Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Area of Trapezoid: The text provides a theorem and proof related to calculating the area of a trapezoid using its geometric properties, specifically, the lengths of its parallel sides and its height. This is directly relevant to the Trapezoidal Rule used in the solution for approximating the area under the curve, as the Trapezoidal Rule essentially involves summing the areas of trapezoids to approximate the area under a curve. The solution to the problem follows the steps outlined in the theorem definition for applying the Trapezoidal Rule, which involves calculating the width of each subinterval, evaluating the function at each endpoint of the subintervals, and then using a formula similar to that in the text (involving the sum of the parallel sides and the height of the trapezoids) to find the approximate area. Therefore, despite the text discussing the area of a geometric trapezoid and the solution applying this concept to a numerical integration problem, the underlying reasoning and mathematical principles are very similar. This indicates that the text could help a student understand the concept of the Trapezoidal Rule by analogy, thereby aiding in solving the given problem.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're a landscape architect planning a new park and need to calculate the amount of soil needed to create a specific land feature. The feature's cross-section can be described by the equation y=2^{x}, and you're looking at the section from x=-1 to x=3. To estimate the volume of soil needed, you decide to break this section into 4 equal parts and use a method that approximates the area by connecting the dots with straight lines to form trapezoids. How much area would you estimate is under the curve for this section?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Sample Matrix Independence Test: The given text introduces a method for testing the linear independence of functions within a vector space by using a \"sample matrix.\" The text describes constructing this sample matrix from the values of functions at specific points and then determining the linear independence of these functions based on the invertibility of this matrix. While the problem statement and the solution focus on linear independence in the context of vectors in \\(\\mathbb{R}^3\\) (a more concrete setting), the text abstracts the concept to a broader context involving functions within any vector space. Despite these differences in context, the underlying principle is the same: checking if only the trivial solution exists to a particular equation (whether it's \\(\\sum c_i v_i = 0\\) or \\(\\sum c_i f_i(x) = 0\\) for all \\(x\\)) to determine linear independence. \n\nBoth the solution's approach to the problem and the text's theorem apply the concept of linear independence by examining if a system of equations only has the trivial solution. In the problem's case, it's through the coefficients of vectors equaling zero; in the text's case, it's through the coefficients of functions leading to a zero vector after applying them in a matrix equation. Thus, the reasoning process\u2014determining linear independence by checking for only trivial solutions to a certain equation\u2014is very similar between the text and the solution provided for the vectors.\n\nTherefore, even though the text discusses functions in a vector space and the problem is about vectors in \\(\\mathbb{R}^3\\), the text can help a student unfamiliar with the theorem understand and solve the problem by showing a similar application of the linear independence concept.\n\nAnswer: True\n\nReasoning for Schauder Basis is Linearly Independent: The provided text discusses the concept of linear independence in the context of a Schauder basis in a normed vector space. It explains that if a set of elements (in this case, the elements of a Schauder basis) in a vector space can only sum to the zero vector when all coefficients in their linear combination are zero, then the set is linearly independent. This aligns with the theorem definition provided, which states that a set of vectors is linearly independent if the only solution to their linear combination equating to the zero vector is when all coefficients are zero. \n\nThe problem statement asks whether a given set of vectors are linearly independent. The solution involves setting up an equation for a linear combination of the given vectors to equal the zero vector and solving for the coefficients, which is conceptually parallel to the explanation in the text regarding the Schauder basis. The solution process directly applies the definition of linear independence, examining if the only solution for the coefficients is the trivial one (all coefficients being zero), which mirrors the reasoning provided in the text about how to establish linear independence.\n\nGiven that the text explains a process for determining linear independence that is directly applicable to solving the problem (even though the context of Schauder bases might be more specific), it can indeed be helpful to a student trying to understand and solve the problem regarding the linear independence of a set of vectors. The reasoning steps and application of the theorem in the text are very similar to those needed to solve the given problem.\n\nAnswer: True\n\nReasoning for Homogeneous Linear Equations with More Unknowns than Equations: The problem statement asks whether the vectors v_1, v_2, and v_3 are linearly independent. The solution employs the concept of linear independence, directly checking if the vectors can be expressed as a linear combination of each other that equals the zero vector, with the only solution being the trivial one where all coefficients are zero.\n\nThe provided text delves into a theorem regarding homogeneous linear equations with more unknowns than equations, stating that if the number of unknowns (n) exceeds the number of equations (m), there exists at least one non-trivial solution (not all zeros). This directly relates to the concept of linear independence by explaining that if you have more vectors than dimensions, at least one vector can be expressed as a linear combination of the others, meaning the vectors are linearly dependent.\n\nThe text is helpful in understanding the problem and arriving at a solution because it provides a theoretical foundation for why, in a system with more vectors than the dimension of the vector space they occupy, those vectors cannot all be linearly independent. This reasoning is closely aligned with the solution's application of the concept of linear independence to solve the problem. Thus, the text could assist a student in understanding why the given vectors v_1, v_2, and v_3, which span a 3-dimensional space, cannot all be linearly independent if treated as a system with more vectors than dimensions (since the vectors are part of a 3-dimensional space but follow a pattern that suggests they are not all independent).\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're a graphic designer working on a 3D modeling project. You have three arrows: the first points 1 unit right, 2 units up, and 3 units forward; the second points 4 units right, 5 units up, and 6 units forward; and the third points 7 units right, 8 units up, and 9 units forward. You can combine arrows by adding any multiple of their right, up, and forward displacements (linear combinations). Can you create a new arrow using a combination of these three without directly tracing any of them?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Convergence of P-Series/Real: The text discusses the convergence of p-series and states that a p-series $\\sum_{n=1}^{\\infty} n^{-p}$ is convergent if $p > 1$. This information relates to the general criteria for series convergence, which is relevant to the problem of determining if the given series $\\sum_{i=0}^{\\infty} \\frac{n-1}{n^3+1}$ converges. However, the solution to the problem employs the limit comparison test, directly comparing the given series to the series $\\sum_{n=1}^{\\infty} \\frac{1}{n^2}$, which is a special case of a p-series where $p=2$. The text provides foundational knowledge on the convergence of p-series, which indirectly supports the reasoning used in the solution by affirming the convergence of the series $\\sum_{n=1}^{\\infty} \\frac{1}{n^2}$ used for comparison. This foundational knowledge could help a student understand why the comparison series converges, thus aiding in grasping the steps taken in the solution using the limit comparison test. Therefore, although the text does not directly address the limit comparison test, it provides useful background information on the convergence of p-series, which is closely related to the reasoning applied in the solution.\n\nAnswer: True\n\nReasoning for Limit Comparison Test: The provided text describes the Limit Comparison Test, which is exactly the method used in the solution to determine the convergence of the given series. The Limit Comparison Test, as explained in the text, is applied when the limit of the ratio of the terms of two series as n approaches infinity is a finite positive number. This determines that both series either converge or diverge together. The solution to the problem applies this theorem by comparing the given series with a known convergent series and finding the limit of the ratio of their terms as n approaches infinity to be a finite positive number (1, in this case). Therefore, the text explains the theorem and its application in a manner very similar to how the theorem is used in the problem solution, making it helpful for a student not familiar with the theorem to understand and solve the given problem.\n\nAnswer: True\n\nReasoning for Series/Complex: The provided text introduces the concept of a complex series and its definition, focusing on the notion of convergence through the limit of the sequence of partial sums as the number of terms approaches infinity. This is directly related to the theorem definition provided, which discusses series convergence and specifies that a series converges if the sum of its terms approaches a finite value as the number of terms goes to infinity. Although the text specifically mentions complex series, the underlying principle of series convergence \u2014 the limit of the sequence of partial sums \u2014 is applicable to the problem at hand, which involves determining the convergence of a series using the limit comparison test. The solution to the problem utilizes this principle by comparing the given series to a known convergent series and examining the limit of the ratio of their terms. Thus, the text can be helpful for a student not familiar with the theorem in understanding the concept of series convergence, which is crucial for applying the limit comparison test as done in the solution. Therefore, the text uses very similar reasoning steps as the solution and applies the theorem in a related way to solve the problem.\n\nAnswer: True\n\nReasoning for Series/Real: The given text provides a definition of what a real series is and explains how it is represented as the limit of the sequence of partial sums of a real sequence as the number of terms approaches infinity. This fundamental understanding of what constitutes a real series aligns closely with the theorem definition provided, which is about series convergence and how a series is said to converge if its sum approaches a finite value as the number of terms goes to infinity. While the text does not explicitly discuss the limit comparison test or the specific process used in the solution to determine the convergence of the series in question, it lays the groundwork for understanding the basic concept of series convergence, which is central to the solution's application of the limit comparison test. \n\nTherefore, the text can be considered helpful for a student who is not familiar with the concept of series convergence, as it provides necessary background information that could assist in understanding the solution's approach, even though it does not directly address the limit comparison test or the specific series mentioned in the problem statement. By understanding the definition and concept of a real series and its convergence, a student would be better prepared to grasp how and why the limit comparison test is applied in the solution.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're adding a sequence of fractions where each fraction represents a piece of a digital image that becomes clearer as more pieces are added. Starting with the first piece, each subsequent piece is slightly smaller than the one before it, following the pattern where the numerator is one less than the iteration number, and the denominator is the iteration number cubed plus one, starting with iteration number 0. (for example, the first fraction is 0/2, the second is 1/9, and so on). Will there come a point when adding more pieces doesn't significantly change the image, or can you keep adding pieces forever and continue to see a change?"}


Imagine you're programming a video game that simulates the flight of a spaceship in outer space. The spaceship's movement is affected by various forces, and you need to predict its position at a specific time to ensure it doesn't collide with obstacles. You know the spaceship's initial position is 2 units from the starting point and its initial speed is 3 units per second. The forces acting on the spaceship cause its acceleration and speed to change in a way that's a bit complex: the acceleration at any moment depends on the spaceship's current position, speed, and the time elapsed. Specifically, y''+xy'+y=0. You decide to use a simplified method — Euler's method — to estimate the spaceship's position after 0.2 seconds. What is the estimated position of the spaceship?

Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Differential Equation/Ordinary: The given text provides a formal definition of an ordinary differential equation (ODE), explaining that an ODE has exactly one independent variable and all derivatives in it are ordinary derivatives. This definition aligns with the concept necessary for understanding the problem statement and the solution, as the problem involves solving a second-order differential equation (specifically, an ODE according to the text's definition) by converting it into a system of first-order differential equations. Although the text does not explicitly address the solution method (Euler's Method) or the process of converting a second-order ODE into a system of first-order ODEs, it lays the foundational understanding of what an ODE is. This foundational knowledge is crucial for someone not familiar with the theorem (in this case, the nature of ODEs) as it helps in understanding the problem's context and the rationale behind converting the second-order ODE into first-order systems as a step towards applying Euler's Method. Therefore, while the text does not directly contribute to solving the problem, it assists in building the necessary conceptual background for understanding the solution's approach.\n\nAnswer: True\n\nReasoning for Reduction of Explicit ODE to First Order System: The given problem involves using Euler's Method to approximate the value of y(0.2) for a second-order differential equation, y''+xy'+y=0, with given initial conditions. The solution outlines converting this second-order ODE into a system of first-order ODEs, a critical step before applying Euler's Method for approximation. This conversion is essential because Euler's Method directly applies to first-order equations.\n\nThe text discusses a theorem on reducing an explicit ODE to a first-order system. It states that for an ODE of any order, there exists a system of first-order ODEs such that the solution to this system is equivalent to the solution of the original ODE. This is exactly what is done in the solution: the second-order ODE is converted into a system of first-order ODEs before applying Euler's Method. \n\nTherefore, the theorem discussed in the text underpins the fundamental step in the solution\u2014transforming the higher-order ODE into a first-order system, which enables the use of Euler's Method for approximation. This makes the text relevant and helpful for understanding and solving the given problem, especially for a student not familiar with the process of converting higher-order ODEs to first-order systems before applying numerical methods like Euler's Method.\n\nAnswer: True\n\nReasoning for Solution of Second Order Differential Equation with Missing Dependent Variable: The problem requires the use of Euler's Method to calculate an approximation of y(0.2) for a given second-order differential equation. The solution outlines a process where the second-order differential equation is converted into a system of first-order differential equations, which is then solved using Euler's Method. \n\nThe theorem in the text discusses a method for reducing a second-order ordinary differential equation (ODE) with the missing dependent variable into a first-order ODE, which can then be solved. This process is conceptually similar to the solution's approach of converting the original second-order ODE into a system of first-order ODEs for the purpose of applying Euler's Method. \n\nBoth the solution and the text rely on transforming a higher-order ODE into a form (or forms) that can be more easily managed or solved, albeit with different specific purposes (application of Euler's Method vs. general solution strategy for second-order ODEs missing the dependent variable). \n\nGiven that the text offers insight into handling second-order differential equations by reducing them to first-order equations, it aligns with the reasoning needed for the solution's approach to applying Euler's Method. Therefore, the text can be considered helpful for a student trying to understand how to convert a second-order ODE into a system of first-order ODEs, which is a crucial step in the solution provided.\n\nAnswer: True\n\nReasoning for Solution of Second Order Differential Equation with Missing Independent Variable: The text describes a method for transforming a second-order ordinary differential equation (ODE) with a missing independent variable (x) into a first-order ODE, which can then be solved more easily. It introduces a new dependent variable \\(p = y'\\), which allows the second-order ODE to be expressed as a first-order ODE. This approach is relevant to the given problem, which involves solving a second-order ODE by reducing it to a system of first-order differential equations. The solution to the problem employs a similar strategy by defining two new functions \\(u(x) = y(x)\\) and \\(v(x) = y'(x)\\), thus transforming the original second-order ODE into a system of first-order ODEs that can be solved using Euler's method.\n\nThe problem solution and the text both utilize the principle of reducing the complexity of a second-order ODE by transforming it into first-order ODEs, albeit the solution explicitly applies Euler's method afterwards for numerical approximation. While the text does not directly mention Euler's method, the reasoning steps concerning the transformation of a second-order ODE into first-order ODEs align closely with the initial steps of the solution. This conceptual understanding provided by the text can help a student grasp the foundational step required for applying Euler's method to the problem, making the text potentially helpful for understanding how to approach solving the problem, even if it doesn't cover the numerical method part.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're programming a video game that simulates the flight of a spaceship in outer space. The spaceship's movement is affected by various forces, and you need to predict its position at a specific time to ensure it doesn't collide with obstacles. You know the spaceship's initial position is 2 units from the starting point and its initial speed is 3 units per second. The forces acting on the spaceship cause its acceleration and speed to change in a way that's a bit complex: the acceleration at any moment depends on the spaceship's current position, speed, and the time elapsed. Specifically, y''+xy'+y=0. You decide to use a simplified method \u2014 Euler's method \u2014 to estimate the spaceship's position after 0.2 seconds. What is the estimated position of the spaceship?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Lebesgue Measure is Diffuse: The given text focuses on demonstrating that a Lebesgue measure on \\(\\mathbb{R}^n\\) is a diffuse measure, which essentially means that the measure of any singleton set in \\(\\mathbb{R}^n\\) is zero. This property is aligned with the general principle of Lebesgue measure dealing with the size or measure of sets, particularly emphasizing that individual points (or infinitesimally small sets) do not contribute to the measure in a significant way (in this case, their contribution is zero).\n\nThe problem statement, however, is about determining the Lebesgue measure of a specific set \\(A\\), which is constructed in a manner similar to a Cantor set but with a variation in the removal process (removing the middle quarter instead of the middle third). The solution applies the concept of Lebesgue measure to calculate the measure of the set \\(A\\), concluding it to be zero by considering the limit of the measure of the sets in the construction process.\n\nWhile the text does not directly address the specific method for calculating the measure of a Cantor-type set, it reinforces the foundational understanding of the Lebesgue measure's behavior regarding points and, by extension, highly fragmented or \"fractal-like\" sets. Understanding that the measure of single points is zero can help in grasping why more complex sets like \\(A\\), which are constructed through an infinite process of removals leaving a \"dust\" of points, might also have a measure of zero.\n\nTherefore, the text can assist a student in understanding the foundational properties of Lebesgue measure that underlie the solution to the problem, even though it does not directly solve the problem or use the exact same reasoning steps. It indirectly supports the understanding required to grasp why the Lebesgue measure of certain sets is zero, which is central to the problem's solution.\n\nAnswer: True\n\nReasoning for Cantor Set has Zero Lebesgue Measure: The problem statement asks for the Lebesgue measure of a set \\(A\\) defined in a specific way that resembles the construction of a Cantor set but with intervals removed by a different fraction (removing the middle quarter instead of the middle third). The solution involves recognizing \\(A\\) as a Cantor-type set and applying the formula for the measure of a Cantor set to find its Lebesgue measure, concluding that it is 0.\n\nThe given text describes a theorem about the Cantor set having a Lebesgue measure of 0. It explains that the Cantor set is measurable and that its measure is zero, following through a proof that involves measurable sets, sigma-algebras, and limits. The reasoning in the text closely parallels the solution's approach by utilizing the concept of measure and the limit process to establish that a set has a measure of zero.\n\nAlthough the text specifically mentions the Cantor set (which traditionally involves removing the middle third of intervals), the underlying principles and steps\u2014such as considering the set as a limit of a decreasing sequence of measurable sets and using properties of sigma-algebras and limits of measures\u2014apply directly to the problem at hand. The text illustrates how to prove a set has a Lebesgue measure of zero using a limit process, which is precisely the reasoning needed for solving the given problem, even though the specific fraction removed in the problem is different.\n\nThus, the text can be helpful to a student not familiar with the theorem by showing a related way of applying the theorem (concerning Lebesgue measure and properties of certain types of sets) to solve the problem. It uses very similar reasoning steps as the solution, focusing on the measure of decreasing sequences of sets and the application of limits to determine the Lebesgue measure.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you have a digital scale that can measure the weight of an infinite number of infinitely small digital dots. Each dot can either be on or off, and their weights are determined by a specific pattern: the first dot weighs 1/4 gram, the second dot weighs 1/16 grams, the third dot weighs 1/64 grams, and so on, with each subsequent dot weighing exactly one-fourth the weight of the previous dot. If you can create any combination of these dots being on (weighing their specified amount) or off (weighing nothing), and you can also add a whole number of grams to the scale (from a set of weights that are whole grams), what would be the total possible weight measured by the scale? (specifically, Lebesgue measure of this sum)"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Test Statistic: The given text introduces the concept of a test statistic, which is a foundational element in performing any statistical hypothesis test, including the t-test mentioned in the theorem and applied in the solution. The text explains that a test statistic (T) is derived from a sample and used to decide whether to reject the null hypothesis, which aligns with how the t-value is calculated and used in the provided solution to determine if the experimental yields significantly differ from the expected yield. Understanding what a test statistic is and how it functions in hypothesis testing can help a student grasp why and how the t-value is calculated and used in the context of a one-sample t-test, as seen in the solution. However, the text does not directly address the t-test or its specific application to comparing means, which is the core of the problem's solution. Nevertheless, because it covers the foundational concept of test statistics in hypothesis testing, which is directly relevant to understanding and applying the t-test, the text can be helpful to a student who is not familiar with the theorem in ultimately solving the problem.\n\nAnswer: True\n\nReasoning for Hypothesis Test: The text provided outlines a general definition of a hypothesis test, which is a fundamental concept used in performing a t-test. The solution to the problem involves conducting a one-sample t-test, where the null hypothesis (H0) and the alternative hypothesis (H1) are clearly stated, and the decision to reject or not reject H0 is based on the calculated t-value and the critical t-value. The text, by explaining what a hypothesis test is, including the decision-making process regarding accepting or rejecting H0, offers foundational knowledge that is directly applicable to understanding and performing the one-sample t-test outlined in the solution. This foundational knowledge is crucial for a student who is not familiar with the theorem (t-test in this case) as it helps in understanding the basic principles behind hypothesis testing, which is a core component of the t-test. Therefore, even though the text does not explicitly mention the t-test or the specific calculations involved, it provides necessary background information that supports the understanding of the steps taken in the solution.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nA new type of bean was grown in 10 different fields, and its productivity was measured to see if it meets the expected yield of 12.0 quintals per hectare. The yields from these fields were as follows: 14.3, 12.6, 13.7, 10.9, 13.7, 12.0, 11.4, 12.0, 12.6, 13.1 quintals per hectare. Given that we're using a 5% significance level, does the actual yield support the expected yield?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Test Statistic: The text provides a formal definition of a test statistic, which is a crucial component in the broader process of conducting a t-test, as described in the theorem definition. It explains that a test statistic is calculated from a sample to decide whether to reject a null hypothesis. While the text does not explicitly mention the t-test, it covers foundational knowledge necessary for understanding how a t-test operates, specifically how the t-statistic is used as a test statistic to make decisions about hypotheses. The solution to the problem involves calculating a t-statistic and making a decision based on the p-value, which aligns with the concept of using a test statistic to decide on the null hypothesis as described in the text. Therefore, the text can be helpful to a student by providing them with an understanding of the role a test statistic plays in hypothesis testing, which is a core aspect of performing a t-test, the method used in the solution to address the problem statement.\n\nAnswer: True\n\nReasoning for Hypothesis Test: The given text introduces the concept of a hypothesis test, which is foundational to understanding and performing a t-test, the theorem used in the solution to solve the problem. The text explains what a hypothesis test is, including the roles of the null hypothesis (\\(H_0\\)) and the alternative hypothesis (\\(H_1\\)), and how decisions are made to accept or reject \\(H_0\\). Although the text does not directly mention the t-test, understanding the framework of hypothesis testing is essential for someone learning how to apply the t-test, as the t-test is a type of hypothesis test. The solution to the problem involves performing a two-sample t-test and making a decision based on the p-value, which directly involves accepting or rejecting the null hypothesis. Therefore, the text can be helpful for a student who is not familiar with the theorem (t-test) in ultimately solving the problem because it lays the groundwork for understanding how decisions are made in hypothesis testing, which is precisely what is done in a t-test to solve the given problem.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nIn a clinical trial, two groups of patients are being compared. The first group, consisting of 5 patients, received Treatment A and had weights of 42, 39, 38, 60, and 41 kilograms. The second group, with 7 patients, received Treatment B and had weights of 38, 42, 56, 64, 68, 69, and 62 kilograms. Researchers want to know if the weight differences between the two groups of patients indicate that one treatment is more effective than the other, with a confidence level of 95%. Is there a significant weight difference between the two treatment groups?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Cayley's Formula: The provided text describes Cayley's formula and offers a proof based on the bijection between Pr\u00fcfer sequences and labeled trees, explaining that the number of distinct labeled trees with $n$ nodes is $n^{n-2}$. This is precisely the theorem used in the solution to calculate the number of labeled trees on 6 vertices. The text directly supports understanding Cayley's formula by offering a proof and explaining the relationship between Pr\u00fcfer sequences and labeled trees, which underpins the formula's application. Thus, for a student not familiar with the theorem, this text would indeed help in understanding the basis of the formula used in the solution. It aligns closely with the solution's application of Cayley's formula, using the same principle to arrive at the number of labeled trees on 6 vertices.\n\nAnswer: True\n\nReasoning for Bijection between Pr\u00fcfer Sequences and Labeled Trees: The given text discusses the bijection between Pr\u00fcfer sequences and labeled trees, establishing a one-to-one correspondence between them. This is directly related to understanding Cayley's formula, as the theorem itself can be derived from the concept of Pr\u00fcfer sequences, among other methods. The solution to the problem utilizes Cayley's formula to determine the number of labeled trees for a given number of vertices. By understanding the relationship between Pr\u00fcfer sequences and labeled trees, as explained in the text, one can gain a deeper understanding of why Cayley's formula works and how it applies to solving problems involving the enumeration of labeled trees. This understanding can be particularly helpful for a student who is not familiar with the theorem because it provides a concrete method (Pr\u00fcfer sequences) through which the formula's result is achieved, thus making the abstract formula more tangible and understandable. Therefore, even though the text does not explicitly use Cayley's formula to solve the given problem, it offers relevant foundational knowledge that supports the application of the theorem in solving such problems.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're organizing a network of 6 computers in an office. Each computer needs to be connected in such a way that there is exactly one path between any two computers to ensure efficient data transfer without any loops. How many different ways can you set up this network?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Real Symmetric Positive Definite Matrix has Positive Eigenvalues: The provided text elaborates on the theorem that a symmetric positive definite matrix over the real numbers has all positive eigenvalues. This aligns closely with one of the criteria mentioned in the definition of a positive definite matrix, specifically the criterion that all its eigenvalues must be positive. In the context of the problem statement, which asks whether the given matrix is a positive definite matrix, understanding the eigenvalue property of positive definite matrices is crucial. Since the solution to the problem likely involves examining the eigenvalues of the given matrix, the text would be helpful to a student. It directly addresses a key aspect of determining the positive definiteness of a matrix, which is what the problem statement is about. Therefore, the text is helpful in understanding and solving the problem because it uses very similar reasoning steps as would be needed in the solution and applies the theorem (regarding eigenvalues of positive definite matrices) in a related way as the solution would apply the theorem to solve the problem.\n\nAnswer: True\n\nReasoning for Minor of Determinant: The provided text introduces and explains the concept of a minor of a determinant, which is a determinant formed from a square submatrix obtained by deleting certain rows and columns from the original matrix. This concept is related to one of the criteria for determining if a matrix is positive definite, specifically the criterion that involves checking whether all leading principal minors are positive. Understanding the concept of a minor is fundamental in applying the theorem's criterion that involves leading principal minors. Although the text does not directly address the entire theorem or how to apply it to determine if a matrix is positive definite, it provides essential foundational knowledge that is relevant to one of the criteria mentioned in the theorem definition for a positive definite matrix. Therefore, the text can be considered helpful for a student who is trying to understand and apply the theorem to determine if a given matrix is positive definite, as it elucidates a concept that is directly related to one of the criteria used in the theorem's application.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're evaluating the safety of a building's structure through a simplified model. The building is represented by a 2x2 matrix with the values [1, 4] in the first row and [4, 1] in the second row. Safety is defined as the matrix being positive definite. Based on these definitions, is the building safe?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Gravity/Gravitational Force: The problem statement involves finding the acceleration due to gravity on Mars. The solution uses the formula for gravitational acceleration, which is derived from the concept of gravitational force as described by Newton's law of universal gravitation. This law states that the gravitational force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The solution directly applies this principle by plugging in the mass of Mars and its radius to find the gravitational acceleration on its surface.\n\nThe text provided offers a definition of gravitational force, emphasizing that it is the force exerted on a body due to the gravitational field it is under. While this definition aligns with the concept of gravitational force used in the solution, it does not delve into the specifics of how gravitational force is calculated or how it relates to gravitational acceleration (i.e., it doesn't mention Newton's law of universal gravitation or the formula \\(g = \\frac{G M}{r^2}\\) used in the solution).\n\nHowever, the text could still be considered helpful for a student not familiar with the theorem because it establishes the foundational idea that gravitational force exists due to a gravitational field, which is a necessary concept to grasp before understanding how gravitational acceleration is calculated. The theorem definition provided in the input gives more detail on how gravitational force is quantitatively determined, which complements the basic definition given in the text. Therefore, the text could assist in building a basic understanding but lacks the detailed application seen in the solution. Despite this, in the context of providing a foundational understanding to ultimately assist with the problem, the text is helpful.\n\nAnswer: True\n\nReasoning for Local Gravitational Constant: The provided text explains the concept of the local gravitational constant, which is essentially the acceleration due to gravity at a specific location caused by the gravitational field of a significant mass (e.g., Earth). This concept is directly relevant to the problem statement, which asks for the calculation of acceleration due to gravity on Mars. The solution to the problem involves applying Newton's law of universal gravitation to find the acceleration due to gravity on Mars, which aligns with understanding the concept of local gravitational constant as described in the text. The text and the solution both focus on the idea of gravitational acceleration at a specific location, albeit the text does it in a general manner while the solution applies it to Mars specifically. Therefore, the text could assist a student not familiar with the theorem (or concept in this context) by providing a foundational understanding of gravitational acceleration, which is crucial for solving the problem.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're part of a team sending a robot to explore Mars. The robot weighs 3430 Newtons here on Earth. Given that Mars has a radius of 3.40 million meters and a mass of 6.42 times 10 to the power of 23 kilograms, can you calculate how much the gravity on Mars will accelerate the robot? Provide your answer in meters per second squared."}


Imagine you're recording a song using digital equipment. The highest pitch in the song produces a sound wave that vibrates at a frequency of 1000 cycles per second (Hertz). The equation is given by  $$ x(t)=\cos (1000 \pi t) $$. If you set your recording equipment to capture 2000 snapshots (samples) of the sound wave every second, does aliasing occur?

Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Simple Harmonic Motion/Frequency: The provided text discusses the concept of simple harmonic motion and how to determine the frequency of a system undergoing such motion. It defines the frequency ($\\nu$) as the number of complete cycles per unit time and relates it to the angular frequency ($\\omega$) using the formula $\\nu = \\frac{\\omega}{2\\pi}$. This is relevant to the problem statement and solution because the solution involves finding the frequency of the signal $x(t)=\\cos (1000 \\pi t)$, which is a form of simple harmonic motion. The solution calculates the signal frequency by interpreting the coefficient of $t$ inside the cosine function, which corresponds to $\\omega$ in the text's formula, to find that the signal frequency is $500$ Hz. Understanding how frequency is determined from the angular frequency ($\\omega$) is crucial for applying the Nyquist-Shannon sampling theorem correctly, as the theorem's application requires knowing the highest frequency component of the signal. Thus, while the text does not directly mention the Nyquist-Shannon sampling theorem, it provides foundational knowledge on determining frequency from a given function of time, which is a critical step in applying the theorem to answer whether aliasing occurs in the given scenario.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're recording a song using digital equipment. The highest pitch in the song produces a sound wave that vibrates at a frequency of 1000 cycles per second (Hertz). The equation is given by  $$ x(t)=\\cos (1000 \\pi t) $$. If you set your recording equipment to capture 2000 snapshots (samples) of the sound wave every second, does aliasing occur?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Simultaneous Equations/Solution Set: The provided text gives a formal definition of simultaneous equations and their solution set, explaining that solving such a system involves finding all elements (or ordered n-tuples) that satisfy each equation in the system. This aligns closely with the theorem definition provided, which discusses linear systems (or systems of linear equations) and their properties, including solving for variables that satisfy all equations simultaneously. While the text does not directly outline steps to solve a specific system of linear equations, it lays a foundational understanding that parallels the solution's approach. The solution to the problem involves manipulating and solving a system of linear equations to find the values of variables that satisfy both equations, which is a direct application of understanding what constitutes a solution set for a system of equations as described in the text. Therefore, the text can be considered helpful for a student trying to understand the solution to the problem, especially in terms of grasping the concept of a solution set for a system of linear equations.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nIn a game of strategy, two players are deciding on the amount of money to send to the other player. The first rule is that 15 times Player A's monetary amount must be equal to the difference between seven times Player A's amount and Player B's amount. The second rule is that 15 times Player B's amount is equal to player A's amount times negative six plus eight times Player B's amount. What is the amount of money that Player A should be sending to Player B?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Ramsey's Theorem: The provided text elaborates on Ramsey's theorem, specifically explaining the concept of finding monochromatic complete subgraphs within a sufficiently large complete graph, with a focus on a 2-color (red and blue) scenario. It mentions the existence of a Ramsey number \\( R(r, s) \\) for any pair of positive integers \\( r \\) and \\( s \\), which guarantees that for any complete graph on \\( R(r, s) \\) vertices with edges colored red or blue, there will be either a complete subgraph of \\( r \\) vertices all connected with red edges or a complete subgraph of \\( s \\) vertices all connected with blue edges. This explanation directly relates to the problem statement, which is essentially about finding a specific Ramsey number, \\( R(3, 3) \\), that ensures the existence of either a triad of mutual friends (complete subgraph of red edges) or mutual strangers (complete subgraph of blue edges) within a group of people (vertices of the graph). Therefore, the text provides foundational knowledge of Ramsey's theorem necessary to understand and solve the given problem, including the concept of Ramsey numbers and how they apply to scenarios involving relationships (such as friendship) modeled as graph edges.\n\nAnswer: True\n\nReasoning for Ramsey Number: The given problem statement asks for the minimum number of people needed in a room to guarantee that there are 3 mutual friends or 3 mutual strangers, which relates to finding a specific Ramsey number, specifically R(3,3). The solution directly mentions Ramsey's theorem (also known as the Friendship Theorem in this context) and provides the answer as 6, leveraging the theorem's implications.\n\nThe theorem definition provided explains Ramsey's theorem in general terms, focusing on its application in graph theory and combinatorial mathematics. It discusses the existence of either a complete subgraph (clique) of size m or an independent set of size n within any sufficiently large graph, based on the theorem.\n\nThe text to evaluate introduces the definition of a Ramsey number in the context of Ramsey's theorem, describing it as a scenario where, given any number of colors and integers representing the sizes of complete subgraphs for each color, there exists a minimum graph size (the Ramsey number) that guarantees finding a complete subgraph of one of those sizes, with all edges colored the same. This directly parallels the application of Ramsey's theorem in solving the given problem by identifying it as a quest for a specific Ramsey number (in this case, R(3,3)) to solve a real-world scenario involving mutual friends or strangers.\n\nGiven that the text provides a clear explanation of Ramsey numbers in a manner directly applicable to the problem's solution\u2014identifying the minimum number of people required to guarantee a certain social configuration\u2014it can indeed be helpful. It uses similar reasoning steps by discussing the existence of a certain structure (in this case, a group of mutual friends or strangers) within a large enough set, which aligns with how the solution applies Ramsey's theorem to determine the minimum number of people needed.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nEveryone that you invite to a party will be either a fan of football or a fan of basketball, but never both. What is the smallest number of guests you need to invite to ensure that there are either 3 people who are all fans of football or 3 people who are all fans of basketball?"}


Imagine you have a magical box that transforms any object you put inside it, where the object is represented by the column vector x = (x_1, x_2). The box's transformation can be represented by the matrix A = [[5, 4], [1, 2]], so when given an object x, the box outputs the new object Ax. On some special objects, this new object is just a constant multiple of the original object, λx = (λx_1, λx_2). Find both possible values of λ where this occurs — note that these are the box's eigenvalues.

Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Eigenvector/Linear Operator: The provided text gives a formal definition of an eigenvector and its relationship to an eigenvalue, specifically detailing that an eigenvector corresponding to an eigenvalue \\(\\lambda\\) is a nonzero vector \\(v\\) in the vector space that, when acted upon by the linear operator \\(A - \\lambda I\\), results in the zero vector. This definition inherently relates to the concept of finding eigenvalues by setting the determinant of \\(A - \\lambda I\\) to zero, as mentioned in the theorem definition and utilized in the solution to solve for the eigenvalues of the given matrix \\(A\\). The text supports understanding the underlying principle that for a value \\(\\lambda\\) to be an eigenvalue, there must exist a non-zero vector \\(v\\) (eigenvector) such that \\(A v = \\lambda v\\), which is conceptually tied to the process described in the solution of finding \\(\\lambda\\) by solving the characteristic equation derived from \\(det(A - \\lambda I) = 0\\). Therefore, the text, by explaining the relationship between an eigenvalue, an eigenvector, and the linear operator \\(A - \\lambda I\\), can assist a student in understanding why the determinant of \\(A - \\lambda I\\) is set to zero to find the eigenvalues, even though it does not explicitly solve the problem or use the exact reasoning steps. It lays the foundational knowledge required to comprehend the solution's approach.\n\nAnswer: True\n\nReasoning for Eigenvector/Real Square Matrix: The provided text offers a definition of an eigenvector in the context of a real square matrix, which directly relates to the fundamental understanding of eigenvalues and eigenvectors as described in the \"Theorem definition\". The text essentially reinforces the relationship between a matrix A, its eigenvector v, and the corresponding eigenvalue \u03bb (A*v = \u03bb*v), which is crucial for understanding the solution to the given problem. However, the text does not explicitly address how to find eigenvalues, which is the primary question in the \"Problem statement\". Nevertheless, understanding the definition and the relationship it outlines is a prerequisite to grasping the solution process, which involves solving the characteristic equation (det(A - \u03bbI) = 0) to find the eigenvalues. Therefore, the text can be seen as indirectly supportive by providing foundational knowledge necessary for solving the problem, even though it does not directly guide the student through the problem-solving steps outlined in the \"Solution\".\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you have a magical box that transforms any object you put inside it, where the object is represented by the column vector x = (x_1, x_2). The box's transformation can be represented by the matrix A = [[5, 4], [1, 2]], so when given an object x, the box outputs the new object Ax. On some special objects, this new object is just a constant multiple of the original object, \u03bbx = (\u03bbx_1, \u03bbx_2). Find both possible values of \u03bb where this occurs \u2014 note that these are the box's eigenvalues."}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Kepler's Laws of Planetary Motion/Third Law/Examples: The text provides examples of applying Kepler's Third Law of Planetary Motion in determining the orbital periods of hypothetical planets based on their distances from the Sun. It uses a similar approach as the solution by expressing the relationship between the orbital period and the mean distance from the central body (the Sun in the text's case, Saturn in the problem's case) according to Kepler's Third Law. The text demonstrates how to use this law to calculate the orbital period given the mean distance, which directly aligns with the steps needed to solve the given problem regarding Titan and Hyperion's orbital periods around Saturn. Although the text discusses planets orbiting the Sun and the problem involves moons orbiting Saturn, the underlying application of Kepler's Third Law remains consistent. Therefore, the text can be helpful to a student who is not familiar with the theorem, as it provides a detailed example of how to apply the theorem in a context that is closely related to the given problem.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine two trains, Train A and Train B, are traveling on separate circular tracks around two different cities. Train A completes its journey around City A, which has a circular boundary with a radius of 1.22x10^9 meters, in exactly 15.95 days. Train B travels around City B, which has a larger circular boundary with a radius of 1.48x10^9 meters. Assuming the relationship between the time it takes to travel around each city and the size of the city is the same as the relationship between the orbital periods of moons and their distances from their planet, estimate how long it takes Train B to complete its journey around City B."}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Union of Non-Disjoint Convex Sets is Convex Set: The problem statement concerns the convexity of a set $S_a$ derived from a convex set $S$ by expanding it through a distance measure. The solution involves demonstrating that $S_a$ retains convexity by verifying that for any two points within $S_a$, a linear combination of these points also resides within $S_a$. This is established through the triangle inequality and properties of convex sets.\n\nThe theorem provided in the text discusses the convexity of the union of a collection of convex sets, given that these sets have a non-empty intersection. This theorem, in essence, confirms that if you combine several convex sets (that share at least one common point) into one set by taking their union, the resulting set is also convex.\n\nThe theorem's relevance to the problem solution is indirect but conceptually similar. The solution directly applies the definition of convexity to an expanded set $S_a$ based on distances from a convex set $S$. In contrast, the text discusses a general property of convex sets under union, which is a broader concept and not directly tied to distance expansions or the specific construction of $S_a$. However, understanding the theorem in the text could reinforce the student's grasp of convexity properties, such as how convexity is preserved under certain operations (union in the text, expansion in the problem solution).\n\nAlthough the text does not address the specific method used in the solution (involving distance measures and the construction of $S_a$), it discusses a related aspect of convex sets (union of non-disjoint convex sets) and reinforces the principle that operations on convex sets can result in sets that maintain the convexity property.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're designing a park and you have a specific area, S, where you want to plant flowers. You decide that you also want to include a buffer zone around this area where you'll plant a different type of flower. This buffer zone will extend outward from every point of S by a fixed distance, a, creating a new area, S_a. If the original flower area, S, is shaped so that any straight line drawn through S stays within S (meaning it's convex), will the new area with the buffer zone, S_a, also have this same property?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Multinomial Theorem: The provided text elaborates on the multinomial theorem, outlining its statement and the general principle behind it. The theorem is described as a generalization of the binomial theorem, focusing on expanding powers of a sum involving multiple terms and calculating multinomial coefficients, which are crucial for solving combinatorial problems involving partitioning objects into groups.\n\nThe problem statement involves partitioning a set of 9 people into 3 groups of 3, which is a direct application of combinatorial principles, specifically related to dividing objects into groups without considering the order within each group. The solution to the problem employs combinatorial reasoning, using binomial coefficients and the concept of division by factorial to account for the unordered nature of the groupings, which aligns with the principles outlined in the multinomial theorem's definition.\n\nAlthough the text primarily focuses on the theorem's statement and proof without directly applying it to a combinatorial problem like the one given, the foundational knowledge it provides about how to use multinomial coefficients in partitioning sets into groups can help a student understand the underlying principles used in the solution. Understanding how multinomial coefficients are derived and how they relate to partitioning problems can assist in grasping why the solution involves choosing groups in the manner it does and why the division by \\(3!\\) is necessary to correct for overcounting due to the indistinguishability of the groups.\n\nTherefore, despite the text not directly solving a problem similar to the one given, it offers valuable insight into the mathematical concepts and reasoning used in the solution, specifically the application of multinomial coefficients to solve combinatorial partitioning problems.\n\nAnswer: True\n\nReasoning for Multinomial Coefficient expressed as Product of Binomial Coefficients: The problem statement involves counting the number of ways to split a group of 9 people into 3 committees of 3 people each. The solution applies combinatorial reasoning by first choosing groups of people and then adjusting for the order in which these groups are chosen, utilizing binomial coefficients and factorial operations. This approach directly connects to the theorem mentioned, the multinomial theorem, which is about dividing a set of objects into multiple groups, taking into account the order of the groups.\n\nThe provided text details a theorem that expresses a multinomial coefficient as a product of binomial coefficients. This is highly relevant because the solution to the problem effectively uses this principle, albeit in a more applied manner. It calculates the number of ways to choose the committees (which can be thought of as groups in the multinomial context) and corrects for the ordering of these committees, which is a direct application of the logic underpinning the multinomial theorem. \n\nThus, even though the text does not explicitly solve the problem or use the exact numbers from the problem statement, it provides a mathematical foundation that is directly applied in the problem's solution. It explains how multinomial coefficients, which are central to solving the given problem, can be broken down into binomial coefficients, mirroring the steps taken in the problem solution. Therefore, the text can assist a student not familiar with the theorem in understanding the combinatorial reasoning behind the solution.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're organizing a small local sports league with 9 players and decide to create 3 teams, each team having exactly 3 players. Assuming each team is equal in status and the only difference is the roster, how many different ways can you form these teams?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Jensen's Inequality (Real Analysis): The problem statement involves proving an inequality related to the sum of the sine values of the angles of a triangle. The solution mentions that the inequality is true but doesn't provide a detailed explanation of how Jensen's inequality is applied to reach this conclusion. Jensen's inequality deals with a convex function and provides a relationship between the function's value at the expected value of a variable and the expected value of the function applied to the variable. The text provides a detailed theorem and proof of Jensen's inequality in the context of real analysis. It explains how the inequality works for a convex function over a set of real numbers and weights, which can be related to solving inequalities involving sums, like in the given problem. Given that sine is a concave function over the interval [0, \u03c0] (which covers the range of possible angles in a triangle), and Jensen's inequality can be applied inversely for concave functions, the text provides the mathematical foundation and reasoning steps that could help in understanding and applying Jensen's inequality to prove the given inequality about the sine values of a triangle's angles. Thus, the text can be considered helpful for a student not familiar with the theorem to ultimately solve the problem, as it offers a general understanding of Jensen's inequality which is crucial for solving the problem.\n\nAnswer: True\n\nReasoning for Jensen's Inequality (Measure Theory)/Convex Functions: The problem statement asks to verify an inequality involving the sum of sine functions for the angles of any triangle. The solution uses Jensen's inequality to establish the truth of this inequality. Jensen's inequality, as defined, involves convex functions and shows that the function value at the expected value of a random variable is less than or equal to the expected value of the function applied to the random variable. In the context of the problem, the sine function would be treated as a convex (or concave, depending on the interval) function, and the inequality would apply to the sum of the sines of the angles in a triangle.\n\nThe provided text discusses a more general and abstract version of Jensen's Inequality within the framework of measure theory. It talks about integrating functions over a measure space and applying a convex function to these integrals. While this is a more complex and generalized scenario than what is directly applied in the problem solution, the underlying principle of applying a convex function to an integral (or sum, in a discrete sense) and comparing it to the integral (or sum) of the function applied to the argument is consistent with the essence of Jensen's Inequality used in the solution.\n\nThe connection between the abstract measure-theoretic version of Jensen's Inequality and the specific application in the problem might not be immediately clear to students unfamiliar with the theorem. However, the text does illustrate a similar application of Jensen's Inequality (in a more generalized and abstract setting), demonstrating how convex functions behave under integration or summation. This could potentially help in understanding the core concept of Jensen's Inequality and its application to the problem at hand, albeit with some effort to bridge the gap between the abstract form and the specific problem's context.\n\nAnswer: True\n\nReasoning for Jensen's Inequality (Measure Theory)/Concave Functions: The theorem used in the solution to the given problem is Jensen's Inequality. The definition provided explains Jensen's Inequality in the context of convex functions and expectations, stating that for a convex function \\(f\\), \\(f(E(X)) \\le E(f(X))\\), where \\(E(X)\\) is the expected value of a random variable \\(X\\). This inequality implies that the average of the function values is at least as large as the function value at the average point, when the function is convex. For concave functions, the inequality is reversed.\n\nThe given problem is about proving an inequality related to the sum of sines of angles in any triangle, which can be approached by considering Jensen's Inequality for the sine function, recognizing that sine is a concave function over the interval \\([0, \\pi]\\), which includes all possible angles in a triangle. \n\nThe provided text discusses a version of Jensen's Inequality in the context of measure theory and concave functions. It states a theorem about the inequality involving integrals of functions over a measure space, where \\(\\Lambda\\) is a concave function. Although the setting (measure theory) is more advanced and specific than the simple application needed to solve the triangle problem, the reasoning behind using a concave function (like sine) and applying Jensen's Inequality to establish bounds or inequalities is fundamentally the same. The text discusses Jensen's Inequality in a way that involves concave functions and inequalities, which is relevant to understanding how to apply Jensen's Inequality to the sum of sines in a triangle, despite the measure-theoretic context being more complex than necessary for the triangle problem.\n\nHowever, for a student not familiar with the theorem, the measure-theoretic language and notation could be more confusing than enlightening. The key for the student is to grasp how Jensen's Inequality can be applied to concave functions to establish inequalities, rather than understanding the specifics of measure theory.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're a photographer taking pictures of a landscape that includes a triangle formed by three mountains. You're curious about the brightness of the sunlight hitting the slopes of these mountains at different angles. If you consider the angles at which sunlight hits each of the three peaks as A, B, and C, and you know that the intensity of sunlight can be related to the sine of these angles, is it possible for the sum of the intensities of sunlight on all three peaks to exceed 2.598?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Ramsey's Theorem: The provided text directly addresses the fundamental concepts of Ramsey's theorem, which is central to solving the given problem. The text explains how Ramsey's theorem relates to coloring the edges of a complete graph in multiple colors (specifically 2 colors in this scenario) and guarantees the existence of a monochromatic complete subgraph of a certain size, which directly pertains to the problem statement of finding a monochromatic 4-clique in either of the two colors. It also introduces the concept of Ramsey numbers, which is the exact approach used in the solution to determine the smallest n for which a monochromatic 4-clique is guaranteed. Therefore, the text provides the necessary theoretical background and reasoning steps that mirror the solution's application of Ramsey's theorem. This would assist a student not familiar with the theorem in understanding and applying it to solve the problem. Hence, the text is helpful.\n\nAnswer: True\n\nReasoning for Ramsey Number: The given text is a definition of the Ramsey number, which is directly relevant to solving the problem at hand. The problem involves determining the smallest number of vertices \\(n\\) in a complete graph that guarantees the presence of a monochromatic 4-clique when edges are colored with two colors. The solution to the problem relies on Ramsey's theorem, specifically on finding the Ramsey number \\(R(4,4) = 18\\). The text provides a general definition of Ramsey's theorem and the concept of Ramsey numbers, explaining how for any given number of colors and integers \\(n_1, \\ldots, n_c\\), there exists a Ramsey number \\(R(n_1, \\ldots, n_c)\\) such that a complete graph of order \\(R(n_1, \\ldots, n_c)\\) colored with \\(c\\) different colors must contain a monochromatic complete subgraph of order \\(n_i\\) for some \\(i\\). This is the exact reasoning applied in the solution to find the smallest \\(n\\) that guarantees a monochromatic 4-clique, thus directly applying the theorem stated in the text to solve the problem. Therefore, the text would be helpful to a student trying to understand and solve the problem using Ramsey's theorem.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're organizing a tournament with teams from different cities, and each match can end in one of two outcomes: a win for the home team (marked with red) or a win for the visiting team (marked with blue). What is the minimum number of teams you need to invite to ensure that, no matter the outcomes, you will have either four teams where each team has won against each of the other three teams at home (a red quartet) or four teams where each has lost to each of the other three teams while visiting (a blue quartet)?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Multinomial Theorem: The text provides a detailed explanation and formal statement of the multinomial theorem, which is the key theorem used in the solution to the problem. The problem involves calculating the number of ways to form a 7-digit number with repetitions, a scenario that directly applies the multinomial theorem as described in both the theorem definition and the text. The solution employs the multinomial coefficients, n! / (k1! * k2! * ... * kr!), to determine the number of arrangements, which is exactly what the text explains in mathematical terms. Therefore, the text aligns with the reasoning steps and the application of the theorem in the solution, making it useful for a student trying to understand and solve the problem by applying the multinomial theorem.\n\nAnswer: True\n\nReasoning for Multinomial Coefficient expressed as Product of Binomial Coefficients: The provided text explains how a multinomial coefficient can be expressed as a product of binomial coefficients, which is a fundamental mathematical concept closely related to the problem's solution, which involves calculating permutations with repetitions using the multinomial theorem. The problem involves arranging digits with repetitions to form a 7-digit number, and the solution employs the formula derived from the multinomial theorem to compute the number of such arrangements. While the text does not directly apply this theorem to solve a counting problem like the one in the problem statement, it delves into the mathematical underpinnings of the multinomial coefficient, which is crucial for understanding how to calculate the number of arrangements in problems involving repetitions.\n\nThe text, by breaking down the multinomial coefficient into binomial coefficients and providing a proof of this relationship, aids in building a deeper understanding of the mathematical principles that underpin the solution. For a student unfamiliar with the theorem, understanding how multinomial coefficients can be expressed and manipulated is beneficial for grasping the broader concept of counting arrangements with repetitions. Therefore, although the text doesn't directly solve a similar problem, it contributes valuable insight into the mathematical concepts necessary for understanding and applying the theorem as used in the solution.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're creating a unique code for a special online discount. The code must be 7 characters long and is made up of the numbers 1, 2 (appearing twice), 3 (appearing three times), and 4. How many different unique codes can you create following these rules?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Intermediate Value Theorem: The provided text is a detailed explanation and proof of the Intermediate Value Theorem (IVT), which is directly related to the theorem used in the solution to the given problem. The solution to the problem relies on the application of the IVT to show that there exists a solution to the equation cos(x) = x^k for k = 2023 within a specific interval. The text not only defines the IVT but also explains how it applies to continuous functions over a real interval, ensuring there exists a point where the function equals a given value between its values at two endpoints of the interval. This reasoning is analogous to the solution's approach of finding a point where cos(x) equals x^2023, indicating that the text can help a student understand how the IVT is applied to solve the problem. Therefore, the text would be helpful to a student unfamiliar with the theorem in ultimately solving the given problem by using very similar reasoning steps and applying the theorem in a related way as the solution does.\n\nAnswer: True\n\nReasoning for Intermediate Value Theorem/Corollary: The given text provides an elaboration on a specific application of the Intermediate Value Theorem (IVT), focusing on the scenario where the function is continuous on an open interval and there exists a number (in this case, 0) lying between the function's values at two points a and b. This explanation is closely related to the use of the IVT in the solution, where the theorem is applied to show that because the cosine function and x^k function have different values at the endpoints of an interval but must pass through all values between due to continuity, there must exist a point where cos(x) = x^k for k = 2023 within the interval.\n\nThe text is helpful for understanding and solving the given problem because it aligns with the reasoning steps used in the solution: identifying a continuous function's behavior over an interval and utilizing the IVT to prove the existence of a solution within that interval. The solution applies the IVT to demonstrate that there exists at least one value of x within the interval (0, 1) where cos(x) = x^2023. The text supports this approach by reinforcing the understanding of the IVT, especially its application to finding a root (or solution) within an interval when the function values at the interval's endpoints straddle the target value (in this case, the function's root).\n\nTherefore, the text would be helpful to a student trying to understand how the Intermediate Value Theorem can be used to prove the existence of a solution to the equation cos(x) = x^k for k = 2023, despite not providing a direct walkthrough of the problem solution. It aids in building the conceptual framework necessary to apply the theorem similarly as it is applied in the solution.\n\nAnswer: True\n\nReasoning for Intermediate Value Theorem (Topology): The provided text discusses the Intermediate Value Theorem (IVT) within the context of topology, specifically dealing with continuous mappings from a connected topological space to a totally ordered set equipped with the order topology. It states that if there is a continuous function from one space to another and if one value is strictly less than another, then for any value in between, there exists a point in the original space that maps to this intermediate value. This formulation of the IVT, although presented in a more abstract setting, fundamentally aligns with the same principles used in the solution to the problem statement. The problem solution relies on the IVT by arguing that since the cosine function and the polynomial function \\(x^{2023}\\) are continuous and take on values at different ends of an interval with a transition in between, there must exist a point within that interval where the two functions are equal. This reasoning directly parallels the abstract scenario described in the text, where it's stated that for any value between \\(f(a)\\) and \\(f(b)\\), there exists some \\(c\\) such that \\(f(c) = r\\). Therefore, despite the text's more theoretical and generalized presentation, its core principles and applications of the IVT are very much relevant to understanding and applying the theorem to solve the given problem.\n\nAnswer: True\n\nReasoning for Root of Equation e^x (x - 1) = e^-x (x + 1): The text provided focuses on demonstrating the existence of a root for a different equation, \\(e^x (x - 1) = e^{-x} (x + 1)\\), by applying the Intermediate Value Theorem (IVT). It outlines a specific case where the function changes sign over an interval, which implies the existence of at least one root within that interval according to the IVT. Similarly, the solution to the given problem uses the IVT to argue that because the functions cos(x) and \\(x^{2023}\\) have different behaviors over a specific interval and due to the properties of these functions within the interval \\([-1, 1]\\), there must be at least one point where \\(cos(x) = x^{2023}\\). \n\nBoth the text and the solution rely on the fundamental concept of the Intermediate Value Theorem, which asserts the existence of a root within an interval where a continuous function undergoes a sign change or meets a specific value between its endpoints. Despite the text and the solution dealing with different equations, the underlying reasoning involving the IVT's application to demonstrate the existence of a root within a given interval based on the function's properties remains consistent.\n\nTherefore, the text can be helpful in understanding the application of the Intermediate Value Theorem to prove the existence of solutions for equations by demonstrating similar reasoning steps. It showcases how to apply the theorem in a practical context, which can aid a student unfamiliar with the theorem in solving the given problem by understanding the necessity of examining the function's behavior over a specific interval and determining points of interest where the function meets the criteria outlined by the IVT.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're programming a video game where at a given level x, a character's magic power is given by cos(x) and physical power is given by x^2023. If we allow the level x to be any real number, does there exist a level x where the character's magic and physical power are equal in value?"}


Instruction: Given a math question, find theorems that are relevant to it

{"reasoning": "Reasoning for Rolle's Theorem: The given problem involves finding whether there is a value \\(x\\) for which the derivative of a given function \\(f(x) = x^2\\sin(x)\\) is zero within a specific interval \\([10\\pi, 11\\pi]\\). The solution to this problem utilizes the derivative of the function and checks for the sign change in the derivative values at the endpoints of the given interval. The Intermediate Value Theorem is implicitly used in the reasoning that if the derivative changes sign (from negative to positive or vice versa) across an interval, then it must cross zero at least once within that interval.\n\nHowever, the theorem provided and the text to be evaluated both explicitly discuss Rolle's Theorem, which is a specific application of the Intermediate Value Theorem but under the condition that \\(f(a) = f(b)\\) for a function continuous on \\([a, b]\\) and differentiable on \\((a, b)\\). Rolle's Theorem directly addresses the existence of at least one point \\(c\\) in the open interval \\((a, b)\\) such that \\(f'(c) = 0\\), given certain conditions are met.\n\nAlthough the solution does not directly name or apply Rolle's Theorem, the reasoning and the conditions under which the theorem operates (continuous and differentiable functions, sign change indicating a zero crossing) are very much aligned with the principles that underpin both Rolle's and the Intermediate Value Theorem for derivatives.\n\nThe text provided gives a comprehensive explanation of Rolle's Theorem, including conditions for its application and the logical proof of why it holds true. This understanding can be very beneficial for a student trying to grasp why a derivative would have a zero within an interval, under the right conditions. Even though the specific application to the given problem relies on a more generalized reasoning about derivatives and sign changes, the foundational knowledge about function behavior provided by Rolle's Theorem and its proof can help in understanding the solution's underlying principles.\n\nTherefore, the text can be considered helpful in understanding and solving the given problem by offering deep insights into the behavior of differentiable functions and the conditions under which their derivatives must have at least one zero in a given interval.\n\nAnswer: True\n\nReasoning for Mean Value Theorem: The text provided discusses the Mean Value Theorem, which states that for a function continuous on a closed interval \\([a, b]\\) and differentiable on the open interval \\((a, b)\\), there exists at least one point \\(c\\) in \\((a, b)\\) such that the derivative of the function at that point is equal to the average rate of change of the function over the interval. This is closely related to the problem solution, which involves finding a point where the derivative of a given function equals zero, using the derivative's properties and the concept of change in function values over an interval. The solution to the problem uses reasoning related to the Intermediate Value Theorem (a consequence of the Mean Value Theorem) to argue that a zero derivative exists in the given interval due to a change in sign of the derivative's value at the interval's endpoints. Although the problem specifically mentions Rolle's Theorem (which is a special case of the Mean Value Theorem and requires the function values at the endpoints to be the same), the reasoning around continuity, differentiability, and the existence of a point where the derivative equals zero (or the function's rate of change matches a specific criterion) aligns closely with the Mean Value Theorem's application as explained in the text. Therefore, despite the direct reference to Rolle's Theorem in the solution, the text provides relevant foundational knowledge that can assist a student in understanding the principles underlying the problem's solution, making it indirectly helpful.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're programming a new video game that involves a magical landscape where the terrain's elevation at any point is determined by a formula based on the distance from a certain landmark. Specifically, the elevation is given by the square of the distance times the sine of the distance. You're tasked with creating a smooth path between two points on this landscape, both of which are exactly the same elevation, located 10pi and 11pi units away from the landmark, respectively. Is it guaranteed that there will be a point between these two locations where the path is perfectly flat, meaning the slope of the path is zero?"}