source string | kind string | messages list |
|---|---|---|
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "In the \"Super Brain\" quiz organized by the study group, the scores of 11 participants are shown in the table below. What are the median and mode of the scores of these 11 participants?\n\n| Score | 60 | 80 | 90 | 95 |\n|-------|----|----|----|----|\n| Number of Participants |... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The condition \"$x > 1$\" is a __________ for \"$2^{x} > 1$\".\nA: Sufficient but not necessary condition\nB: Necessary but not sufficient condition\nC: Suffici... |
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 an exam, 49% of candidates failed in English and 36% failed in Hindi, and a certain percentage failed in both subjects. The total number of candidates wh... |
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 the volume of two cubes are in the ratio 64 : 1 , the ratio of their edges is :"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "if a tire rotates at 400 revolutions per minute when the car is traveling 120 km / h , what is the circumference of the tire ?"
},
{
"role": "assistant... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Eden, Mary, and Iris gather sand to fill their sandbox. Eden carried some buckets of sand. Mary carried 3 more buckets of sand than Eden. Iris carried 1 less b... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Harly's animal shelter has some dogs. She adopts out 40% of them but then has to take back 5 because of personality conflicts with other dogs in their adopt... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Given circle $C_{1}: (x+2)^{2}+(y-3)^{2}=1$, and circle $C_{2}: (x-3)^{2}+(y-4)^{2}=9$, where $A$ and $B$ are moving points on circles $C_{1}$ and $C_{2}$ respectively, and point $P$ is a moving point on the $y$-axis, then the maximum value of $|PB|-|PA|$ is __( ).\n\nA: $\\sqr... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "a can do a piece of work in 8 days . he works at it for 4 days and then b finishes it in 6 days . in how many days can a and b together finish the work ?"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nop... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Suppose there is a bag containing 4 red balls and 2 white balls, all of the same size and texture. If balls are drawn one after another with replacement from th... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Find $\\cot 30^\\circ$."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOpe... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Arjun started a business with Rs. 20,000 and is joined afterwards by Anoop with Rs. 40,000. After how many months did Anoop join if the profits at the end of t... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Suppose the even function $f(x)$ is an increasing function on the interval $[1,3]$ and has a minimum value of $0$. Then on the interval $[-3,-1]$, $f(x)$ is:\nA... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Let $x$ and $y$ be complex numbers such that\n\\[\\frac{x + y}{x - y} + \\frac{x - y}{x + y} = 2.\\]\nFind \n\\[\\frac{x^4 + y^4}{x^4 - y^4} + \\frac{x^4 - y^4}{x^4 + y^4}.\\]"
... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Which of the following statements is exclusively incorrect?\n(A) Increasing, decreasing, multiplying, or dividing each side of a true inequality by the sa... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Let set $A = \\{x | x^2 - 3x + 2 = 0\\}$, and set $B = \\{x | x^2 + 2(a + 1)x + (a^2 - 5) = 0\\}$.\n(1) If $A \\cap B = \\{2\\}$, find the value of the real number $a$.\n(2) If $... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Compute the number of nonempty subsets $S \\subseteq\\{-10,-9,-8, . . . , 8, 9, 10\\}$ that satisfy $$ |S| +\\ min(S) \\cdot \\max (S) = 0. $$"
},
{
... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Find all triples $(x, y, z)$ of nonnegative integers such that\n$$ x^5+x^4+1=3^y7^z $$"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\n /-Find all triples $(x,... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Compute $\\sin 60^\\circ$."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOpe... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "What is the 150th letter in the pattern ABCDEFGHIJKLMNOPQRSTUVWXYZABC... repeating indefinitely?"
},
{
"role": "assistant",
"content": "```lean\nimp... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "How many different four-letter arrangements can be formed using the letters $A, B, C, D, E$, and $G$, if the last letter must be $D$, one of the other lette... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Given a sequence $\\{ a_{n} \\}$ that satisfies $a_{1}=1$ and $a_{n+1}=\\frac{a_{n}}{1+{a_{n}}^{2}}$, let $S_{n}$ and $T_{n}$ be the sum of the first $n$ terms of the sequences $... |
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 function $f(2x) = 4x - 1$, find the value of $f(2)$."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\ns... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "There are five numbered weights, each with unique identifiers. The set includes one weight each of 5g, 3g, and 1g, and two weights of 2g. If three weights are randomly selected, what is the probability that the total mass of these three weights is 9g? (Express the result as a s... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "a) Prove that \\(\\cos \\alpha+\\cos (\\alpha+x)+\\cos (\\alpha+2 x)+\\ldots+\\cos (\\alpha+n x)=\\frac{\\sin \\left(\\alpha+\\left(n+\\frac{1}{2}\\right) x\\right)-\\sin \\left(\\alpha-\\frac{1}{2} x\\right)}{2 \\sin \\frac{1}{2} x}\\).\n\nb) Prove that if \\(\\varphi=\\frac{2... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Find the curvature, radius of curvature, and coordinates of the center of the curve \\( y = \\frac{4}{x} \\) at the point \\( x = 2 \\)."
},
{
"role... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "$AD$ and $BC$ are both perpendicular to $AB$ , and $CD$ is perpendicular to $AC$ . If $AB = 4$ and $BC = 3$ , find $CD$ .\n\n![Image](https://cdn.artofproblemsolving.c... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Find the area of a trapezium whose one parallel side is 24 cm long, the other parallel side is some length, and the distance between them is 18 cm. The ar... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Participant A competes in a triathlon, swimming 1.5 kilometers, then cycling 40 kilometers, and finally running 10 kilometers. A's running speed is 5 times their swimming speed, and their cycling speed is 2.5 times their running speed. The combined time for swimming and running... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "The measures of the interior angles of a convex pentagon form an increasing arithmetic sequence. How many such sequences are possible if the pentagon is not equiangular and all o... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Water boils at 212 °F or a certain temperature in °C and ice melts at 32 °F or 0 °C. If the temperature of a pot of water is 40 °C, and the temperature of the p... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Tobias went to a swimming pool for 3 hours. Swimming every 100 meters took him 5 minutes, but every 25 minutes he had to take a 5-minute pause. How many meters ... |
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 $$\\overrightarrow{a} = (1, 0)$$, and $$\\overrightarrow{b} = (1, 2\\sqrt{3})$$.\n\n(1) Find the angle between $$\\overrightarrow{a}$$ and $$\\overrig... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Let $a \\diamond b = a \\cdot \\sqrt{b + \\sqrt{b + \\sqrt{b + \\sqrt{b + ...}}}}$. If $2 \\diamond h = 8$, find the value of $h$."
},
{
"role": "as... |
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 imaginary part of the complex number $\\dfrac{1-i}{1-2i}$ is ( ).\n\nA: $\\dfrac{1}{5}$ \nB: $\\dfrac{3}{5}$ \nC: $-\\dfrac{1}{5}$ \nD: $-\\dfrac{3}{... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Zoe bought an app that cost $5 dollars. Then each month she has to pay $8 to play online. If she played the game online for 2 months, how much money would sh... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "During the preparation period of the Shanghai Expo, 5 volunteers and 2 foreign friends line up for a photo. The 2 foreign friends stand next to each other but not at either end of the line. The total number of different arrangements is ( ).\nA: 1440\nB: 960\nC: 720\nD: 480"
}... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Let \\( f(x) \\) be defined on the interval \\((-1,1)\\). Then \\( f(x) \\) can be expressed as the sum of an even function \\( g(x) \\) and an odd function \\( h(x) \\). Find \\... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Source: 2017 Canadian Open Math Challenge, Problem A3\n-----\nTwo $1$ × $1$ squares are removed from a $5$ × $5$ grid as shown.\n\n[asy]\nsize(3cm);\nfor(int i ... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "In an 80 meters race, A beats B by 56 meters or a certain amount of time. A's time over the course is 3 seconds. By how many seconds does A beat B?"
},
{
... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "In a 50-question test with two marks for each question, Meghan learned that she scored 20 marks less than her friend Jose. Jose scored some more marks than their friend Alisson, ... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The function $g(x)$ satisfies\n\\[c^3 g(a) = a^3 g(c)\\] for all real numbers $a$ and $c.$ If $g(3) \\neq 0,$ find\n\\[\\frac{g(6) - g(2)}{g(3)}.\\]"
},
{
... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "What is the greatest common divisor of $3^{1200}-1$ and $3^{1210}-1$?"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "Given that a sphere's surface area and volume are equal, its radius is \\_\\_\\_\\_\\_\\_."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib... |
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 complex number $z$ is purely imaginary, and $\\frac{z+2}{1-i}+z$ is a real number, where $i$ is the imaginary unit.<br/>$(1)$ Find the complex n... |
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 ratio of the larger number to the smaller number if their sum is 14 and the smaller number is 3.5."
},
{
"role": "assistant",
"content": "`... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "A standard eight-sided fair die with faces numbered from 1 to 8 is rolled five times. The probability that the product of all five numbers rolled is a perfect cube is given as $\... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Let $M$ be the set of all integers from $1$ to $2013$. Each subset of $M$ is given one of $k$ available colors, with the only condition that if the union of two different subsets... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "in how many ways can an answer key for a quiz be written if the quiz contains 3 true - false questions followed by 3 multiple - choice questions with 4 answer choices each , if the correct answers to all true - false questions can not be the same ?"
},
{
"role": "assist... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "Mr. Smith has 8 children. Assuming that the gender of each child is determined independently and with equal likelihood of male and female, what is the probability th... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "A line passing through the point (0, 4) can be drawn such that it intersects the hyperbola $y^2 - 4x^2 = 16$ at exactly one point."
},
{
"role": "assis... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Find the number of real solutions of the equation\n\\[\n\\frac{x}{50} = \\sin x.\n\\]"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\n... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Cory has $4$ apples, $1$ orange, and $2$ bananas. If Cory eats one piece of his fruit per day for a week and the pieces of fruit within each category are indist... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Two identical smaller cubes are stacked next to a larger cube. Each of the two smaller cubes has a volume of 8. The combined height of the smaller cubes equals... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Find all positive integers $b$ for which there exists a positive integer $a$ with the following properties:\n\n- $a$ is not a divisor of $b$.\n\n- $a^a$ is a divisor of $b^b$"
... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Which of the following statements is correct?\n\nA: If $(\\frac{\... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "In the Cartesian coordinate system $Oxyz$, given the point $A(2, 1, -1)$, find the coordinates of the point $A_1$ which is symmetrical to point $A$ with respec... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "In the lab , Chang has two solutions that contain alcohol and is mixing them with each other . He uses 500 milliliters less of Solution A than Solution B.... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Simplify $\\left(\\sqrt[4]{81} - \\sqrt{8\\frac{1}{2}}\\right)^2$\n**$\\textbf{(A)}\\ \\frac{35}{2} - 3\\sqrt{34} \\qquad \\textbf{(B)}\\ 17 - 3\\sqrt{34} \\... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Which of the following radicals is the simplest quadratic radical?\n\nA: $\\sqrt{0.5}$\n\nB: $\\sqrt{\\frac{1}{7}}$\n\nC: $-\\sqrt{3}$\n\nD: $\\sqrt{8}$"
}... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "if 2 x + y = 7 and x + 2 y = 8 , then ( x + y ) / 3 ="
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeart... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Daria wants to go to a concert by her favorite band. She wants to buy tickets for her and for three of her friends. One ticket cost is $90. How much money does Daria need to earn if she currently has only $189?"
},
{
"role": "assistant",
"content": "```lean\nimport ... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Solve the fractional equation $\\frac{5}{x} = \\frac{7}{x-2}$, and its root is $\\_\\_\\_\\_\\_\\_.$"
},
{
"role": "assistant",
"content": "```lean\... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Bob is purchasing marbles, costing $20$ cents each. However, if he buys more than 10 marbles, the cost for each additional marble (beyond the 10th) drops to $15$ cents per marble... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "There are 7 natives seated in a circle: knights, liars, and bores (there is at least one representative of each tribe at the table). A knight always tells the truth, a liar always lies, and a bore lies if at least one knight is seated next to them; otherwise, the bore can say a... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Compute \\(\\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix}^{10}.\\)"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aeso... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Consider the cyclic quadrilateral $EFGH$ where the side lengths are distinct integers all less than $20$ such that $EG \\cdot GH = EF \\cdot FH$. What is ... |
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 ratio of $3x - 4$ to $y + 7$ is constant, and $y = 5$ when $x = 3$, then, when $y = 20$, $x$ equals:\nA) $5.1$\nB) $5.0833$\nC) $6$\nD) $7$"
... |
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 grasshopper, the frog and the mouse had a jumping contest. The grasshopper jumped 19 inches. The frog jumped 10 inches farther than the grasshopper and the... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Simplify\n\\[\\frac{\\tan^3 45^\\circ + \\cot^3 45^\\circ}{\\tan 45^\\circ + \\cot 45^\\circ}.\\]"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\ntheorem tan_cot... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Formulate a detailed proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "Willie starts with 36.0 stickers. Emily gives some stickers to him. Willie ends with 43 stickers. How many stickers did Emily give to Willie?"
},
{
"... |
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 integer $n,$ $0 \\le n \\le 180,$ such that $\\cos n^\\circ = \\cos 1534^\\circ.$"
},
{
"role": "assistant",
"content": "```lean\nimport Ma... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "The Pythagoras High School band has $90$ female and $70$ male members. The orchestra has $70$ female and $90$ male members, with $50$ females being members of b... |
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 the function $f(x) = 2\\cos\\left(\\frac{\\pi}{2} - \\omega x\\right) + 2\\sin\\left(\\frac{\\pi}{3} - \\omega x\\right)$ ($\\omega > 0$, $x \\in \\mat... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "Find the coordinates of the point in the complex plane corresponding to the complex number $z= \\frac {1-i}{2+i}$.\nA: (1, -3)\nB: $\\left(\\frac {1}{5}, ... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "If the complex number $z=(m^2-1)+(m+1)i$ is purely imaginary, then the value of the real number $m$ is $\\boxed{1}$."
},
{
"role": "assistant",
"co... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "The complex number $(m+2)-i$ is a pure imaginary number, then the real number $m=$ ."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimpo... |
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 triangle PQR below, $\\cos Q = \\frac{5}{13}$, where PQ = 13. What is QR?\n\n[asy]\npair P,Q,R;\nP = (0,12);\nQ = (0,0);\nR = (5,0);\ndraw(P--Q--R--cycle)... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The marked price of a book was 40% less than the suggested retail price. Alice purchased the book for 40% off the marked price during a special promotional event. What percent of the suggested retail price did Alice pay?\nA) 30%\nB) 34%\nC) 36%\nD) 40%\nE) 45%"
},
{
"ro... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Each point of a plane is painted in one of three colors. Show that there exists a triangle such that:\r\n$ (i)$ all three vertices of the triangle are of the same color;\r\n$ (ii... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "A bag contains 3 red, 2 white, and 1 black ball. If two balls are randomly selected from the bag, identify the pair of mutually exclusive but not contradictory events.\n\nA: At l... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Two numbers have a certain HCF and a product of 2560. The LCM of the two numbers is 128. What is the HCF of the two numbers?"
},
{
"role": "assistant",
"content": "``... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Using Lean, provide a formal proof for the stated mathematical proposition."
},
{
"role": "user",
"content": "A train starts from city A at a certain time and travels towards city B, which is 465 km away. The train travels at 60 km/hr. Another train starts from ci... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "What is the smallest positive integer that can be written in the form $5013m + 111111n$, where $m$ and $n$ are integers?"
},
{
"role": "assistant",
"content": "```lea... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Find all $\\alpha>0$ and $\\beta>0$ that for each $(x_1,\\dots,x_n)$ and $(y_1,\\dots,y_n)\\in\\mathbb {R^+}^n$ that:\\[(\\sum x_i^\\alpha)(\\sum y_i^\\beta)\\g... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Let \\( A \\) be a set with \\( n \\) elements. Suppose there are \\( m \\) subsets \\( A_{1}, A_{2}, \\cdots, A_{m} \\) of \\( A \\) such that no two subsets intersect. Prove th... |
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 rectangular coordinate system of space, the coordinates of the point that is symmetric to point $P(1, 3, -5)$ about the $xOy$ plane are ( ).\nA: $(-1... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "The shortest path from Atown to Cetown runs through Betown. Two of the signposts that can be seen on this path are shown, but one of them is broken and a number missing. \n\nWhat distance was written on the broken sign?\nA) $2 \\text{ km}$\nB) $3 \\text{ km}$\nC) $4 \\text{ km}... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Prove that:\n$$\n2^{4} \\times 5^{7} \\times 2003 \\mid (2^{286}-1)(3^{500}-1)(1978^{100}-1).\n$$"
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen BigOperators Real Nat Topology Rat\n\n\n /-\nProve th... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "user",
"content": "Let the sum of the first $n$ terms of a geometric sequence $\\{a_n\\}$ be $S_n$. If $a_1 + 8a_4 = 0$, then $\\frac{S_3}{S_4} =$ \nA: $\\frac{6}{5}$\nB: $\\frac{14}{15}$\nC: $\\frac{7}{15}$\nD: $-\\frac{3}{5}$"
},
{
"role": "assistant",
"content": "```lean\nimport Ma... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "If 9 engines consume 24 metric tonnes of coal when each is working 8 hours a day, and 3 engines of the former type consume as much as a certain number of engines of ... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Let $\\mathbf{A} = \\begin{pmatrix} a & 2 \\\\ -3 & d \\end{pmatrix}$ and $\\mathbf{B} = \\begin{pmatrix} 2a & 1 \\\\ -1 & d \\end{pmatrix}$ where $a$ and $... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Develop a rigorous proof in Lean for the stated mathematical proposition."
},
{
"role": "user",
"content": "Three boys and two girls are to stand in a row according to the following requirements. How many different arrangements are there? (Answer with numbers) \n... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "In triangle \\(ABC\\), with sides opposite angles \\(A\\), \\(B\\), and \\(C\\) being \\(a\\), \\(b\\), and \\(c\\) respectively, it is given that \\((2a+b)\\si... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Demonstrate the proof of the given mathematical assertion using Lean."
},
{
"role": "user",
"content": "Prove that \\((x + a)(x + 2a)(x + 3a)(x + 4a) + a^4\\) is a perfect square."
},
{
"role": "assistant",
"content": "```lean\nimport Mathlib\nimport A... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "Given that $I$ is the incenter of $\\triangle ABC$, and $P$ is a point inside $\\triangle ABC$ such that\n$$\n\\angle PBA + \\angle PCA = \\angle PBC + \\angle PCB,\n$$\n\nprove ... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Write the proof of the following mathematical statement in Lean."
},
{
"role": "user",
"content": "A cobbler can mend 3 pairs of shoes in an hour. From Monday to Thursday, the cobbler works for some hours each day, and on Friday, he only works from 8am to 11am. Th... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Construct a formal proof in Lean for the following mathematical claim."
},
{
"role": "user",
"content": "There are seven unmarked envelopes on a table, each with a letter for a different person. If the mail is randomly distributed among these seven people, with ea... |
Goedel-LM/Goedel-Pset-v1/train | code | [
{
"role": "system",
"content": "Use Lean to formally prove the following statement."
},
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_n+a_{n+1}+a_{n+2}=(\\sqrt{2})^{n}$. Find the sum of the first $9$ terms of the sequence $\\{a_n\\}$ (express the answer as a numerical value)."
... |
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