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value | conversations listlengths 2 2 | problem stringlengths 67 28.8k ⌀ | reasoning stringlengths 1.08k 118k ⌀ | deepseek_solution stringlengths 4 15.6k ⌀ | id stringclasses 230
values | ground_truth_solution stringclasses 177
values | source stringclasses 1
value | code null | correct bool 1
class | judge_reasoning stringclasses 230
values | __original_row_idx int64 6 2.26k ⌀ |
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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... | [
{
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"value": "Return your final response within \\boxed{}. Find all permutations $a_1, a_2, \\ldots, a_9$ of $1, 2, \\ldots, 9$ such that \\[ a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1 \\]\r\nand\r\n\\[ a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_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{}. Solve the system of equations:\n\\[ \n\\begin{cases} \nx^{2} y - x y^{2} - 3 x + 3 y + 1 = 0, \\\\ \nx^{3} y - x y^{3} - 3 x^{2} + 3 y^{2} + 3 = 0.\n\\end{cases} \n\\]"
},
{
"from": "assistant",
"value": "<|begin_of_thought... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In an isosceles triangle \\( ABC \\) where \\( AB = BC \\), side \\( AC = 10 \\). A circle with a diameter of 15 is inscribed in the angle \\( ABC \\) such that it touches the side \\( AC \\) at its midpoint. Find the radius of the cir... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "1. e4 1... d5 2. exd5 2... Nf6 3. d4 3... Nxd5 4. Nf3 4... Bf5 5. Be2 5... e6 6. O-O 6... Be7 7. c4 7... Nb4 8. Na3 8... O-O 9. Be3 9... N8c6 10. Qd2 10... Bf6 11. Rfd1 11... Qe7 12. Qc3 12... e5 13. dxe5 13... Nxe5 14. Nxe5 14... Bxe5 15. Qb3 15... Qh4 16. g3 16... Qh3 17. Qxb4 ... | 1. e4 1... d5 2. exd5 2... Nf6 3. d4 3... Nxd5 4. Nf3 4... Bf5 5. Be2 5... e6 6. O-O 6... Be7 7. c4 7... Nb4 8. Na3 8... O-O 9. Be3 9... N8c6 10. Qd2 10... Bf6 11. Rfd1 11... Qe7 12. Qc3 12... e5 13. dxe5 13... Nxe5 14. Nxe5 14... Bxe5 15. Qb3 15... Qh4 16. g3 16... Qh3 17. Qxb4 17... Be4 18. Bf1 18... Qf5 19. Bg2 19..... | Okay, let's see. I need to analyze this chess position and figure out the best move for White here. The game has gone through a lot of moves, and now it's Black's turn after 29... something. Wait, the last move was 29... Rf3, right? Wait, no, let me check the notation again.
The moves given are up to 29. Rae1, then Bl... | **Final Answer**
The best move for Black is **29... Rxg3+**, leading to a forced checkmate.
**Analysis:**
1. **29... Rxg3+**
- Black's rook captures the pawn on g3, delivering check. White has two responses:
- **30. fxg3**: Capturing the rook with the f-pawn.
- Followed by **30... Qxg3+** (check),... | Ga6LdwjM | Rxg3+ | chess | null | true | The solution correctly identifies the move Rxg3+ as the best move for Black, and the analysis validates that this move leads to a forced checkmate in the subsequent moves. | 1,228 |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given the function \\( f(x)=\\frac{a^{x}}{1+a^{x}} \\) with \\( a>0 \\) and \\( a \\neq 1 \\), and \\( [m] \\) representing the greatest integer less than or equal to a real number \\( m \\) (i.e., the floor function), find the range o... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. A gardener needs to plant 10 trees over the course of three days. In how many ways can he distribute the work over the days if he plants at least one tree each day?"
},
{
"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 acute-angled $\\triangle ABC$ be inscribed in circle $\\odot O$. Let $H$ be the orthocenter and $M$ be the midpoint of $BC$. The line passing through point $A$ and parallel to $BC$ intersects $\\odot O$ at point $X$. The line passi... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 guiding normal vector to the ellipsoid \\(x^{2}+2y^{2}+3z^{2}=6\\) at the point \\(M_{0}(1,-1,1)\\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the guiding normal vector 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": "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 running through a rectangular field. This field can be represented as a matrix with 3 rows 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{}. In a $7 \\times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid?"
},
{
"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{}. Given an isosceles triangle with a base of 12 and a side length of 18, what lengths should be marked on the sides from the vertex to form a trapezium with a perimeter of 40 when connecting the marked points?"
},
{
"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 $x_1, x_2, ..., x_{2004}$ be a sequence of integer numbers such that $x_{k+3}=x_{k+2}+x_{k}x_{k+1}$ , $\\forall 1 \\le k \\le 2001$ . Is it possible that more than half of the elements are negative?"
},
{
"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{}. A 5 by 5 grid of unit squares is partitioned into 5 pairwise incongruent rectangles with sides lying on the gridlines. Find the maximum possible value of the product of their areas."
},
{
"from": "assistant",
"value": "<|be... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the smallest real number \\( M \\) such that\n\\[ \n\\left|\\sum a b\\left(a^{2}-b^{2}\\right)\\right| \\leqslant M\\left(\\sum a^{2}\\right)^{2} \n\\]\nholds for all real numbers \\( a \\), \\( b \\), and \\( 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 a triangle \\(ABC\\) with angle bisectors \\(AA_1\\), \\(BB_1\\), and \\(CC_1\\). The angles \\(A\\), \\(B\\), and \\(C\\) are in the ratio \\(4:2:1\\). Prove that \\(A_1B_1 = A_1C_1\\)."
},
{
"from": "assistant",
"va... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "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.Gautam Buddha University is organising a car racing competition \"GBU F1 Racers\". Each contestant h... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "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 only difference between easy and hard versions is constraints.\n\nThe BerTV channel every day br... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 pairs of integers \\((x, y)\\) such that\n\n\\[ x^4 - 2y^2 = 1 \\]"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this equation to solve: \\( x^4 - 2y^2 = 1 \\). I need to find all int... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. \nGiven a convex polygon \\( P_{0} P_{1} \\ldots P_{n} \\) with \\( n+1 \\) sides, it is divided into \\( n-1 \\) triangles using \\( n-2 \\) non-intersecting diagonals. Prove that it is possible to number the triangles with the number... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 geometric sequence $\\left\\{a_{n}\\right\\}$, $a_{2}=\\sqrt{2}, a_{3}=\\sqrt[3]{3}$. Find the value of $\\frac{a_{1}+a_{2011}}{a_{7}+a_{2017}}$."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Tímár Mihály is in a difficult situation because the red crescent marking on the treasure bag has worn off. He knows that the treasure hidden in the wheat is in the heaviest of the four bags. From three weighings, the following informa... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. From point $A$ to point $B$ at 13:00, a bus and a cyclist left simultaneously. After arriving at point $B$, the bus, without stopping, returned and met the cyclist at point $C$ at 13:10. Returning to point $A$, the bus again without st... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 how many three-digit whole numbers does the sum of the digits equal $25$?\n$\\text{(A)}\\ 2 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 6 \\qquad \\text{(D)}\\ 8 \\qquad \\text{(E)}\\ 10$"
},
{
"from": "assistant",
"v... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find $k$ if\n\\[(\\sin \\alpha + \\csc \\alpha)^2 + (\\cos \\alpha + \\sec \\alpha)^2 = k + \\tan^2 \\alpha + \\cot^2 \\alpha.\\]"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's see. I need to find ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 triples of natural numbers such that the product of any two numbers in the triple, increased by 1, is divisible by twice the third number."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, 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{}. Find all pair $(m,n)$ of integer ( $m >1,n \\geq 3$ ) with the following property:If an $n$ -gon can be partitioned into $m$ isoceles triangles,then the $n$ -gon has two congruent sides."
},
{
"from": "assistant",
"va... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "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 is a binary string a of length n. In one operation, you can select any prefix of a with an equ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Quartet $(p,a,b,c)$ is called \"interesting\" if $p$ is an odd prime, no two of $a,b,c$ are equal, $p|ab+1$ , $p|ac+1$ , $p|bc+1$ . Prove that all \"interesting\" quartets satisfy inequality $p+2 \\leq \\frac{a+b+c}{3}$ ."
... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In the isosceles triangle $ABC$, the vertex angle is $B A C = \\alpha$. The base $BC$ is divided into $n$ equal segments by points $D_{1}, D_{2}, \\ldots, D_{n-1}$, moving from $B$ to $C$. Point $E$ divides side $AB$ in the ratio $1:(n... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Find the number of integers \\( n \\) greater than 1 such that for any integer \\( a \\), \\( n \\) divides \\( a^{25} - a \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find all intege... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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}, \\ldots, a_{n} \\) be real numbers such that \\(\\sum a_{i} = 0\\) and \\(\\sum \\left|a_{i}\\right| = 1\\). Show that \\(\\left|\\sum i \\cdot a_{i}\\right| \\leq \\frac{n-1}{2}\\)."
},
{
"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{}. Given $\\alpha$ and $\\beta$ in the interval $\\left(0, \\frac{\\pi}{2}\\right)$, prove that $\\alpha + \\beta = \\frac{\\pi}{2}$ is a necessary and sufficient condition for $\\frac{\\sin^{4} \\alpha}{\\cos^{2} \\beta} + \\frac{\\sin^{... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 two arrays of integers *a* and *b*. For each element of the second array *b**j* you sh... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 \\( \\cos(x^2) + \\cos(y^2) - \\cos(xy) < 3 \\) for real numbers \\( x \\) and \\( y \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to show that the expression cos(x²) + cos(y²) ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let the function \\( f(x) \\) be defined for \\( x \\geq 0 \\) and satisfy the conditions:\n(1) For any \\( x, y \\geq 0 \\), \\( f(x) f(y) \\leq y^{2} f\\left(\\frac{x}{2}\\right) + x^{2} f\\left(\\frac{y}{2}\\right) \\),\n(2) 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{}. Find a rational number that differs from \na) $\\alpha = \\sqrt{2}$;\nb) $\\alpha = 2+\\sqrt{5}$;\nc) $\\alpha = 3+\\sqrt{7}$ by no more than 0.0001."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I 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 perimeter of a rectangle is 48. What is the largest possible area of the rectangle?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out the largest possible area of a rectangle wh... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "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 grasshopper is in a flower field. The field contains $N \\cdot N$ flowers arranged in $N$ rows 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{}. \\(\\frac{\\sin^{2} x - 2}{\\sin^{2} x - 4 \\cos^{2} \\frac{x}{2}} = \\tan^{2} \\frac{x}{2}\\)"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to solve this trigonometric equation: (sin²x - 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 vector field \\( a=\\frac{2 \\cos \\theta}{r^{3}} \\mathbf{e}_{r}+\\frac{\\sin \\theta}{r^{3}} \\mathbf{e}_{\\theta} \\) is solenoidal."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I n... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Polycarpus has n markers and m marker caps. Each marker is described by two numbers: xi is the color... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 call a number *perfect* if the sum of its positive integer divisors(including $1$ and $n$ ) equals $2n$ . Determine all *perfect* numbers $n$ for which $n-1$ and $n+1$ are prime numbers."
},
{
"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": "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.Jon fought bravely to rescue the wildlings who were attacked by the white-walkers at Hardhome. On hi... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given point \\( A(0,2) \\) and the parabola \\( y^2 = x + 4 \\), with two points \\( B \\) and \\( C \\) on the parabola such that \\( AB \\perp BC \\), find the range of the y-coordinate of point \\( C \\)."
},
{
"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{}. Find all real polynomials \\( p(x) \\) such that \\( 1 + p(x) \\equiv \\frac{p(x-1) + p(x+1)}{2} \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find all real polynomials p(x) that satis... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two.\n\n*Proposed by Morteza Saghafian*"
}... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 a regular hexagon with an area of $144 \\mathrm{~cm}^{2}$, all its diagonals were drawn. The hexagon thus \"split\" into several triangles and quadrilaterals.\n\na) Determine into how many parts the diagonals have divided the hexa... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "The Great Pyramid of Giza, the largest pyramid in Egypt, is not only the oldest of the Seven Wonders of the Ancient World, but it is also the only one to remain largely intact. The Egyptian Pyramids can take up to 20 years to build and are so big that Al-Aziz Uthman, son of the g... | The Great Pyramid of Giza, the largest pyramid in Egypt, is not only the oldest of the Seven Wonders of the Ancient World, but it is also the only one to remain largely intact. The Egyptian Pyramids can take up to 20 years to build and are so big that Al-Aziz Uthman, son of the great Saladin who crushed the Crusaders, ... | Okay, I need to solve this programming problem. Let's read the task carefully. The input is a sequence of distances separated by spaces. The output should be 10x10 text pyramids separated by those distances. Each distance of 1 equals two characters. The pyramids to the left are displayed as if in front, so overlapping ... | To solve this problem, we need to generate a series of 10x10 text pyramids separated by specified distances. Each distance is given in units where 1 unit equals 2 characters. The pyramids should be displayed such that pyramids to the left appear in front of those to the right.
### Approach
1. **Input Parsing**: Read t... | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 \\in \\mathbb{N}^{*} \\) and \\( P \\in \\mathbb{R}[X] \\) be a polynomial of degree at most \\( n \\) such that \\( P(k) = \\frac{k}{k+1} \\) for all \\( k \\in \\llbracket 0, n \\rrbracket \\). What is the value of \\( P(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{}. Let \\( P(x) \\) be a polynomial of degree at most 3 such that \\( P(x) = \\frac{1}{1 + x + x^2} \\) for \\( x = 1, 2, 3, 4 \\). What is \\( P(5) \\)?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let's... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Rectangle A has length \\( 6 \\, \\text{cm} \\) and area \\( 36 \\, \\text{cm}^2 \\). Rectangle B has length \\( 12 \\, \\text{cm} \\) and area \\( 36 \\, \\text{cm}^2 \\). Rectangle C has length \\( 9 \\, \\text{cm} \\) and area \\( 3... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let positive integers \\(a\\), \\(b\\), \\(c\\), and \\(d\\) satisfy\n$$\na^{2} - ab + b^{2} = c^{2} - cd + d^{2}.\n$$\nProve that \\(a + b + c + d\\) is a composite number."
},
{
"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{}. There are three piles of books, totaling 240 books. The number of books in pile A is 30 more than three times the number of books in pile B, and pile C has 15 fewer books than pile B. How many books are in pile A?"
},
{
"from":... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In the rectangle \\(ABCD\\), a point \\(M\\) is taken on the arc \\(\\overparen{AB}\\) of the circumcircle of \\(ABCD\\), distinct from vertices \\(A\\) and \\(B\\). Points \\(P\\), \\(Q\\), \\(R\\), and \\(S\\) are the projections 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{}. In the country of East Westmore, statisticians estimate there is a baby born every $8$ hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year?\n$\\textbf{(A)}\\hspace... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 language of the Tru-la-la tribe, any sequence of 10 digits consisting of 0s and 1s is considered a word. Two words are defined as synonyms if and only if one can be obtained from the other through a series of operations where 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{}. Laura went for a training ride on her bike. She covered the first 10% of the total distance in 20% of the total time of the ride. What was the ratio of her average speed over the first 10% of the distance to her average speed over 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{}. Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?\n$\\textbf{(A)}\\ 15 \\qquad \\textbf{(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{}. \nIn how many ways can 1000000 be represented as the product of three factors, if the products that differ by the order of the factors:\n\na) are considered distinct?\n\nb) are considered identical?"
},
{
"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": "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.If there is an integer not less than 0 satisfying the following conditions, print the smallest such ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 median of $\\mbox{M}$ numbers is defined as the middle number after sorting them in order 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 $a$ and $b$ be positive whole numbers such that $\\frac{4.5}{11}<\\frac{a}{b}<\\frac{5}{11}$ . \nFind the fraction $\\frac{a}{b}$ for which the sum $a+b$ is as small as possible. \nJustify your answer"
},
{
"from":... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. ABCD is a quadrilateral with vertices in that order. Prove that AC is perpendicular to BD if and only if \\( AB^2 + CD^2 = BC^2 + DA^2 \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this g... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 fraction \\( \\frac{1}{7} \\) is added to another fraction, and the result is a proper fraction with a denominator less than 5. What is the largest fraction that could have been added?"
},
{
"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{}. Given two spheres \\(\\left(O_{1}, r_{1} ; O_{2}, r_{2}\\right)\\) and a point \\(P\\) on their intersection circle. Construct a line through point \\(P\\) that has an initial angle of \\(30^{\\circ}\\) with the horizontal plane and wh... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Solve the equation\n\\[\n\\sqrt{x+\\sqrt{4 x+\\sqrt{16 x+\\sqrt{\\ldots+\\sqrt{4^{2008} x+3}}}}} \\sqrt{x}=1 .\n\\]\nExpress your answer as a reduced fraction with the numerator and denominator written in their prime factorization."
... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 an integer such that \\( q = 2^p - 1 \\) is a prime, and let \\( n = 2^{p-1} q \\). Prove that the sum of the reciprocals of the divisors of \\( n \\) is 2."
},
{
"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{}. Petia has a credit card with a cashback of $1 \\%$ on all purchases. Additionally, the bank offers Petia the option to choose up to three categories with a higher cashback each month. Which categories are most beneficial to choose cons... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 2N balls, N white and N black, arranged in a row. The integers from 1 through N are writte... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 unit square lies across two parallel lines a unit distance apart, so that two triangular areas of the square lie outside the lines. Show that the sum of the perimeters of these two triangles is independent of how the square is placed... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 this problem you will write a simple code generator for a 2D programming language derived from [B... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established 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 all f:N $\\longrightarrow$ N that:\n[list]**a)** $f(m)=1 \\Longleftrightarrow m=1 $ **b)** $d=gcd(m,n) f(m\\cdot n)= \\frac{f(m)\\cdot f(n)}{f(d)} $ **c)** $ f^{2000}(m)=f(m) $ [/list]"
},
{
"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{}. \\( p \\) is a prime. Find the largest integer \\( d \\) such that \\( p^d \\) divides \\( p^4! \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to find the largest integer \\( d \\) such 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{}. Inside a cylinder with a base radius of 6, there are two spheres each with a radius of 6. The distance between the centers of the spheres is 13. If a plane is tangent to these two spheres and intersects the surface of the cylinder form... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 points on plane $\\alpha$, and on the same side of the line $AB$ there are $n$ different points $P_1, P_2, \\ldots, P_n$. Prove that the set of distances from $A$ and $B$ to $P_i$ $(i=1, 2, \\ldots, n)$ contains ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. There are two equiangular convex polygons \\( P_{1} \\) and \\( P_{2} \\) with different numbers of sides. Each angle of \\( P_{1} \\) measures \\( x \\) degrees, and each angle of \\( P_{2} \\) measures \\( kx \\) degrees, where \\( 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 a regular triangular pyramid, a cross-section is made that forms a square. Find the volume of the pyramid if the side of the base is $a$ and the side of the square cross-section is $b$."
},
{
"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{}. Prove that if in any triangle\n\n$$\n\\sin C - \\cos B = \\cos A,\n$$\n\nor\n\n$$\na + \\cot \\left(45^{\\circ} - B \\right) = \\frac{2}{1 - \\cot C},\n$$\n\nthen the triangle is a right triangle. Conversely, in every right triangle,\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 center of the circle with equation $x^2+y^2=4x+12y-39$ is the point $(h,k)$. What is $h+k$?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I have this equation of a circle: x squared plus y square... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$?\n$\\text{(A) } 15\\quad \\text{(B) } 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{}. Let the set \\( M = \\{1, 2, \\cdots, 10\\} \\) have five-element subsets \\( A_1, A_2, \\cdots, A_k \\) that satisfy the condition: any two elements of \\( M \\) appear together in at most two subsets \\( A_i \\) and \\( A_j \\) (wher... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 equation \\( 65x - 43y = 2 \\) in natural numbers \\( x \\) and \\( y \\)."
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to solve the equation 65x - 43y = 2 where x and y are natural... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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, three points $A$, $B$, and $C$ are chosen such that $A B=3$ and $B C=5$. What can $A C$ be equal to?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, let me try to figure out this problem. So, 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{}. \\( p(x, y, z) \\) is a polynomial with real coefficients such that:\n1. \\( p(tx, ty, tz) = t^2 f(y - x, z - x) \\) for all real \\( x, y, z, t \\) (and some function \\( f \\));\n2. \\( p(1, 0, 0) = 4 \\), \\( p(0, 1, 0) = 5 \\), 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{}. A circumference with radius \\(r\\) is inscribed in a circular sector of radius \\(R\\). The length of the chord \\(AB\\) is equal to \\(2a\\). Prove that\n\n$$\n\\frac{1}{r} = \\frac{1}{R} + \\frac{1}{a}\n$$"
},
{
"from": "ass... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 can it be expressed using a single equality sign that each of the three real numbers \\(a, b, c\\) is separately equal to zero?"
},
{
"from": "assistant",
"value": "<|begin_of_thought|>\n\nOkay, so I need to figure out ... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and 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 an array $a_1, a_2, \\dots, a_n$ consisting of $n$ distinct integers. You are allowed to pe... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 all ellipses with center at the origin, exactly one such ellipse is tangent to the graph of the curve $x^3 - 6x^2y + 3xy^2 + y^3 + 9x^2 - 9xy + 9y^2 = 0$ at three distinct points. The area of this ellipse is $\\frac{k\\pi\\sqr... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides (with each line distinct and... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Let \\( O \\) be the circumcenter, and \\( \\Omega \\) be the circumcircle of an acute-angled triangle \\( ABC \\). Let \\( P \\) be an arbitrary point on \\( \\Omega \\), distinct from \\( A, B, C \\), and their antipodes in \\( \\Ome... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. What is the smallest number of sides that a polygon (not necessarily convex) with an odd number of sides can have, if it can be divided into parallelograms?"
},
{
"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": "Return your final response within \\boxed{}. Uncle Fyodor makes sandwiches with sausage, and Matroskin and Sharik eat them. It is known that Matroskin ate three times more sandwiches than Sharik, Sharik ate 21 sandwiches less than Uncle Fyodor made, and Uncle Fyodor made twice as... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... | [
{
"from": "user",
"value": "Return your final response within \\boxed{}. Does there exist a strictly increasing function from the set of positive integers $\\mathbb{N}^{*}$ to itself such that $f(1)=2$ and for every positive integer $n$, the following equation holds?\n\n$$\nf(f(n)) = f(n) + n\n$$\n\n(IMO 19... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 passenger can easily determine the speed of a train: They measure how many seconds it takes to cover $200 \\mathrm{~m}$! (There are distance markers every 200 meters along the track.) If the number of seconds is $p$, then the train'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{}. For the quadratic equation in $x$, \\( x^2 + z_1 x + z_2 + m = 0 \\), where \\( z_1 \\), \\( z_2 \\), and \\( m \\) are all complex numbers, and \\( z_1^2 - 4z_2 = 16 + 20i \\). Suppose the two roots of this equation are \\( \\alpha \\... | null | null | null | null | null | null | null | null | null | null |
Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in 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 \\( x=1.9 \\dot{8} \\dot{9} \\) and \\( x-1=\\frac{K}{99} \\), find \\( K \\).\n\nThe average of \\( p, q \\) and \\( r \\) is 18. The average of \\( p+1, q-2, r+3 \\) and \\( t \\) is 19. Find \\( t \\)."
},
{
"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 \\( x \\) be a real number, and let \\( y = |x-1| + |x+1| \\). Consider the following four conclusions:\n(I) \\( y \\) does not have a minimum value.\n(II) There is only one \\( x \\) for which \\( y \\) attains its minimum 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 volume of an ideal gas increases by $n=2$ times in an isobaric process, and then decreases to its initial value in a process where the pressure $P$ of the gas is directly proportional to its volume $V$.\n1. By what factor does 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{}. You decided to buy yourself a New Year's gift worth 8,000 rubles. You have the option to pay with one of two bank cards: a debit card or a credit card. If you pay with the credit card (the credit limit allows it), you will need to repa... | null | null | null | null | null | null | null | null | null | null |
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