwise "NO".
Sample Input
4 3 5 5 3
Sample Output
YES
Explanation
Array can be divided in tw... | Okay, let's see. I need to determine if an integer array can be split into two parts with equal sums. Hmm. So the total sum of the array must be even, right? Because if the sum is odd, it's impossible to split into two equal halves. Wait, that's a good starting point. Like, if the total sum is S, then S has to be even;... | To solve this problem, we need to determine if an integer array can be divided into two non-empty parts such that the sum of elements in each part is equal. The solution involves checking if the total sum of the array is even, and then using dynamic programming to determine if a subset with half of the total sum exists... | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 $\\mathcal{H}$ be the unit hypercube of dimension 4 with a vertex at $(x, y, z, w)$ for each choice of $x, y, z, w \\in \\{0,1\\}$. Note that $\\mathcal{H}$ has $2^4 = 16$ vertices. A bug starts at the vertex $(0,0,0,0)$. In how ma... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Can the quadratic equation \\( a x^{2} + b x + c = 0 \\) with integer coefficients have a discriminant equal to 23?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out if a quadratic 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 base of a right prism is an isosceles trapezoid where the diagonal is equal to $a$, and the angle between the diagonal and the longer base is $\\alpha$. The prism's diagonal is inclined to the base at an angle $\\beta$. Find the vo... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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}{1 \\times 2} + \\frac{x}{2 \\times 3} + \\frac{x}{3 \\times 4} + \\ldots + \\frac{x}{999 \\times 1000} = 999\\), find \\(x\\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see.... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 an island, there are 100 knights and 100 liars. Each of them has at least one friend. One day, exactly 100 people said: \"All my friends are knights,\" and exactly 100 people said: \"All my friends are liars.\" What is the smallest ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 $\\triangle ABC$, $AB = AC$, and $D$ is a point inside $\\triangle ABC$ such that $\\angle DCB = \\angle DBA$. Let $E$ and $F$ be points on segments $DB$ and $DC$, respectively. Prove that the necessary and sufficient condition for ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 perpendicular bisector of the side $\\mathrm{BC}$ of triangle $\\mathrm{ABC}$ intersects the side $\\mathrm{AC}$ at point $\\mathrm{D}$ and the bisector of the angle $\\mathrm{ABD}$ at point $\\mathrm{P}$. Prove that the points $\\... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.Mohit(Ex GenSec ) is the most active member of the roasting club who loves giving tasks to other mem... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Compose equations whose roots are the numbers:\n\na) \\(\\operatorname{ctg}^{2} \\frac{\\pi}{2 m+1}, \\operatorname{ctg}^{2} \\frac{2 \\pi}{2 m+1}, \\operatorname{ctg}^{2} \\frac{3 \\pi}{2 m+1}, \\ldots, \\operatorname{ctg}^{2} \\frac{... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can 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": "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.Given a Binary Tree of size N, your task is to complete the function countNonLeafNodes(), that shoul... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.One day, during daily web surfing, you encountered a web page which was written in a language you've... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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{}. Fedia has three sticks. If it is impossible to form a triangle with them, Fedia shortens the longest stick by the sum of the lengths of the other two. If the length of the stick does not become zero and the triangle still cannot be for... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.Given a palindromic string palindrome, replace exactly one character by any lowercase English letter so that the string becomes the lexicogra... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 each cell of an \\(8 \\times 8\\) square, one of the diagonals is drawn. Consider the union of these 64 diagonals. It consists of several connected components (where points belong to the same component if one can move between them u... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.An area named Renus, is divided into $(N \\times M)$ cells. According to archaeological survey the a... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "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.Archimedes is the name of a new 2D video game, which is very simple: the player hits start, at which... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.Brain training puzzle games are popular from children to adults. We decided to make a puzzle game an... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. The diagonals of a trapezoid divide it into triangles, two of which have an area ratio of $5:1$. What can be said about the trapezoid?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, let's tackle this ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 sequence \\(\\{a_n\\}\\) such that \\(a_1 = 1\\), \\(a_2 = 3\\), and \\(a_{n+2} = 3a_{n+1} - a_n\\):\n\n(1) Prove that \\(1 + a_n a_{n+2}\\) \\((n \\in \\mathbb{N}^*)\\) is a perfect square.\n\n(2) Let \\(b_n = \\left\\lfloor \... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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\\) with bases \\(AD=20\\) and \\(BC=14\\), the circles constructed on the sides \\(AB\\), \\(BC\\), and \\(CD\\) as diameters intersect at one point. The length of diagonal \\(AC\\) is 16. Find the length of \\(BD\... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 of positive integers $\\left\\{a_{n}\\right\\}_{n=1}^{+\\infty}$ is defined as follows: $a_{1}=1$, and for any $n>1$, $a_{n}$ is the smallest positive integer different from $a_{1}$, $a_{2}$, $\\ldots$, $a_{n-1}$ 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{}. Select the dimensions of a regular tetrahedral pyramid such that the ratio of the radii of the circumscribed and inscribed spheres is minimized."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need t... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( A \\) be a positive integer, and let \\( B \\) denote the number obtained by writing the digits of \\( A \\) in reverse order. Show that at least one of the numbers \\( A+B \\) or \\( A-B \\) is divisible by 11."
},
{
"... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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.Find the number of integers between 1 and K (inclusive) satisfying the following condition, modulo 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 a convex octagon, the number of obtuse angles is \\( m \\) and the number of acute angles is \\( n \\). Determine the relationship between \\( m \\) and \\( n \\):\n(A) \\( m < n \\).\n(B) \\( m > n \\).\n(C) \\( m = n \\).\n(D) Any... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 an undirected graph consisting of n vertices. A number is written on each vertex; 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{}. Inside an angle of $90^{\\circ}$, there is a given point located at distances of 8 and 1 from the sides of the angle. What is the minimum length of the segment passing through this point, whose ends lie on the sides of the angle?"
},... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 set $S$ of $2n+1$ distinct irrational numbers, find the largest positive integer $m$ such that there exist $m$ different numbers in $S$ where the sum of any two of them is still an irrational number."
},
{
"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{}. Let \\( a \\) and \\( b \\) be two positive numbers such that \\( \\frac{a}{1+a} + \\frac{b}{1+b} = 1 \\). Show that \\( \\frac{a}{1+b^{2}} - \\frac{b}{1+a^{2}} = 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": "Return your final response within \\boxed{}. Points $A$ and $B$ are on a circle of radius $5$ and $AB = 6$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$?\n$\\textbf{(A)}\\ \\sqrt {10} \\qquad \\textbf{(B)}\\ \\frac {7}{2} \\qquad \\... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 three numbers $a_{1}, a_{2}, a_{3}$ are chosen from the numbers $1,2, \\cdots, 14$ in increasing order such that they satisfy the conditions $a_{2} - a_{1} \\geqslant 3$ and $a_{3} - a_{2} \\geqslant 3$, how many different ways are ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 value of\n\\[\\frac{n}{2} + \\frac{18}{n}\\]is smallest for which positive integer $n$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the positive integer n that makes the expressi... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 quadratic function \\( f(x) = ax^2 + bx + c \\) (where \\( a, b, c \\in \\mathbb{R} \\) and \\( a \\neq 0 \\)) satisfy the following conditions:\n1. For all \\( x \\in \\mathbb{R} \\), \\( f(x-4) = f(2-x) \\);\n2. For \\( x \\i... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Valentine's Day is coming and our friend Mr.IDC has fallen in love with a girl. He just could not 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.Write a program which finds the greatest common divisor of two natural numbers a and b\n\nHint\n\nYo... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 numbers of the form \\(2^{n}\\) for different positive integers \\(n\\) can start with any predetermined combination of digits."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nAlright, so I need to 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{}. The function \\( f(x) \\) is defined as \\( f(x) = \\frac{x-1}{x+1} \\). \nThe equation \\( f(x^2) \\times f(x) = 0.72 \\) has two solutions \\( a \\) and \\( b \\), where \\( a > b \\). What is the value of \\( 19a + 7b \\)?"
},
... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 N real values A_1, A_2, \\ldots, A_N. Compute the number of pairs of indices (i, j) su... | null | null | null | null | null | null | null | null | null | null |
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