source string | kind string | messages list | num_tokens int64 | cluster int64 |
|---|---|---|---|---|
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of the $x$ term in the expansion of $(1-x)(1+x)^{5}$ is (__).\n\nA: $4$\nB: $6$\nC: $8$\nD: $12$"
},
{
"role": "assistant",
"c... | 278 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of the $x$ term in the expansion of $(x^2-x-2)^3$ is ( )\n\nA: -12 \nB: 12 \nC: 4 \nD: -4"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\n ... | 267 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of the $x^3$ term in the expansion of $(1+ \\frac{2}{x})(1-x)^4$ is ( )\n\nA: -2 \nB: 2 \nC: -3 \nD: 3"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topolo... | 266 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "What is the coefficient of \\(x^{4}\\) in the product \n\n$$\n\\left(1-2x+4x^{2}-8x^{3}+16x^{4}\\right) \\cdot \\left(1+2x+4x^{2}+8x^{3}+16x^{4}\\right)?\n$$"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n... | 380 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Given $a=-\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\cos x dx$, find the coefficient of the $x^3$ term in the expansion of the binomial $(x^2+\\frac{a}{x})^6$.\nA: $20$\nB: $-20$\nC: $160$\nD: $-160$"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\n... | 377 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Find the coefficient of $x^{4}$ in the expansion of the expression $x{{(x-\\\\dfrac{2}{x})}^{7}}$. (Answer with a number only)"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat To... | 209 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Determine the coefficient of the term containing $x^{3}$ in the expansion of $(1-x)^{5}+(1-x)^{6}+(1-x)^{7}+(1-x)^{8}$.\nA: $121$\nB: $-74$\nC: $74$\nD: $-121$... | 299 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of the $x^3$ term in the expansion of $x(x+3)^5$ is ( ).\nA: 270\nB: 180\nC: 90\nD: 45"
},
{
"role": "assistant",
"content"... | 251 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Determine the number of terms in the simplified expression of \\([ (a + 2b)^3 (a - 2b)^3 ]^2\\).\n**A)** 4 \n**B)** 5 \n**C)** 6 \n**D)** 7 \n**E)** 8... | 359 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Consider the expansion of $[(a+4b)^{2}(a-2b)^{2}]^{3}$ when fully simplified. How many distinct terms does it have?\nA) 6 \nB) 7 \nC) 8 \nD) 9 \nE) 1... | 316 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of $x^3$ in the expansion of $(1-x)(1+x)^8$ is $\\boxed{\\text{answer}}$."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\n /-The coefficient of... | 238 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The coefficient of the term containing $x^{2}$ in the expansion of $(x- \\frac {1}{x})^{6}$ is (__).\n\nA: $-20$\nB: $20$\nC: $-15$\nD: $15$"
},
{
"role": "a... | 246 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Determine the coefficient of the $x^{2}$ term in the expansion of \\\\((1+2x)^{5}\\\\).\n\nA: $80$\nB: $40$\nC: $20$\nD: $10$"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Top... | 262 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(x^2 - 3x + 2)^5$ is ( ).\n\nA: $-240$\nB: $-120$\nC: $0$\nD: $120$"
},
{
"role": "assis... | 307 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of $x^{5}y^{2}$ in the expansion of $(x^{2}+x+y)^{5}$ is ( ).\nA: $10$\nB: $20$\nC: $30$\nD: $60$"
},
{
"role": "assistant",
"... | 299 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(1-2x)^4$ is ( )\n\nA: 32 \nB: 4 \nC: -8 \nD: -32"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Ra... | 249 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "The coefficient of $x^3$ in the expansion of $(1-3x)^5$ is ( )\n\nA: -270 \nB: -90 \nC: 90 \nD: 270"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib... | 260 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Determine the coefficient of the third term and the binomial coefficient in the expansion of \\\\((3b+2a)^{6}\\\\)."
},
{
"role": "assistant",
"c... | 188 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(1-2x)^4$ is ( )\n\nA: 32 \nB: 4 \nC: -8 \nD: -32"
},
{
"role": "assistant",
... | 244 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The number of terms in the expansion of $(\\sqrt{x} - \\frac{1}{x})^{10}$ that contain positive integer powers of $x$ is ( ) \nA: 0 \nB: 2 \nC: 4 \nD: 6... | 290 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "In the expansion of $(x^{2}+ \\frac {1}{x^{2}}+2)^{3}(x-2)$, the constant term is $\\boxed{\\text{answer}}$."
},
{
"role": "assistant",
"content": "... | 333 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Given the expansion of the expression $\\left(1-\\frac{1}{x}\\right)\\left(1+x\\right)^{5}$, determine the coefficient of the $x^3$ term."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators ... | 188 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of $x^8$ in the expansion of $(x^2+1)(x+a)^8$ is 113, then the value of the real number $a$ is \\_\\_\\_\\_\\_\\_."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Na... | 246 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The constant term in the expansion of \\\\(\\left(x^{2}+2\\right)\\left( \\frac{1}{x^{2}}-1\\right)^{5} \\\\) is ( )\nA: \\\\(2\\\\)\nB: \\\\(3\\\\)\nC: \\\\(-2\\\\)\nD: \\\\(-3\\\\)"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\n... | 326 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The coefficient of $x^2$ in the expansion of $(1- \\frac {1}{x^{2}})(1+x)^{4}$ is $\\_\\_\\_\\_\\_\\_$."
},
{
"role": "assistant",
"content": "```lean\ni... | 170 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "The coefficient of $\\frac{1}{x}$ in the expansion of $(1-x^2)^4\\left(\\frac{x+1}{x}\\right)^5$ is ( )\n\nA: 5 \nB: 11 \nC: -21 \nD: -29"
},
{
"role": "assistant",
... | 264 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The constant term in the expansion of the binomial $$( \\sqrt {x}- \\frac {1}{ \\sqrt[3]{x}})^{5}$$ is ( )\nA: -10\nB: 6\nC: 10\nD: 20"
},
{
"role": "a... | 306 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "The constant term in the expansion of $\\left(x- \\frac{1}{2x}\\right)^6$ is ( )\n\nA: $\\frac{15}{16}$\n\nB: $-\\frac{15}{16}$\n\nC: $\\frac{5}{2}$\n\nD: $-\\frac{5}{2}$"
},
... | 297 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of \\\\(x^{4}y^{2}\\\\) in the expansion of \\\\((x-2y)^{6}\\\\) is \\\\((\\\\) \\\\()\\\\)\nA: \\\\(15\\\\)\nB: \\\\(-15\\\\)\nC: \\\\(60\\\\)\nD: \\\\(-60\\\\)"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_opt... | 350 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Given the expansion of the binomial $\\left(1-2x\\right)^{8}$, which of the following statements is true?\n\nA: The coefficient of the term containing $x^{2}$ is 28.\n\nB: The sum of the coefficients of all terms is 1.\n\nC: The term with the largest coefficient is the fifth te... | 348 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of $x$ in the expansion of the binomial $(x^{2}+ \\frac {1}{x})^{5}$ is ______."
},
{
"role": "assistant",
"content": "```lean\n... | 168 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(1-2x)^4$ is __( ).\n\nA: 32\nB: 4\nC: -8\nD: -32"
},
{
"role": "assistant",
"conte... | 226 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of $x^{4}y^{2}$ in the expansion of $(3x-y)(x+2y)^{5}$ is ( )\nA: $110$\nB: $120$\nC: $130$\nD: $150$"
},
{
"role": "assistant",
... | 265 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of $x^{2}y^{3}$ in the expansion of $\\left(\\frac{1}{2}x-2y\\right)^{5}$ is\nA: $-20$\nB: $-5$\nC: $5$\nD: $20$"
},
{
"role": "assista... | 253 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "Find the fourth term in the expansion of ${(x-2y)}^{7}$.\n\nThe options are:\nA: $-35{x}^{4}{y}^{3}$\nB: $35{x}^{4}{y}^{3}$\nC: $280{x}^{4}{y}^{3}$\nD: $-280{x}^{4}{... | 368 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The coefficient of $x^2$ in the expansion of $(1-x)^4$ is ( )\nA: -6\nB: -12\nC: 12\nD: 6"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\... | 237 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "In the expansion of $(3x-2)(x-1)^{6}$, the coefficient of $x^{3}$ is ( ).\n\nA: $85$\n\nB: $5$\n\nC: $-5$\n\nD: $-85$"
},
{
"role": "assistant",
... | 279 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of the $x^3$ term in the expansion of $(1+x)(1- \\sqrt {x})^{6}$ is ( )\n\nA: 14\nB: 15\nC: 16\nD: 17"
},
{
"role": "assistant",
... | 294 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of $x^3$ in the expansion of $(1-2x)^6$ is (__).\nA. 240\nB. 15\nC. -15\nD. -160"
},
{
"role": "assistant",
"content": "```lean\nim... | 257 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The coefficient of $x^2$ in the expansion of $(x+1)^5(x-2)$ is ( )\n\nA: 25 \nB: 5 \nC: -15 \nD: -20"
},
{
"role": "assistant",
"content": "```l... | 252 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of the third term in the expansion of $$(1+ \\frac {1}{2}x)^{5}$$ is ( )\n\nA: 5 \nB: $$\\frac {5}{2}$$ \nC: $$\\frac {5}{4}$$ \nD: $$\\frac {5}{8}$$"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option ma... | 319 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The total number of terms in the expansion of $(x+y+z)^4$ is ( ).\nA: 10\nB: 15\nC: 20\nD: 21"
},
{
"role": "assistant",
"content": "```lean\nimpo... | 244 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The coefficient of $x^3y^3$ in the expansion of $(x-y)(2x-y)^5$ is $\\_\\_\\_\\_\\_\\_$. (Answer with a number)"
},
{
"role": "assistant",
"content": "``... | 253 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The coefficient of $x^{3}$ in the expansion of $\\left(x- \\frac {1}{x}\\right)^{9}$ is ( ).\n\nA: $-36$\n\nB: $36$\n\nC: $-84$\n\nD: $84$"
},
{
"role":... | 291 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The term with the largest binomial coefficient in the expansion of $(1-2x)^8$ is:\nA: The 4th term\nB: The 5th term\nC: The 7th term\nD: The 8th term"
},
{
... | 321 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "In the expansion of $\\left(a-\\dfrac{1}{\\sqrt{a}}\\right)^7$ the coefficient of $a^{-\\dfrac{1}{2}}$ is:\n\n\n\n$\\textbf{(A)}\\ -7 \\qquad\n\\textbf{(B)}\\ 7 \\qquad\n\\textbf... | 440 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The term with the largest coefficient in the expansion of $(x- \\frac {1}{x})^{6}$ is the \\_\\_\\_\\_\\_\\_ term."
},
{
"role": "assistant",
"co... | 330 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The coefficient of the term containing $x^{2}$ in the expansion of $(1-2x)^{4}$ is:\n\nA: $-24$\n\nB: $24$\n\nC: $-16$\n\nD: $16$"
},
{
"role": "assistant",
... | 184 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(1-2x)^4$ is ( )\n\nA: 32\nB: 4\nC: -8\nD: -32"
},
{
"role": "assistant",
"cont... | 233 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Expand the expression \\\\((x+ \\frac {1}{x}-2)^{5}\\\\) and find the constant term.\n\nA: 252\nB: -252\nC: 160\nD: -160"
},
{
"role": "assistant"... | 254 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The number of terms in the expansion of \\([(a+2b)^3(a-2b)^3]^2\\) when simplified is:\nA) 5\nB) 6\nC) 7\nD) 8\nE) 9"
},
{
"role": "assistant",
"... | 373 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of the third term in the expansion of $( \\sqrt {2}x-1)^{5}$ is ( )\nA: $-20 \\sqrt {2}$\nB: $20$\nC: $-20$\nD: $20 \\sqrt {2}$"
},
{
... | 337 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(1-2x)^5(1+3x)^4$ is \\_\\_\\_\\_\\_\\_."
},
{
"role": "assistant",
"content": "`... | 288 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of $x^7$ in the polynomial expansion of \n\\[ (1+2x-x^2)^4 \\]\nis\n\n$ \\textbf{(A)}\\ -8 \\qquad\\textbf{(B)}\\ 12 \\qquad\\textbf{(C)}\\ ... | 371 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "If the expansion of $(x^{2}- \\frac {1}{2x})^{n}$ has only its fourth term's binomial coefficient as the maximum, then the sum of the coefficients of all ... | 262 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The coefficient of $x^2$ in the expansion of $(1+2x)^3$ is ( )\nA: 80\nB: 12\nC: 20\nD: 10"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem lean_workbook_... | 168 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "The coefficient of $x^4$ in the expansion of $(x+1)(x-2)^6$ is ( )\nA: -100\nB: -15\nC: 35\nD: 220"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nim... | 326 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "The coefficient of the term containing $x^2$ in the expansion of $\\left(x- \\frac {1}{x}\\right)^{6}$ is ( ) \nA: -20 \nB: 20 \nC: -15 \nD: 15"
},
... | 317 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "In the expansion of $(x-2y+3z)^{6}$, the coefficient of $x^{3}y^{2}z$ is ( ).\n\nA: $-60$\n\nB: $240$\n\nC: $-360$\n\nD: $720$"
},
{
"role": "assistant",
"content": "... | 314 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "The coefficient of the term containing $x$ in the expansion of $(1-2x)^{4}$ is ( )\nA: $32$\nB: $4$\nC: $-8$\nD: $-32$"
},
{
"role": "assistant",
"content": "```lean... | 242 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "The coefficient of the term $x^{2}y^{2}$ in the expansion of $(x-2y)^{4}$ is:\n\nA: $24$\n\nB: $-24$\n\nC: $6$\n\nD: $-6$"
},
{
"role": "assistant",... | 259 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The constant term in the expansion of $\\left( 2+\\frac{1}{{x}^{3}} \\right)\\left( 1-x \\right)^{5}$ is ( )\nA: $12$\nB: $-12$\nC: $-8$\nD: $-18$"
},
{
"... | 286 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "In the binomial expansion of $\\left(2x^{2}-\\frac{1}{x}\\right)^{5}$, the coefficient of $x$ is ( ).\n\nA: $10$\n\nB: $-10$\n\nC: $-40$\n\nD: $40$"
},
... | 294 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The coefficient of $x^3$ in the expansion of $(1-2x)(1-x)^5$ is ( )\n\nA: 10 \nB: -10 \nC: -20 \nD: -30"
},
{
"role": "assistant",
"content": "`... | 268 | 0 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfy $a^{x} < a^{y} (0 < a < 1)$, which of the following relationships always holds? \nA: $x^{3} > y^{3}$ \nB: $\\sin x > \\... | 289 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Given that the line $y=2x+m$ intersects the $x$-axis at the point $(-1,0)$, the solution set of the inequality $2x+m\\leqslant 0$ with respect to $x$ is ( ).\n\nA: $x \\gt -1$\n\... | 253 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "(1) Find the range of real number $a$ such that for any real number $x$, $|x-1|+|x-3| \\geqslant a^{2}+a$ always holds;\n(2) If $a+b=1$, find the minimum value... | 289 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "When $x \\in (1, 2)$, the inequality $(x-1)^2 < \\log_a x$ always holds. Then, the range of the real number $a$ is."
},
{
"role": "assistant",
"cont... | 207 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be constants, and suppose that the inequality \\[\\frac{(x-a)(x-b)}{x-c} \\le 0\\] is true if and only if either $x < -6$ or $|x-31| \\le 1.$ Given that $a ... | 272 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Find the value of $k$ if the length of the interval of solutions for the inequality $1 \\leq x^2 - 3x + k \\leq 5$ is $8$.\nA. $\\frac{9}{4}$\nB. $\\frac{11}{4}$\nC. 3\nD. $\\fra... | 316 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Given that the maximum value of $x$ that satisfies the inequality $|x^{2}-4x+k|+|x-3|\\leqslant 5$ is $3$, find the value of the real number $k$."
},
{
"role": "assistant... | 193 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Examine the increasing and decreasing behavior of the function \\( y = a x^n \\)."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\... | 175 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "If the inequality system about $x$ $\\left\\{\\begin{array}{l}x＞3-m\\\\ x≤5\\end{array}\\right.$ has five integer solutions, then the range of values for $m$ is ____."
},
{
... | 226 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Given \\(a \\in [-1,1]\\), find the solution to \\(x^{2} + (a-4)x + 4 - 2a > 0\\).\n\n(A) \\(x > 3\\) or \\(x < 2\\)\n\n(B) \\(x > 2\\) or \\(x < 1\\)\n\n(C) \... | 283 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Given proposition $p$: A real number $x$ satisfies the system of inequalities $\\begin{cases} \\log_{\\frac{1}{3}}x > -1 \\\\ x^{2}-6x+8 < 0\\end{cases}$,... | 302 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "For any real number $m\\in[0,1]$, $mx^{2}-2x-m\\geqslant 2$. Find the range of values for $x$."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport ... | 181 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Find the largest positive integer $n$ such that\n\\[(\\sin x + \\cos x)^n \\geq \\frac{2}{n}\\] for all real numbers $x$."
},
{
"role": "assistant",
... | 192 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Given the inequality $|x+m| \\leqslant 4$ has a solution set $A$, and $A \\subseteq \\{x | -2 \\leqslant x \\leqslant 8\\}$, then the range of the real numb... | 271 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "For the fractional equation $\\frac{m}{x-2}=3$ with positive solutions for $x$, the range of $m$ is ______."
},
{
"role": "assistant",
"content": "```lean\nimport Mat... | 179 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "If the diagonal of a rectangle is $a + b$ and the ratio of its sides is 2:1, then the area of the rectangle is:\nA. $\\frac{(a+b)^2}{2}$\nB. $\\frac{2(a+b)^... | 281 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Given that $(a+1)x - 1 - \\ln x \\leqslant 0$ holds for any $x \\in [\\frac{1}{2}, 2]$, find the maximum value of $a$."
},
{
"role": "assistant",
"... | 218 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "If the fractional equation in terms of $x$, $\\frac{1-x}{x-2}=\\frac{m}{2-x}-2$, has a positive root, then the value of $m$ is ( ).\n\nA: $-7$\n\nB: $-1$\n\... | 225 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "If there exists a point on the graph of the function $y= \\frac{k}{x}$ ($k>0$) that is at a distance of 1 from the origin, then the range of values for $k$ i... | 205 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "In the elective course 4-5, Advanced Topics on Inequalities, it is given that the inequality $a^2 - 3a \\leq |x+3| + |x-1|$ holds true for all real numbers $x$. The range of the ... | 242 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Find the nonzero value of $c$ for which there exists exactly one positive value of $b$ such that the equation $x^2 + \\left(2b + \\frac{2}{b}\\right)x + c = 0$ ... | 209 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Given that the derivative of the function $g(x)$ is $g'(x)=e^{x}$, and $g(0)g'(1)=e$, (where $e$ is the base of the natural logarithm), if there exists $x\\in(... | 343 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Given a quadratic equation in $x$: $x^{2}-4x+m=0$ where $m \\lt 4$. \n$(1)$ Determine the situation of the roots of the equation and explain the reason. \n... | 234 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Given a constant \\( a \\in (0,1) \\), and \\( |x| + |y| \\leqslant 1 \\), what is the maximum value of the function \\( f(x, y) = a x + y \\)?"
},
{
... | 199 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\left(a^{2}+x^{2}\\right) / x\\), where \\(a>0\\) is a constant and \\(x>0\\) is a variable."
},
{
"role": "assi... | 187 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Given $p: -2 \\leq x \\leq 11$, $q: 1 - 3m \\leq x \\leq 3 + m$ ($m > 0$), if $\\neg p$ is a necessary but not sufficient condition for $\\neg q$, then th... | 264 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Given $(x+2)^n = x^n + ax^{n-1} + \\ldots + bx + c$ (where $n \\in \\mathbb{N}^*$, $n \\geq 3$), and $b = 4c$, find the value of $a$."
},
{
"role": "... | 246 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Given the function $f(x)=|x+1|+|x-2|-m$.\n\n$(1)$ When $m=5$, find the solution set for $f(x) > 0$.\n\n$(2)$ If the solution set for the inequality $f(x) \\geq... | 270 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Elective 4-1: Selected Topics in Geometric Proofs\n\nGiven the inequality $|x+3|-2x-1 < 0$, its solution set is $(x_{0},+\\infty)$.\n\n(Ⅰ) Find the value of $x_{0}$;\n\n(Ⅱ) If the function $f(x) = |x-m|+\\left| x+\\frac{1}{m} \\right|-x_{0}$ ($m > 0$) has a root, find the value... | 304 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Given real numbers $x$ and $y$ satisfy $a^{x} < a^{y} (0 < a < 1)$, then which of the following relations always holds? \nA: $\\dfrac{1}{{x}^{2}+1} > \\dfrac{1}{{y}^{2}+1}$ \nB... | 260 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "If the solution of the equation $\\frac{{2x+m}}{{x-1}}=1$ with respect to $x$ is a non-negative number, then the range of values for $m$ is ( ).\n\nA: $m\... | 279 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "If $(x + y)\\left( \\frac {1}{x} + \\frac {a}{y}\\right) \\geq 16$ holds true for any nonzero real numbers $x$ and $y$, then the minimum value of the positive r... | 232 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Given that the equation $2x+4=m-x$ has a negative solution for $x$, what is the range of values for $m$?\n\nA: $m＜\\frac{4}{3}$\n\nB: $m＞\\frac{4}{3}$\n\nC:... | 217 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "For any real number $x$, the inequality $ax^2 + 2ax - (a+2) < 0$ always holds. The range of the real number $a$ is (__).\n\nA: $-1 \\leq a \\leq 0$\n\nB: $-1 < a < 0... | 282 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Given the quadratic inequality about $x$, $x^2 - ax + a + 3 \\geq 0$ holds true in the interval $[-2, 0]$. Determine the range of values for the real numb... | 220 | 1 |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "If the real number $a$ makes the solution set of the inequality system about $x$ $\\left\\{\\begin{array}{l}{-2＜x-1＜3}\\\\{x-a＞0}\\end{array}\\right.$ be $-1 \\lt x \\lt 4$, then... | 228 | 1 |
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