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

{"reasoning": "Reasoning for Conservation of Angular Momentum: The given text provides a detailed explanation of the conservation of angular momentum, beginning with a fundamental principle derived from Newton's Laws of Motion, specifically Newton's Third Law. It explains how in a system of interacting masses where no external forces are acting, the forces between the bodies occur in equal and opposite pairs, leading to a situation where the total torque on the system is zero, thereby conserving angular momentum.\n\nThe solution to the problem relies on the conservation of angular momentum to find the door's angular speed after being struck by a bullet. It calculates the initial angular momentum of the bullet, considers the final angular momentum of the door and bullet system, and uses the principle that angular momentum is conserved (initial equals final) to find the angular speed.\n\nThe text is helpful for understanding and solving the given problem because it lays out the foundational principle that the solution applies \u2013 the conservation of angular momentum in the absence of external forces. This principle directly supports the reasoning used in the solution, where the conservation of angular momentum is applied to calculate the angular speed of the door after the bullet embeds in it. Even though the text does not provide a step-by-step application to the specific problem of a bullet striking a door, it gives a solid understanding of the underlying principle used in the solution. Therefore, the text would assist a student not familiar with the theorem by grounding them in the fundamental concept needed to approach the problem.\n\nAnswer: True\n\nReasoning for Angular Momentum: The provided text gives a definition of angular momentum, which is central to understanding and solving the given problem. The problem involves calculating the angular speed of a door after being struck by a bullet, which embeds itself in the door. The solution to the problem utilizes the conservation of angular momentum principle, which directly relates to the definition of angular momentum provided in the text. The text explains angular momentum as the product of a body's moment of inertia about a point and its angular velocity about that point, which is exactly the concept needed to understand the solution that involves calculating the door's moment of inertia and then using the conservation of angular momentum to find the door's angular speed. Although the text does not directly provide the steps to solve the problem, it lays the foundational understanding of angular momentum, which is crucial for a student not familiar with the theorem to begin solving the problem. Thus, the text can be helpful in understanding and approaching the solution to the given problem by explaining a fundamental concept that the solution is based on.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're playing a game where you have a toy cannon that shoots small metal balls. You decide to aim at a lightweight door that can swing open on its hinges. The door is 1 meter wide and weighs 15 kilograms. You shoot a metal ball weighing 10 grams at a speed of 400 meters per second straight at the middle of the door, and the ball sticks to it. What is the new angular speed of the door after the ball sticks to it? (Unit: rad/s)"}


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

{"reasoning": "Reasoning for Derivative of Composite Function: The text provides a formal definition and proof of the chain rule for differentiation, which is directly relevant to the solution of the given problem. In the problem's solution, the derivative of the y-coordinate of the rocket's trajectory is found by differentiating the expression 200t - 4.9t^2 with respect to t. Although the solution itself does not explicitly mention using the chain rule, the process of differentiating this expression involves principles that are foundational in the chain rule, especially considering the expression as a composite function (even if it appears straightforward). The text explains how the derivative of a composite function can be determined by using the chain rule, which is essential for understanding how to find the derivative of any complex or composite function. Therefore, the text can be helpful in understanding the steps taken in the solution, particularly for a student who is not familiar with how derivatives, especially in the context of maximizing a function, are calculated using the chain rule.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine launching a toy rocket straight up into the sky, and its path through the air can be tracked with a formula based on time since launch. The formula for its flight path is given by the equation where the height of the rocket at any time 't' (in seconds) is determined by $h=200t - 4.9t^{2}$, while the horizontal distance at any time 't' is determined by 'd=80t'. How high does the rocket go at its highest point before it starts to come back down?"}


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

{"reasoning": "Reasoning for Multinomial Theorem: The provided text explains the multinomial theorem, which directly relates to the solution of the given problem. The theorem describes how to calculate the number of ways to partition a set into multiple groups, considering the order of the groups, which is exactly what is needed to solve the problem of forming a 7-digit number with given repeated digits. The formula in the solution, \\(n! / (n1! * n2! * ... * nk!)\\), is a direct application of the multinomial coefficients (\\(\\dbinom n {k_1, k_2, \\ldots, k_m} = \\dfrac {n!} {k_1! \\, k_2! \\, \\cdots k_m!}\\)) mentioned in the text. This means the text provides the theoretical foundation and the mathematical expression used in the problem's solution. Therefore, the text would be helpful for a student not familiar with the theorem to understand and ultimately solve the problem using the theorem's application as demonstrated in the solution.\n\nAnswer: True\n\nReasoning for Multinomial Coefficient expressed as Product of Binomial Coefficients: The provided text delves into the relationship between multinomial coefficients and binomial coefficients, presenting a theorem that expresses a multinomial coefficient as a product of binomial coefficients. The problem statement involves determining the number of ways to form a 7-digit number using a given set of digits with repetitions, which is a direct application of the multinomial theorem. The solution utilizes the multinomial theorem by calculating the multinomial coefficient for the given digits, which represent the groups into which the set of digits can be partitioned. Although the text does not directly provide a step-by-step method for solving the specific problem stated, it does offer insight into the underlying principles of multinomial coefficients, which are central to understanding and applying the solution's approach. By illustrating how multinomial coefficients can be broken down into a product of binomial coefficients, the text indirectly supports the comprehension of the solution's methodology, especially in terms of understanding how the permutations with repetitions are calculated using the formula derived from the multinomial theorem. Therefore, for a student unfamiliar with the theorem, the text can be helpful in grasping the fundamental concepts necessary to understand and apply the multinomial theorem in solving counting problems similar to the one given.\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 is 7 characters long and must be made using the numbers 1, 2 (twice), 3 (four times). How many different codes can you create ensuring each code uses all the numbers exactly as specified?"}


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

{"reasoning": "Reasoning for L'H\u00f4pital's Rule: The provided text elaborates on L'H\u00f4pital's rule, offering both a theorem statement and a proof that closely aligns with the theorem definition given. It discusses conditions under which the rule can be applied (functions being continuous and differentiable on certain intervals) and asserts that under these conditions, the limit of a function as x approaches a certain point can be evaluated by taking the limit of the derivatives of the numerator and denominator. This is directly relevant to solving the problem statement, which involves finding the limit of an expression as x approaches 0 - a scenario that invokes L'H\u00f4pital's rule due to an indeterminate form. The solution to the problem utilizes L'H\u00f4pital's rule by taking derivatives and evaluating the limit, which is in line with the reasoning and application explained in the text. Therefore, the text can assist a student not familiar with the theorem in understanding and applying L'H\u00f4pital's rule to solve the problem at hand.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're trying to tune a guitar. To do this, you need to figure out the vibration amplitude: for each tuning setting x that you try, the amplitude is given by (sin(2x) / x)^(1+x). What does the amplitude approach as you try settings x that approach 0?"}


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

{"reasoning": "Reasoning for Entire Function with Bounded Real Part is Constant: The provided text discusses a theorem that states an entire function with a bounded real part is constant. It uses a similar principle related to Liouville's theorem by constructing a function \\(g\\) from \\(f\\) and then showing that \\(g\\) is bounded and entire, which implies \\(g\\) (and hence \\(f\\)) must be constant. This parallels the reasoning used in the solution, which directly applies Liouville's theorem to conclude that if an entire function \\(f\\) is bounded, it must be constant. Both the solution and the text leverage the concept of boundedness to derive that a function must be constant, though the text approaches it through the boundedness of the real part specifically and then generalizes the result. \n\nGiven that the text explains how boundedness (in this case, of the real part) leads to a function being constant, and considering that the problem solution relies on the theorem stating a bounded entire function must be constant, the text provides a related application of Liouville's theorem. It could help a student understand how boundedness in the complex plane, whether directly or indirectly applied, leads to constancy, and therefore can assist in understanding and solving the given problem.\n\nAnswer: True\n\nReasoning for Liouville's Theorem (Complex Analysis): The provided text outlines a proof of Liouville's Theorem, which states that if an entire function is bounded, then it must be constant. This directly aligns with the theorem used in the solution to the given problem, which involves an entire function with a magnitude that does not fall below 1, implying it is bounded above by some value (as it doesn't grow arbitrarily large). The solution asserts that by Liouville's Theorem, the function must be constant, which is the same principle discussed in the text. The text explains the theorem's proof by showing that the derivative of any scaled version of the function is zero, leading to the conclusion that the function itself must be constant. This reasoning process directly supports the solution's steps and applies the theorem in a similar manner to deduce that the function in the problem statement is constant. Therefore, the text can be helpful for a student to understand and apply Liouville's Theorem to solve the given problem, especially in understanding why the condition given in the problem (|f(z)| \u2265 1) leads to the conclusion that f must be constant.\n\nAnswer: True\n\nReasoning for Removable Singularity at Infinity implies Constant Function: The given text presents a theorem and proof concerning entire functions that have a removable singularity at infinity, concluding that such functions must be constant. This aligns closely with the solution to the problem statement, which relies on Liouville's theorem to argue that an entire function bounded in magnitude (|f(z)| \u2265 1 for every z in C) must be constant. Both the solution and the text apply the concept of boundedness in the complex plane to reach a conclusion about the constancy of the function. Liouville's theorem is directly applied in the text's proof, just as it is in the problem's solution, albeit the text approaches the boundedness through the concept of a removable singularity at infinity. Despite this different approach, the core reasoning involving boundedness and the application of Liouville's theorem to deduce constancy of the function is very similar. Therefore, the text can be considered helpful for a student not familiar with the theorem, as it provides an alternative yet relevant application of Liouville's theorem to infer that an entire function must be constant under certain conditions.\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 software developer working on a new graphics editing tool. You've created a function that applies a special filter to images. However, no matter what image you input, the intensity of the filter effect seems to remain the same across the entire digital canvas. Assuming the filter's effect intensity is determined by a mathematical function that takes complex numbers as inputs and the effect intensity never drops below a certain level for any part of an image, does this mean your filter function is applying a uniform effect across all images?"}


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

{"reasoning": "Reasoning for Test Statistic: The text provides a definition of a test statistic, which is a foundational concept used in hypothesis testing, including t-tests. A test statistic is calculated from sample data and is used to decide whether to reject the null hypothesis. While the text does not specifically mention the t-test, the definition of a test statistic is relevant to understanding how a t-test operates, as the t-statistic is a type of test statistic used to compare the means of two groups. In the solution provided, a t-statistic is calculated and used to test the hypothesis regarding the mean difference between two types of rubber plants. Thus, the text introduces a fundamental concept (test statistic) that is directly applied in the t-test procedure described in the solution. Therefore, it can assist a student in understanding the underlying principles of how the solution applies the t-test to solve the problem. However, it is important to note that the text does not explicitly explain how to conduct a t-test or interpret its results, which means it may not fully equip a student with all the knowledge needed to solve the problem without additional information. Yet, it contributes to a foundational understanding necessary for grasping the solution's methodology.\n\nAnswer: True\n\nReasoning for Hypothesis Test: The provided text outlines the basic concept of a hypothesis test, which is foundational to understanding how a t-test functions as a specific type of hypothesis test. In the given solution, a two-sample t-test is performed, which inherently involves the process of hypothesis testing as described in the text. The text explains that a hypothesis test involves deciding when to accept or reject a null hypothesis ($H_0$) based on sample values, which directly relates to the solution's process of using t-test results to reject the null hypothesis that there is no significant difference between the means of two types of rubber plants. Although the text does not explicitly mention the t-test or delve into its specifics, understanding the concept of hypothesis testing is crucial for someone learning about or applying t-tests. Thus, the text can be helpful as an introductory or complementary explanation for a student unfamiliar with the theorem but trying to understand the solution's methodology.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nA farmer is comparing the productivity of two different types of rubber trees on his farm. He collects samples of rubber from 16 trees of Type 1 and 12 trees of Type 2. He wants to know if there's a significant difference in the average rubber production between the two types of trees. The rubber production (in kilograms) for Type 1 trees are: 6.21, 5.70, 6.04, 4.47, 5.22, 4.45, 4.84, 5.84, 5.88, 5.82, 6.09, 5.59, 6.06, 5.59, 6.74, 5.55. For Type 2 trees, the production is: 4.28, 7.71, 6.48, 7.71, 7.37, 7.20, 7.06, 6.40, 8.93, 5.91, 5.51, 6.36. Is there a statistically significant difference in rubber production between the two types of trees?"}


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

{"reasoning": "Reasoning for Cayley's Formula: The provided text explains Cayley's formula, which is the theorem used in the solution to determine the number of trees on 5 labeled vertices. The text not only states the theorem clearly\u2014\"The number of distinct labeled trees with $n$ nodes is $n^{n - 2}$.\"\u2014but it also gives a brief insight into the proof of the theorem through a bijection between Pr\u00fcfer sequences and labeled trees. This aligns well with the solution, where Cayley's formula is applied directly to find the number of trees with 5 labeled vertices by substituting n = 5 into the formula, yielding \\(5^{5-2} = 5^3 = 125\\).\n\nMoreover, the text's mention of the bijection provides a deeper understanding of why Cayley's formula works, which could help a student not familiar with the theorem to grasp not just the how but the why, enhancing their understanding of the solution provided.\n\nGiven that the text applies the theorem in a directly related way as the solution applies it to solve the problem, it can assist a student in understanding the steps taken in the solution and why those steps are valid. Therefore, the text is helpful for someone trying to solve the problem by applying Cayley's formula.\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 5 computers, each computer being a point in the network. You want to connect these computers in a way that every computer is connected to the network but without creating any loops, essentially forming a direct or indirect path between any two computers. How many different ways can you organize this network?"}


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

{"reasoning": "Reasoning for Intermediate Value Theorem: The text provided offers an in-depth explanation of the Intermediate Value Theorem (IVT), including its formal statement and proof. This theorem is precisely the one utilized in the solution to the given problem, which aims to determine if the polynomial $p(x) = x^5 + x - 1$ has any real roots. The solution applies the IVT by evaluating the polynomial at two points and using the change in sign to infer the existence of a root between those points, which aligns with the core concept of the IVT as explained in the text. The text's explanation includes the necessary conditions for applying the IVT, namely the continuity of the function on an interval and the presence of values of opposite signs at two points within the interval. This information can assist a student in understanding why the IVT is applicable to the problem and how it leads to the conclusion that there is at least one real root. Therefore, the text can be helpful for a student not familiar with the theorem in understanding and solving the given problem.\n\nAnswer: True\n\nReasoning for Intermediate Value Theorem/Corollary: The given text presents a corollary of the Intermediate Value Theorem (IVT) that specifically focuses on the existence of a root (where the function equals zero) in an open interval, given that the function takes on values with opposite signs at the endpoints of the interval. This is precisely the concept used in the solution to the problem, where the solution applies the IVT to show that the polynomial $p(x) = x^5 + x - 1$ must have a real root between $x = -1$ and $x = 1$ because $p(-1) < 0$ and $p(1) > 0$. The text and the solution both rely on the fundamental idea of the IVT\u2014that a continuous function which takes values of opposite signs at two points in an interval must cross zero at least once within that interval. Therefore, the text can be considered helpful for a student trying to understand how to apply the IVT to prove the existence of a real root in a given interval for a continuous function.\n\nAnswer: True\n\nReasoning for Intermediate Value Theorem (Topology): The given text presents a more general form of the Intermediate Value Theorem within the context of topology, specifically focusing on connected topological spaces and continuous mappings. It outlines that if a continuous function maps from a connected space to a totally ordered set, and if this function takes two values such that one is strictly less than the other, then for any value between these two, there exists a preimage in the original space. This reasoning aligns closely with the application of the Intermediate Value Theorem in the solution to the problem statement, which involves determining the existence of a real root for a polynomial by evaluating the function at two points and using the fact that the function values at these points have opposite signs. The topological version of the theorem provided in the text emphasizes the same core principle that the solution employs: if a function is continuous and takes on values on either side of a given value, there must be a point where the function equals that given value. This principle is applied in the problem solution to show that the polynomial must have a real root between two points where the function values have opposite signs. Therefore, despite the more abstract setting, the text can be helpful in understanding and solving the given problem by applying the Intermediate Value Theorem in a closely related way.\n\nAnswer: True\n\nReasoning for Existence of Positive Root of Positive Real Number/Positive Exponent: The given text discusses a theorem related to the existence of a positive root for a given positive real number with a positive exponent. It essentially says that for any positive real number x and any positive integer n, there exists a non-negative real number y such that y raised to the power of n equals x. This discussion involves using the Intermediate Value Theorem (IVT) to prove that a function defined by raising y to the power of n can achieve any positive value x, given that the function starts at a value less than or equal to x and is unbounded above.\n\nThe problem statement, on the other hand, is about determining whether a specific polynomial has any real roots. The solution applies the Intermediate Value Theorem by evaluating the polynomial at two points and showing that it must cross the x-axis (thus having a real root) because it takes on values of opposite signs at these points.\n\nBoth the text and the solution apply the Intermediate Value Theorem to show the existence of a certain value (a root in the problem statement, and y^n = x in the text). While the contexts are different\u2014one is finding a root of a polynomial, and the other is finding a root of a real number\u2014the underlying principle of using IVT to establish the existence of a value between two points remains the same. \n\nGiven this, the text could help a student understand the application of the IVT in a slightly different context, potentially aiding in grasping how IVT can be used to prove the existence of a value (in the problem's case, a real root) within a certain interval. Hence, it can be considered helpful for a student not familiar with the theorem to understand and solve the problem by showing a similar application of IVT.\n\nAnswer: True\n\nReasoning for Brouwer's Fixed Point Theorem/One-Dimensional Version: The given text discusses Brouwer's Fixed Point Theorem, specifically its one-dimensional version, which states that for a real function that is continuous on a closed interval and maps from that interval to itself, there exists at least one point in the interval where the function value equals the point itself. This theorem fundamentally relies on the concept of continuity and the existence of a point meeting a specific condition within a closed interval, akin to how the Intermediate Value Theorem (IVT) is applied in the solution of the given problem. Although Brouwer's Fixed Point Theorem is not the same as the IVT, the reasoning in the text ultimately appeals to the IVT to prove the existence of a fixed point. The proof in the text makes a transition to using the IVT by defining a new function that demonstrates the change of signs over an interval, thereby allowing for the conclusion that there exists a point where the function equals zero. This is conceptually similar to how the solution uses the IVT to show the existence of a real root for the polynomial \\(p(x)\\) by demonstrating a sign change over an interval. Therefore, the text can be helpful for a student by showing a related application of the IVT, even though it discusses a different theorem. It illustrates the continuity principle and the reasoning that when a continuous function changes signs over an interval, there is a point where the function takes an intermediate value (in this case, zero), which is directly relevant to understanding and solving the given problem using the IVT.\n\nAnswer: True\n\nReasoning for Positive Real has Real Square Root: The given text discusses a theorem regarding positive real numbers having real square roots and utilizes the Intermediate Value Theorem (IVT) as part of its proof. However, the problem statement asks whether a specific polynomial has any real roots, which is a different question from proving that a positive real number has a real square root. Despite this difference, the text does show an application of the Intermediate Value Theorem in a somewhat related context, as it demonstrates the theorem's utility in proving the existence of a value within a certain range that satisfies a specific condition (in the text's case, finding a real number whose square equals another positive real number).\n\nThe solution to the problem uses the Intermediate Value Theorem directly to show that a polynomial must have a root within a certain interval, based on the change of sign of the polynomial's values at the endpoints of the interval. While the context and specific function (polynomial vs. square root function) are different, the underlying application of the Intermediate Value Theorem is conceptually similar: both use the theorem to deduce the existence of a value within an interval that satisfies a given equation (either $x^5 + x - 1 = 0$ or $y^2 = x$).\n\nTherefore, while the text does not directly address polynomials or the specific problem statement, it does provide a relevant illustration of how the Intermediate Value Theorem can be applied to demonstrate the existence of a solution within a certain interval for a continuous function. This could be helpful for a student not familiar with the theorem, as it offers an example of the theorem's application in a proof, even though the context is slightly different. Thus, the text can be helpful in understanding the reasoning and methodology behind using the Intermediate Value Theorem to solve the given problem, despite the difference in the specific function being examined.\n\nAnswer: True\n\nReasoning for Root of Equation e^x (x - 1) = e^-x (x + 1): The given text illustrates an application of the Intermediate Value Theorem (IVT) to determine the existence of a root for a different equation, $e^x (x - 1) = e^{-x} (x + 1)$. The text demonstrates using the theorem by evaluating the function at two points to show that it takes on values of opposite signs, thereby concluding that a root exists between those two points. This process mirrors the solution approach for the original problem of determining if $p(x) = x^5 + x - 1$ has any real roots. The solution to the original problem also involves applying the IVT by evaluating the polynomial at two points to find values of opposite signs and hence concludes the existence of at least one real root within that interval. Thus, the reasoning steps in the text align closely with the steps used in the solution, employing the IVT in a similar manner to solve a problem of finding a real root of an equation. This demonstrates that the text can assist a student in understanding how to apply the IVT to solve such problems, making it helpful for someone not familiar with the theorem to ultimately 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\nImagine you're planning a road trip that starts at sea level and ends at the top of a mountain. Your elevation gain is measured by a function that increases as you travel along the path. If your starting point is marked as the beginning of the function and the peak of the mountain as the end, and you know that at some point during your trip you'll cross a bridge that's exactly at sea level but not at the start, can you conclude that there's a point on your path where you're exactly at sea level again, not counting the beginning?"}


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

{"reasoning": "Reasoning for Girth: The given text introduces the concept of \"girth\" within the context of graph theory, defining it as the smallest length of any cycle in a graph, and states that an acyclic graph is defined as having a girth of infinity, essentially because it contains no cycles. While the problem statement focuses on the absence of directed cycles in a directed graph under a specific condition (an ordered numbering of nodes that prevents links to lower-numbered nodes), the solution confirms this leads to the absence of directed cycles, aligning with the concept of an acyclic graph as mentioned in the theorem definition.\n\nAlthough the text does not explicitly address the problem statement's scenario of ordered numbering of nodes or the specific application of the theorem to directed graphs, it provides a fundamental understanding of acyclic graphs by defining a characteristic (infinite girth) that only acyclic graphs possess\u2014no cycles. This characteristic is directly relevant to understanding why the provided condition (no links to lower-numbered nodes) ensures the absence of directed cycles, because if such a condition holds, the graph inherently cannot have any cycles, thus making it an acyclic graph.\n\nTherefore, despite the text approaching the concept from a slightly different angle (the concept of girth), it contributes to the foundational understanding necessary for a student to grasp why the problem statement is true when considering the absence of cycles (directed or otherwise) in an acyclic graph. This foundational knowledge can help in understanding the theorem and its application to the problem, even if indirectly.\n\nAnswer: True\n\nReasoning for Finite Sequences in Set Form Acyclic Graph: The text introduces a theorem related to the concept of acyclic graphs, focusing on constructing an acyclic graph from a set of finite sequences. The theorem specifies conditions under which a graph remains acyclic, which aligns with the underlying principle of the problem statement about directed cycles and ordered numbering of nodes in a directed graph. Although the text does not directly address the problem statement, it provides a foundational understanding of acyclic graphs and conditions that ensure a graph's acyclicity. This background can be instrumental in understanding why a directed graph with an ordering that prevents links to lower-numbered nodes does not contain directed cycles, as it indirectly supports the reasoning that such ordering helps maintain the acyclic nature of the graph. Therefore, the text can be helpful for a student by reinforcing the concept of acyclic graphs, which is essential for understanding why the problem statement is true.\n\nAnswer: True\n\nReasoning for Acyclic Graph: The provided text offers a definition of an acyclic graph, which directly relates to the problem statement by explaining what an acyclic graph is: a graph without any cycles. The problem statement involves determining the truth of a statement about the existence of directed cycles in a directed graph under certain conditions. Since the solution uses the concept of an acyclic graph to justify its answer (True), understanding what an acyclic graph is would be crucial for someone unfamiliar with the concept to solve the problem. The text does not specifically address the directed aspect detailed in the problem statement and solution, but it lays the foundational knowledge needed to understand why a graph with an ordered numbering of nodes that prevents links to lower-numbered nodes does not contain directed cycles (by virtue of being acyclic). Therefore, while the text might not provide a step-by-step match with the solution's reasoning, it applies the theorem (acyclic graph) in a related way by defining what an acyclic graph is, which is necessary for understanding the problem's solution.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nIn a city's road map, each intersection is connected by one-way streets. If it's possible to assign a unique number to each intersection in such a way that all streets lead from an intersection with a lower number to an intersection with a higher number, does this mean there are no routes that allow you to start from one intersection and return to it without retracing your steps? True or false?"}


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

{"reasoning": "Reasoning for Error Correction Capability of Linear Code: The given text discusses the error correction capability of a linear code by explaining how a linear code with a minimum distance \\(d\\) can correct \\(e\\) transmission errors where \\(2e + 1 \\leq d\\). Although it approaches the topic from the perspective of error correction in linear codes, the underlying concept of utilizing the minimum distance (or Hamming distance, in this context) to infer error correction capabilities is closely related to calculating and understanding Hamming distances among codewords, as in the problem statement. \n\nThe problem solution involves calculating the Hamming distance between pairs of codewords and finding the minimum Hamming distance among them. The text, by explaining how the minimum distance (or Hamming distance) is pivotal in error correction, indirectly supports the understanding of why and how Hamming distances are calculated and utilized, which is the essence of the problem statement.\n\nHowever, the text does not explicitly demonstrate the calculation of Hamming distances or directly solve a problem similar to the given one. Its focus is more on the implications of having a certain minimum distance in the context of error correction rather than on the computation of the distances themselves.\n\nDespite this, for a student not familiar with the concept of Hamming distance, the text could serve as a valuable resource to understand the importance of Hamming distances in error correction and how they can be used to infer the properties of a code or a set of codewords. This conceptual understanding can be helpful in grasping why calculating the Hamming distance is relevant and necessary in the problem at hand.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're comparing three digital images that are extremely simplified down to a grid of 5 pixels each, represented by either black (0) or white (1) pixels. The images are as follows: Image A: 00000, Image B: 10101, and Image C: 01010. By counting the number of pixels that differ between each pair of images, find the smallest number of differing pixels."}


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

{"reasoning": "Reasoning for Multinomial Coefficient expressed as Product of Binomial Coefficients: The given problem involves counting the number of ways to choose a set of balls from a box with color restrictions, specifically focusing on scenarios with at least two red balls. The solution to the problem utilizes binomial coefficients to separate the counting process into distinct cases, summing up the ways to choose balls based on the specified conditions. The theorem mentioned in the solution is the multinomial theorem, which is used for counting the number of ways to partition a set of objects into multiple groups. \n\nThe provided text discusses expressing multinomial coefficients as a product of binomial coefficients. This is highly relevant to solving the given problem because the solution involves breaking down the problem into cases that fundamentally rely on calculating binomial coefficients for combinations of choosing red and non-red balls. The text gives a mathematical framework for understanding how multinomial coefficients, which generalize binomial coefficients to more than two groups, can be related back to the binomial coefficients used in the solution. This understanding is foundational for someone trying to grasp why the solution approach works, as it underlines the relationship between multinomial and binomial coefficients in combinatorial problems.\n\nDespite the solution not explicitly invoking the multinomial theorem in a direct form (it doesn't use multinomial coefficients in the solution steps), the underlying principle of breaking down a problem into cases and using binomial coefficients aligns with the essence of the text. Therefore, the text can be helpful for a student not familiar with the theorem by bridging their understanding of binomial coefficients used in the solution to the broader context of multinomial coefficients and their applications in combinatorics. Thus, the text assists in deepening the understanding of the combinatorial tools 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 organizing a display case for a toy store, and you have 4 unique superhero figurines, 3 unique dinosaur figurines, and 2 unique robot figurines. You want to select 4 figurines for the display, but you've promised a big superhero fan that at least 2 of the figurines will be superheroes. In how many different ways can you arrange the display under these conditions?"}


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

{"reasoning": "Reasoning for Triangles with Proportional Sides are Similar: The provided text discusses the concept of similarity specifically in the context of triangles, emphasizing that if two triangles have corresponding sides that are proportional, then they are similar, which means their corresponding angles are equal. This is directly related to the theorem definition provided, which states that similarity in geometry refers to figures having the same shape but possibly different sizes, with corresponding sides being proportional and corresponding angles being congruent. The solution to the problem relies on the concept of similarity to set up a proportion between corresponding sides of the given polygons to find the value of 'y'. While the text focuses on triangles, the underlying principle of corresponding sides being proportional in similar figures is applicable to all polygons, including the ones mentioned in the problem statement. Therefore, the reasoning steps in the text about how proportions between sides indicate similarity, and the way similarity is used to solve the problem, are very much aligned. Thus, the text can be considered helpful for someone trying to understand and solve the given problem using the concept of similarity, even though it specifically discusses triangles, because the fundamental principles of similarity are the same across all geometric shapes.\n\nAnswer: True\n\nReasoning for Equiangular Triangles are Similar: The problem statement deals with finding the value of \\(y\\) in two similar polygons by using the sides' lengths. The solution relies on the concept of similarity to establish a proportion between corresponding sides of the two polygons and then solves for \\(y\\).\n\nThe given text discusses the concept of similarity within the context of equiangular triangles, demonstrating that if two triangles have the same corresponding angles, their sides are proportional, and thus, they are similar. This is underpinned by geometric proofs involving concepts like parallel lines, the Parallel Postulate, and properties of parallelograms and transversal lines to establish the proportionality of sides in similar triangles.\n\nAlthough the text specifically addresses triangles and uses geometric proofs to establish similarity and proportionality of sides, the underlying principle of corresponding sides being proportional in similar figures is directly applicable to the solution of the given problem. The solution uses this principle of proportionality between corresponding sides of similar polygons to set up an equation and solve for \\(y\\). \n\nTherefore, even though the text focuses on triangles, the reasoning about similarity and proportionality of sides it provides can help a student understand the foundational theorem used in the problem's solution. This comprehension can assist in solving the given problem and others like it, even if the specific geometric figures differ (polygons in the problem vs. triangles in the text).\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine two different models of the same type of toy car, Model A and Model B. Model A has a front bumper to rear bumper length of 12 units and a door width of 9 units. Model B, which is a scaled version of Model A, has a front bumper to rear bumper length of 3y-1 units, a door width of 6 units, and a hood length of 10 units. If the proportions between corresponding parts of these two models are the same, what is the value of y?"}


Imagine you're a treasure hunter, and you're integrating the size of a strange field. To do so, find the value of \int_a^b \frac{dx}{\sqrt{(x-a)(b-x)}}, rounded to the thousands' place.

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

{"reasoning": "Reasoning for Integration by Substitution/Definite Integral: The text provides a detailed explanation of the integration by substitution method, which is directly applicable to the problem given. The problem statement involves finding the value of an integral with a square root in the denominator, a scenario that often requires a substitution to simplify. The solution employs a specific substitution to transform the integral into a more manageable form and ultimately solve it. The text outlines the theoretical foundation and general process of integration by substitution, demonstrating how to replace variables and integrate with respect to the new variable, which aligns with the steps used in the solution. Therefore, the text can be helpful for a student not familiar with the theorem as it applies the integration by substitution in a related way to solve the integral, which is what the solution does as well. The fact that the text and solution both apply integration by substitution to solve an integral problem indicates that the text could assist a student in understanding and applying this method 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\nImagine you're a treasure hunter, and you're integrating the size of a strange field. To do so, find the value of \\int_a^b \\frac{dx}{\\sqrt{(x-a)(b-x)}}, rounded to the thousands' place."}


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

{"reasoning": "Reasoning for Linear Bound Lemma: The given text discusses the Linear Bound Lemma, which states that for a simple connected planar graph \\(G_n\\) with \\(n \\geq 3\\) vertices, the number of edges \\(m\\) is bounded by \\(m \\leq 3n - 6\\). This theorem directly supports the reasoning used in the problem's solution, which applies the same bound to conclude that it is not possible to draw a simple connected planar graph with the specified number of vertices and faces because the derived number of edges exceeds the maximum allowed by this bound. Both the theorem in the text and the theorem mentioned in the solution (related to maximal planar graphs and their properties) are applied to establish limits on the number of edges in a planar graph, thus utilizing a similar mathematical foundation based on Euler's formula and properties of planar graphs. Therefore, the text can indeed assist a student in understanding and applying the relevant concepts to solve the given problem by providing a foundational theorem that underpins the solution's logic.\n\nAnswer: True\n\nReasoning for Euler's Theorem for Planar Graphs: The provided text outlines Euler's theorem for planar graphs, stating that for a connected planar graph with \\(V\\) vertices, \\(E\\) edges, and \\(F\\) faces, the equation \\(V - E + F = 2\\) holds. This theorem is directly applied in the solution to the given problem, which involves determining the possibility of drawing a simple connected planar graph with 200 vertices and 400 faces by utilizing Euler's formula to relate vertices, edges, and faces. The solution uses the theorem to conclude that such a graph cannot exist, as the calculated number of edges would exceed the maximum allowed by another property of planar graphs. The text not only introduces Euler's theorem but also hints at its proof, which is foundational for understanding the reasoning behind the solution's mathematical steps. Thus, the text provides the necessary theoretical background and applies the theorem in a related way as the solution, making it helpful for a student unfamiliar with the theorem to understand and 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\nImagine you're designing a park with 200 different spots (like trees, benches, or fountains) and you want to create pathways connecting these spots. The park is designed so that each pathway section directly connects two spots without crossing any other pathway, and the entire park is divided into distinct areas by these pathways. If you aim to design the park with 400 distinct areas, is this achievable without any pathways crossing each other?"}


Imagine you're a scientist studying a special function f that is bounded and entire. You're analyzing how this signal interacts with two specific points on the radius-r ball. What is the value of the integral $\int_{|z|=r} f(z)/(z-z_1)(z-z_2) dz$?

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

{"reasoning": "Reasoning for Cauchy-Goursat Theorem: The given text is a description of the Cauchy-Goursat Theorem, which is a specific formulation of Cauchy's Integral Theorem. The Cauchy-Goursat Theorem states that if a function \\(f\\) is holomorphic (complex-differentiable) everywhere within a simply connected domain \\(D\\) that is bounded by a closed contour \\(\\partial D\\), then the integral of \\(f\\) over the contour \\(\\partial D\\) is zero. This theorem is foundational in complex analysis and directly relates to Cauchy's Integral Theorem by emphasizing the condition of a function being holomorphic over a simply connected domain and the resultant integral around a closed contour being zero.\n\nThe problem provided involves using Cauchy's Integral Formula, which is a direct consequence of Cauchy's Integral Theorem, to solve an integral involving a bounded entire function and a specific contour. The solution outlines how to apply Cauchy's Integral Formula, by considering a function \\(g(z) = \\frac{f(z)}{z - z_2}\\) and applying the formula to calculate the integral.\n\nAlthough the text does not directly provide the solution or talk about Cauchy's Integral Formula, it discusses the foundational theorem (Cauchy-Goursat Theorem) that underpins the solution's approach. Understanding the Cauchy-Goursat Theorem can help in understanding why Cauchy's Integral Formula works the way it does, as it explains the conditions under which the integral of a holomorphic function over a closed contour is zero. This understanding is crucial for a student to grasp why the solution to the problem is valid and how the theorem applies to solving integrals of this nature.\n\nTherefore, even though the text does not directly solve the problem or mention Cauchy's Integral Formula, it provides a necessary understanding of the underlying principles that make such solutions possible. For a student not familiar with the theorem, understanding the Cauchy-Goursat Theorem could indeed be a stepping stone to understanding and applying Cauchy's Integral Formula 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\nImagine you're a scientist studying a special function f that is bounded and entire. You're analyzing how this signal interacts with two specific points on the radius-r ball. What is the value of the integral $\\int_{|z|=r} f(z)/(z-z_1)(z-z_2) dz$?"}


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

{"reasoning": "Reasoning for Finite Semigroup Equal Elements for Different Powers: The given text discusses a theorem related to finite semigroups, focusing on the existence of equal elements for different powers within such structures. The core of this discussion leverages the Pigeonhole Principle to argue that due to the finite nature of the semigroup, there must exist repetitions in the sequence of powers of any element, implying that some elements are equal under the operation of the semigroup. \n\nAlthough the application context (finite semigroups) and the main problem (determining the minimum number of people sharing the same birthday among a group of 1000) differ significantly, the underlying mathematical concept used in both the solution to the problem and the text is the Pigeonhole Principle. This principle is correctly identified and defined in the input as a tool for deducing the existence of repetitions or shared properties among a larger set of objects (people with birthdays) or elements (powers of an element in a semigroup) when these objects or elements are distributed across a finite set of containers or categories (days of the year or semigroup elements).\n\nThe text illustrates a correct application of the Pigeonhole Principle in a different mathematical domain (semigroups) but follows a very similar reasoning step as required for solving the given problem (identifying the necessity of repetitions or overlaps due to the constraints of finite categorization). Hence, for a student not familiar with the Pigeonhole Principle, understanding how it is applied in the text to prove the existence of equal powers within a finite semigroup can help grasp the fundamental concept of the principle. This understanding could then assist in applying the principle to solve the problem of determining the minimum number of people sharing the same birthday.\n\nAnswer: True\n\nReasoning for Pigeonhole Principle: The provided text explicates the Pigeonhole Principle through a mathematical theorem and proof, stating that in any division of a finite set \\(S\\) into \\(k\\) subsets, at least one subset must contain at least \\(\\lceil \\frac{n}{k} \\rceil\\) elements, where \\(\\lceil \\cdot \\rceil\\) denotes the ceiling function. This principle directly aligns with the solution to the problem, which uses the Pigeonhole Principle to determine that in a group of 1000 people (the finite set \\(S\\)), divided by 365 possible birthdays (the subsets \\(S_1, S_2, \\ldots, S_{365}\\)), there must be at least one day where \\(\\lceil \\frac{1000}{365} \\rceil\\) people share the same birthday, which mathematically rounds up to 3. \n\nThe text presents a formal theorem and proof that logically supports the solution's application of the Pigeonhole Principle to conclude that at least 3 people must share a birthday in a group of 1000. It details how to mathematically deduce the minimum number of objects (in this case, people) that must be in at least one of the containers (birthdays), which is precisely the reasoning needed to solve the given problem. Therefore, the text provides a theoretical foundation that can aid a student in understanding and applying the Pigeonhole Principle to solve the problem, even if they were not previously familiar with the 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 massive event that will be attended by 1000 guests. Given that there are only 365 days in a year, what is the minimum number of guests who must have been born on the same day of the year?"}


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

{"reasoning": "Reasoning for Stirling Numbers of the Second Kind/Definition 1: The given text provides a definition of Stirling numbers of the second kind, which are essential for understanding how to count the number of ways to partition a set of n elements into k non-empty subsets. This is directly relevant to solving the given problem, where we are asked to partition a set of 6 distinct letters into 2 non-empty groups, adhering to the criteria that each group must contain at least 2 letters. Although the solution does not explicitly mention using Stirling numbers, the concept of partitioning a set into subsets is fundamental to the problem. The text elaborates on how Stirling numbers are calculated, which can be a crucial step in understanding the underlying principles needed to approach and solve the problem. Therefore, the text's focus on the definition and calculation method of Stirling numbers of the second kind aligns closely with the reasoning steps needed in the solution, where partitions of the set are considered. This can assist a student not familiar with the theorem in grasping the concept of set partitioning required to solve the problem, even though the specific application of the theorem (using its formula directly) isn't showcased 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 collection of 6 unique stickers. You want to divide them into two albums, but each album must have at least 2 stickers. How many different ways can you organize your stickers across these two albums?"}


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

{"reasoning": "Reasoning for Markov Chain: The provided text gives a definition of a Markov Chain, explaining the Markov property where the probability of transitioning to the next state depends only on the current state and not on the sequence of events that preceded it. This foundational concept is critical to understanding and applying the Viterbi algorithm, which is utilized in the solution to solve the problem. The Viterbi algorithm, as outlined in the theorem definition, is a dynamic programming algorithm used for finding the most likely sequence of hidden states in a Hidden Markov Model (HMM), where the Markov property plays a central role. Although the text does not directly explain the Viterbi algorithm, understanding the concept of a Markov chain and the Markov property is essential for grasping how the Viterbi algorithm operates and why it is applicable to the problem at hand. Therefore, the text can be helpful in providing the necessary theoretical background to understand the solution's methodology, especially for a student not familiar with the 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 analyzing data from a remote weather station that can only send binary signals: a signal for 'umbrella' when the observer decides to carry an umbrella, and a signal for 'no umbrella' when they don't. The weather at this location can either be rainy or sunny. Historical data shows that there's a 60% chance of rain and a 40% chance of sunshine on any given day. Additionally, the weather pattern is such that if it's rainy (or sunny) today, there's a 70% chance it will be the same tomorrow. The observer has a habit of carrying an umbrella 60% of the time when it's rainy, and 40% of the time when it's sunny. Over three days, the signals received were: 'umbrella', 'no umbrella', 'umbrella'. Based on this information, what was the most likely weather pattern over these three days? Represent rainy days as 1 and sunny days as 0."}


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

{"reasoning": "Reasoning for Multinomial Theorem: The provided text discusses the Multinomial Theorem, which is a generalization of the Binomial Theorem. The Binomial Theorem is directly applied in the solution to the problem statement, using the binomial coefficient formula for calculating the number of ways to choose 3 toppings out of 8, which is a specific application of combinatorial principles outlined by the binomial coefficients. The text, while more complex and general, outlines the foundation for understanding binomial coefficients (a special case of multinomial coefficients) and provides a mathematical framework that includes the binomial theorem as a subset. Since the Multinomial Theorem encompasses the principles of the Binomial Theorem and the binomial coefficients are a specific case of multinomial coefficients, the text can assist a student in understanding the underlying mathematical principles needed to solve the problem, even though it does not directly address the problem's simpler binomial scenario. Thus, despite the text's broader scope, it is relevant for building foundational knowledge that includes the theorem used in the solution.\n\nAnswer: True\n\nReasoning for Generating Function for Binomial Coefficients: The provided text explains the concept of generating functions for binomial coefficients and relates directly to the Binomial Theorem's application in combinatorics. The solution to the given problem utilizes the combination formula, which is a direct application of binomial coefficients (n C r = n! / (r! * (n-r)!)) to calculate the number of ways to choose 3 toppings out of 8. The text, by discussing binomial coefficients and their generating functions, indirectly supports the understanding of how combinations work, which is fundamental to solving the problem at hand. Although the text does not directly apply the combination formula to solve a counting problem like the problem statement does, it reinforces the underlying mathematical principles (binomial coefficients) that the solution uses. Therefore, the text can be helpful to a student not familiar with the theorem as it provides a deeper insight into binomial coefficients, which are crucial for understanding the solution to the problem.\n\nAnswer: True\n\nReasoning for Binomial Theorem/Examples/4th Power of Sum: The given text focuses on demonstrating an application of the Binomial Theorem by showing how to expand the expression \\((x + y)^4\\) using the theorem. It explains the process of finding coefficients for each term in the expansion, which is a direct application of the theorem. However, the problem statement is about calculating the number of ways a customer can order a pizza with 3 out of 8 toppings, which is a combinatorial problem. The solution uses the concept of combinations, which is indeed related to the Binomial Theorem, as combinations can be calculated using binomial coefficients \\(\\binom{n}{r} = \\frac{n!}{r!(n-r)!}\\). The theorem definition explains that binomial coefficients, represented as \\(C(n, k)\\) or \\(\\binom{n}{k}\\), are used in combinatorics to solve counting problems, including finding the number of ways to arrange objects or subsets of a given size.\n\nAlthough the text does not explicitly address combinatorial problems or the calculation of combinations as presented in the problem statement and solution, it indirectly relates to the concept through the explanation of binomial coefficients. Understanding the expansion and the coefficients within the binomial theorem expansion can provide foundational knowledge necessary for grasping the concept of combinations. Therefore, for a student unfamiliar with the theorem, understanding how binomial coefficients work in the context of the Binomial Theorem could indirectly help them understand the calculations involved in solving the problem, even though the text does not directly address combinatorial counting or the specific formula used in the solution.\n\nAnswer: True\n\nReasoning for Number of Compositions: The given text elaborates on the concept of compositions in combinatorics, specifically focusing on $k$-compositions of a positive integer and how to calculate the number of such compositions using binomial coefficients. This relates directly to the Binomial Theorem, which is central to solving combinatorial problems involving \"n choose k\" scenarios, exactly like the problem statement about choosing 3 toppings out of 8. The text utilizes the binomial coefficient formula in a manner closely aligned with the solution's application of the formula to find the number of ways to choose toppings. This demonstrates a very similar reasoning process and applies the theorem in a related way, by using binomial coefficients to solve combinatorial problems. Therefore, the text can be instrumental in helping someone understand the underlying principles needed to solve the given problem, even though it discusses compositions rather than combinations directly.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're at a build-your-own pizza shop where you can choose from 8 unique toppings for your pizza. If you decide to customize your pizza with exactly 3 different toppings, how many unique combinations of these toppings can you create?"}


You are working with four possible machines, each of which is represented by a matrix that operates on an input vector $v$: (a) $(\Phi^T\Phi)^{-1}$. (b) $\Phi(\Phi^T\Phi)^{-1}$. (c) $\Phi(\Phi^T\Phi)^{-1}\Phi^T$. (d) $\Phi^T(\Phi^T\Phi)^{-1}\Phi^T$. Which of them takes any vector $v$ and projects it onto the space spanned by the columns of $\Phi$?

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

{"reasoning": "Reasoning for Orthogonal Projection onto Closed Linear Span: The given problem statement asks for identifying the matrix that projects any vector \\(v\\) onto the space spanned by the columns of \\(\\Phi\\). The correct solution involves using a specific formula for the projection matrix, \\(P = \\Phi(\\Phi^T\\Phi)^{-1}\\Phi^T\\), which directly corresponds to option (c).\n\nThe text provided discusses the orthogonal projection onto a closed linear span in a Hilbert space and elaborates on how any vector \\(h\\) in \\(H\\) can be projected onto a subspace \\(M\\) using an orthogonal projection that involves inner products and sums over an orthonormal basis of \\(M\\). It focuses on the conceptual and computational mechanisms behind projecting vectors onto a subspace defined by an orthonormal basis in the context of Hilbert spaces.\n\nThe theorem and its application in the solution are related to the concept of projecting vectors onto a subspace using matrices. Although the contexts (abstract Hilbert spaces and finite-dimensional vector spaces spanned by columns of a matrix) and the notations (use of inner products and sums in the text versus matrix multiplications and inversions in the solution) differ, the underlying principle of projecting vectors onto a subspace is common to both. The text explains the mathematical foundation for understanding how projections work in a more general setting, which can enhance comprehension of the specific projection operation performed in the solution. \n\nTherefore, despite the differences in presentation and specificity, the text can help in understanding the rationale behind the formula used in the solution by elucidating the general principles of orthogonal projections, which are directly relevant to understanding how the projection matrix in option (c) operates.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nYou are working with four possible machines, each of which is represented by a matrix that operates on an input vector $v$: (a) $(\\Phi^T\\Phi)^{-1}$. (b) $\\Phi(\\Phi^T\\Phi)^{-1}$. (c) $\\Phi(\\Phi^T\\Phi)^{-1}\\Phi^T$. (d) $\\Phi^T(\\Phi^T\\Phi)^{-1}\\Phi^T$. Which of them takes any vector $v$ and projects it onto the space spanned by the columns of $\\Phi$?"}


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

{"reasoning": "Reasoning for Multinomial Theorem: The provided text details the multinomial theorem, explaining its formula and application in counting the number of ways to distribute objects (or in this case, people) into groups. This theorem is directly relevant to the problem at hand, which involves distributing a set number of people into specified groups. The solution to the problem leverages the concept of choosing combinations of people for each group, which is an application of the multinomial coefficients described in the theorem's definition. Although the solution does not explicitly mention the multinomial theorem by name when calculating the combinations, the process of selecting groups of certain sizes from a total is inherently related to the principles outlined in the multinomial theorem. Therefore, the text provides foundational knowledge on the theorem that could help a student understand the underlying mathematical principles used in the solution. It presents the theorem's formula and a general overview, which can assist in comprehending how the solution applies combinatorial principles to solve the distribution problem. Consequently, despite the text not walking through a problem identical to the one provided, it gives essential background and conceptual understanding that could aid a student in grasping the combinatorial method used in the solution.\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 directly relevant to the solution of the given problem. The problem solution involves dividing a group of 10 people into 3 committees of specific sizes (3, 4, and 3), and it uses binomial coefficients to calculate the number of ways to achieve this division (${10 \\choose 3} \\times {7 \\choose 4}$). The text deepens the understanding of how multinomial coefficients, which are used to solve exactly this type of problem (dividing a set into subsets of specified sizes), relate to binomial coefficients. Since the solution to the problem relies on the application of binomial coefficients in a step-by-step manner to divide the group, and the text provides theoretical support for understanding how these calculations fit into the broader concept of multinomial coefficients, it is aligned with the reasoning used in the problem's solution. Therefore, the text can help a student who is not familiar with the theorem by showing them the underlying principles that justify the method used in the solution. It bridges the gap between the specific application of binomial coefficients in the solution and the more general concept of dividing a set into groups, which is what the multinomial theorem encompasses.\n\nAnswer: True\n\nReasoning for Product Rule for Counting: The given text introduces the Product Rule for Counting, which states that if an element can be chosen from set \\(S\\) in \\(m\\) ways and another element can be chosen from set \\(T\\) in \\(n\\) ways, then the ordered pair can be chosen in \\(m \\times n\\) ways. This principle of counting is fundamentally used in the solution to the problem, where the number of ways to choose committees from a group of people involves selecting subsets of different sizes from a larger set, and the selections are multiplied together to get the total number of combinations. Although the solution explicitly uses combinations (denoted as \\({n \\choose k}\\)) to solve the problem and does not mention the Product Rule for Counting by name, the underlying logic of multiplying the number of ways to make each selection reflects the principle stated in the text. Therefore, even though the solution is framed in terms of combinations, the reasoning about multiplying possibilities to arrive at a total number of outcomes is very much in line with the Product Rule for Counting described in the text. The text can help a student understand the rationale behind the multiplication step in the solution, even if it does not directly apply the multinomial theorem. Hence, the text could be helpful for a student not familiar with the theorem used in the solution because it provides a foundational counting principle that underlies the combinatorial reasoning 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 organizing a school project and you have 10 students to divide into 3 project teams with specific roles. The first team needs 3 students to research, the second team requires 4 students to create the presentation, and the third team also needs 3 students to handle the project documentation. In how many different ways can you assign the students to these teams?"}


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

{"reasoning": "Reasoning for Linear Bound Lemma: The given text is essentially a detailed explanation and proof of the theorem that underpins the solution to the problem statement. The problem asks for the maximum number of edges in a simple planar graph with 30 vertices. The solution uses Euler's formula for planar graphs and combines it with a reasoning about the minimum number of edges per face to derive the formula E \u2264 3V - 6, which is then used to calculate the maximum number of edges for 30 vertices as 84.\n\nThe theorem mentioned in the problem is the \"maximal planar graph,\" and the theorem definition provided explains that a maximal planar graph with 'n' vertices has exactly 3n - 6 edges. This definition is directly related to solving the problem since it provides the exact formula used in the solution to find the maximum number of edges.\n\nThe text provided is a proof of the Linear Bound Lemma, which states that for a simple connected planar graph \\(G_n\\) with \\(n \\ge 3\\) vertices, the number of edges \\(m\\) is bounded by \\(m \\le 3n - 6\\). This lemma directly supports the reasoning in the solution by establishing the theoretical maximum number of edges in a simple planar graph based on the number of its vertices. It uses Euler's formula and reasoning about the minimum number of edges per face, which aligns with how the solution approaches the problem. Thus, the text would help a student understand the theoretical foundation for the solution's formula and how it is derived, making it easier for them to grasp why the solution is correct.\n\nAnswer: True\n\nReasoning for Euler's Theorem for Planar Graphs: The given text provides a detailed exposition of Euler's theorem for planar graphs, which is directly relevant to solving the given problem. The solution to the problem utilizes Euler's formula (V - E + F = 2) as a foundational step in deriving the maximum number of edges in a simple planar graph with 30 vertices. The text explains and proves Euler's theorem, which lays the groundwork for understanding how the number of vertices, edges, and faces in a planar graph are related\u2014an understanding that is crucial for anyone trying to solve the problem without prior knowledge of the theorem.\n\nMoreover, the text and the solution apply the theorem in a related way: both use the formula derived from Euler's theorem to establish a relationship between the components of a planar graph. Although the text primarily focuses on the theorem itself rather than its application in solving specific problems like the maximum number of edges in a planar graph, the fundamental knowledge it provides about Euler's theorem and its proof can assist a student in comprehending the solution's approach. By understanding the theorem's basis, a student could more easily grasp why the solution's steps lead to determining the maximum number of edges in the graph.\n\nTherefore, since the text explicates the theorem that is crucial for understanding the solution to the problem, and because this theorem is applied in a similar way in both the text and the solution, the text can be considered helpful for a student trying to understand and 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\nImagine you're designing a park with 30 different points of interest (like benches, sculptures, and fountains) and you want to connect them with pathways so that you can walk from any one point to any other without leaving the pathways. However, you want to keep the design simple and open, so you decide that no two pathways will cross each other, and you'll use the minimum number of pathways needed to keep everything connected. What's the maximum number of pathways you can build?"}


Suppose you have a square sandbox of dimension 1 inch by 1 inch. You have 19 flags that you are going to plant anywhere in the sandbox. You also have a circular hoop with a radius of $\frac{\sqrt 2}{6}$, which you want to put in the sandbox to include as many flags as possible. What is the minimum number of flags you can ensure will always be within the hoop no matter how the flags are planted in the sandbox?

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

{"reasoning": "Reasoning for Pigeonhole Principle: The text provided gives a formal presentation of the pigeonhole principle, focusing on a partition of a finite set $S$ into $k$ subsets and establishing that at least one subset must contain at least $\\ceiling{\\frac{n}{k}}$ elements. This mathematical proof is grounded in the concept of division and the ceiling function, employing a proof by contradiction to show that it's impossible for all subsets to have fewer than $\\ceiling{\\frac{n}{k}}$ elements without violating the initial condition that $S$ is a finite set with cardinality $n$.\n\nThe problem statement deals with placing 19 points within a unit square and asks for the minimum number of points that can be covered with a circle of a given diameter. The solution utilizes the pigeonhole principle to divide the square into 9 smaller squares, arguing that since there are more points (19) than squares (9), at least one square must contain 3 points, which can then be covered by the circle.\n\nThe text, while not directly addressing the geometric context of the problem, lays down the fundamental reasoning of the pigeonhole principle that underpins the solution. It describes the necessary mathematical background and reasoning steps that someone unfamiliar with the pigeonhole principle would need to understand and apply it in various contexts, including the geometric scenario presented in the problem statement. Therefore, the text can be considered helpful for a student trying to understand how the pigeonhole principle is applied to solve the given problem, as it strengthens the conceptual understanding of why at least one of the smaller squares must contain at least 3 points.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nSuppose you have a square sandbox of dimension 1 inch by 1 inch. You have 19 flags that you are going to plant anywhere in the sandbox. You also have a circular hoop with a radius of $\\frac{\\sqrt 2}{6}$, which you want to put in the sandbox to include as many flags as possible. What is the minimum number of flags you can ensure will always be within the hoop no matter how the flags are planted in the sandbox?"}


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

{"reasoning": "Reasoning for Number of Distinct Parenthesizations on Word: The text provided discusses the number of distinct parenthesizations of a word using Catalan numbers, offering a formula for calculating these numbers. Similarly, the solution to the problem statement uses the concept of Catalan numbers to determine the number of paths from the origin to a point on a grid without crossing a specified line, applying a formula for calculating these numbers. Both the text and the solution apply the concept of Catalan numbers, although in different contexts (parenthesizations of words in the text and paths on a grid in the problem statement). The reasoning steps involve the application of Catalan numbers through a formula, which is akin to how the solution applies the theorem to solve the grid path problem. This indicates that the text uses very similar reasoning steps as the solution and applies the theorem in a related way. Therefore, the text can be helpful in understanding how to apply Catalan numbers to solve problems involving combinatorial structures, which could assist a student who is not familiar with the theorem in ultimately solving the grid path 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 creating a path in a video game from the bottom left corner of a 10x10 square grid to the top right corner. You can only move either one step to the right or one step upwards in each move. However, your path cannot cross over the diagonal line that runs from the bottom left to the top right corner of the grid. How many different paths can you create that follow these rules?"}


Imagine you're a magician who has created a spell that can change the temperature in any room to a specific degree, no matter where it is in the world. This is represented by f = u(z) + iv(z), an entire function in complex plane C. You've discovered that no matter how you cast this spell, the change in temperature you can achieve is always below a certain level, let's say 100 degrees Fahrenheit. I.e., the modulus of the function is bounded. If this spell works everywhere on Earth without fail, does this mean your spell can only ever achieve one specific temperature change, no matter where you use it or how — is the function constant?

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

{"reasoning": "Reasoning for Entire Function with Bounded Real Part is Constant: The text provided elaborates on a theorem that directly relates to the problem statement by considering an entire function with a bounded real part and proving that such a function must be constant. This directly mirrors the application of Liouville's theorem as described in the solution, which is used to conclude that a bounded entire function must be constant. The theorem definition given matches the concept utilized in the text, where it talks about boundedness and entire functions in the context of complex analysis, leading to the conclusion that the function in question is constant. The text follows a similar line of reasoning by considering a transformation of the original function to demonstrate its boundedness and then applying Liouville's theorem to conclude constancy. This approach, albeit slightly more complex due to the introduction of an exponential function, essentially serves the same purpose of demonstrating the applicability of Liouville's theorem to show that a function under certain boundedness conditions must be constant. Therefore, the text can be helpful in understanding and solving the given problem, as it employs a related application of the theorem and explains the reasoning in a step-by-step manner that could assist a student not familiar with Liouville's theorem.\n\nAnswer: True\n\nReasoning for Liouville's Theorem (Complex Analysis): The provided text outlines a proof of Liouville's theorem, which directly relates to the solution of the problem statement. The theorem states that if a function is entire (holomorphic on the entire complex plane) and bounded, it must be constant. The text elaborates on this by assuming a bounded entire function and then utilizing the Cauchy Integral Formula to prove that the derivative of the function is zero for all z in the complex plane, which implies that the function is constant. This reasoning aligns with the solution to the problem, which relies on Liouville's theorem to argue that the given function f is constant because it is bounded (as indicated by |u(z)| < M for every z). Since the text uses similar reasoning steps as the solution and applies the theorem in a related way to solve the problem, it would be helpful for a student not familiar with Liouville's theorem to understand and solve the given problem.\n\nAnswer: True\n\nReasoning for Removable Singularity at Infinity implies Constant Function: The provided text discusses a scenario where an entire function $f : \\mathbb{C} \\to \\mathbb{C}$ has a removable singularity at infinity and concludes that such a function must be constant. This reasoning is closely related to the problem statement and its solution, which relies on Liouville's theorem to assert that an entire function bounded in the complex plane must be constant. Both the solution to the problem and the text use the concept of boundedness as a critical point for leveraging Liouville's theorem to establish that a function is constant. Although the text approaches the topic from the perspective of removable singularities, it converges on the same fundamental principle of boundedness that is central to Liouville's theorem. Thus, the text applies the theorem in a related way to solve a problem that hinges on the same underlying mathematical principles as those in the given problem statement and solution. Therefore, the text could be helpful in understanding and solving the problem, especially in illustrating how boundedness, either directly stated or implied through conditions like removable singularities at infinity, leads to the function being constant according to Liouville'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 a magician who has created a spell that can change the temperature in any room to a specific degree, no matter where it is in the world. This is represented by f = u(z) + iv(z), an entire function in complex plane C. You've discovered that no matter how you cast this spell, the change in temperature you can achieve is always below a certain level, let's say 100 degrees Fahrenheit. I.e., the modulus of the function is bounded. If this spell works everywhere on Earth without fail, does this mean your spell can only ever achieve one specific temperature change, no matter where you use it or how \u2014 is the function constant?"}


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

{"reasoning": "Reasoning for Modulus of Limit: The provided text discusses a theorem related to the modulus (or absolute value) of a sequence converging to a limit, stating that the limit of the modulus of a sequence as n approaches infinity is equal to the modulus of the limit of that sequence. This concept is indeed relevant to the problem statement, which involves the limit of the absolute value of a function as x approaches a certain value c and asks for the limit of the function itself as x approaches c. The solution to the problem relies on understanding that if the absolute value of f(x) approaches 0, then f(x) itself must approach 0, which aligns with the reasoning in the text about the modulus of a convergent sequence. However, the text explicitly applies its theorem in the context of sequences (as n approaches infinity), not in the context of functions as x approaches a certain value. The underlying mathematical principle about limits and absolute values is closely related, but the direct application is in a slightly different context (sequences versus functions). Despite this, the reasoning about limits and absolute values can be analogous and potentially helpful for someone trying to understand the relationship between the limit of the absolute value of a function and the limit of the function itself. Therefore, while the context is different (sequences vs. functions), the fundamental reasoning about limits and absolute values is sufficiently similar to assist in understanding the solution to the problem.\n\nAnswer: True\n\nReasoning for Squeeze Theorem/Functions: The given text elaborates on the Squeeze Theorem, which is mentioned as the theorem used in the solution to the problem. The problem statement deals with finding the limit of a function \\(f(x)\\) as \\(x\\) approaches \\(c\\), given that the limit of the absolute value \\(|f(x)|\\) as \\(x\\) approaches \\(c\\) is 0. The solution concludes that the limit of \\(f(x)\\) as \\(x\\) approaches \\(c\\) is also 0.\n\nThe text provides a detailed explanation and proof of the Squeeze Theorem, which states that if for all \\(x\\) in an interval around \\(a\\) (except possibly at \\(a\\)), \\(g(x) \\leq f(x) \\leq h(x)\\) and the limits of \\(g(x)\\) and \\(h(x)\\) as \\(x\\) approaches \\(a\\) are both \\(L\\), then the limit of \\(f(x)\\) as \\(x\\) approaches \\(a\\) is also \\(L\\). This is directly applicable to the given problem, where the function \\(f(x)\\) is \"squeezed\" between \\(0\\) and \\(0\\) (since the absolute value \\(|f(x)|\\) approaches \\(0\\), \\(f(x)\\) can be thought of as being between \\(0\\) and \\(0\\)), leading to the conclusion that the limit of \\(f(x)\\) is \\(0\\).\n\nFurthermore, the text discusses a special case that mirrors the conditions of the problem statement, where \\(L = 0\\) and it's shown through a proof that the limit of \\(f(x)\\) as \\(x\\) approaches \\(a\\) (or \\(c\\) in the problem statement) is \\(0\\). This closely aligns with the reasoning steps and application of the theorem in the solution to the given problem.\n\nTherefore, the text is helpful in understanding and solving the given problem as it uses very similar reasoning steps as the solution and applies the Squeeze Theorem in a related way as the solution applies the theorem to solve the problem.\n\nAnswer: True\n\nReasoning for Squeeze Theorem/Sequences/Real Numbers: The given text presents a detailed explanation of the Squeeze (Sandwich) Theorem in the context of sequences, focusing on sequences that converge to a limit \\(l\\). It outlines that if a sequence \\(\\{x_n\\}\\) is always between two other sequences \\(\\{y_n\\}\\) and \\(\\{z_n\\}\\) that both converge to the same limit \\(l\\), then the sequence \\(\\{x_n\\}\\) must also converge to that limit \\(l\\). This explanation mirrors the core principle of the Squeeze Theorem as defined for functions within calculus, which is about the behavior of functions rather than sequences. However, the underlying logic of comparison and limits is fundamentally similar.\n\nThe problem statement and solution provided deal with the limit of a function \\(f(x)\\) as \\(x\\) approaches a certain point \\(c\\), specifically when the absolute value of that function approaches 0. The solution leverages the understanding that if \\(|f(x)|\\) approaches 0, then \\(f(x)\\) itself must approach 0, aligning with the general intuition behind the Squeeze Theorem without directly invoking the comparison between two other functions or bounds.\n\nAlthough the text discusses sequences and their limits rather than functions and their limits at a point, the conceptual bridge between sequences and functions regarding limits and the behavior of being \"squeezed\" between two converging behaviors is intact. This demonstrates the application of the theorem in a closely related but distinct mathematical setting (sequences vs. functions). Given this, the text can help a student unfamiliar with the theorem understand the foundational idea behind the solution to the problem: that the behavior of something (whether a sequence or a function) can be determined by its bounds.\n\nTherefore, despite the difference in context (sequences vs. continuous functions), the text provides valuable insight into the reasoning and principles underlying the Squeeze Theorem, which can assist in understanding and solving the given problem by illustrating the concept of limits and the behavior of functions or sequences within those limits.\n\nAnswer: True", "gold_answer": "N/A", "query_with_instruction": "Given a math question, find theorems that are relevant to it\nImagine you're observing the brightness of a light bulb as it's being adjusted. As the control knob is turned to a specific setting, the absolute change in brightness approaches zero. What does the change in brightness become as the knob reaches that specific setting?"}


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

{"reasoning": "Reasoning for Ramsey's Theorem: The text provides an in-depth explanation of Ramsey's theorem, which is the exact theorem used in the solution of the given problem. Ramsey's theorem states that for any pair of positive integers \\(r, s\\), there exists a least positive integer \\(R(r, s)\\) such that any complete graph on \\(R(r, s)\\) vertices, with edges colored in two colors, must contain either a complete subgraph on \\(r\\) vertices which is entirely one color or a complete subgraph on \\(s\\) vertices which is entirely the other color. This theorem underpins the solution to the problem statement by determining the smallest \\(n\\) (in this case, \\(R(3,6) = 18\\)) that guarantees the existence of either a red triangle or a blue 6-clique in a two-color edge coloring of a complete graph. \n\nThe text not only restates Ramsey's theorem in a manner directly relevant to the problem statement but also expands on the theorem's implications for any number of colors and different sized subgraphs, which provides a broader understanding necessary for solving the problem. Furthermore, the proof section in the text begins to outline an approach to demonstrate the existence of such a Ramsey number through inductive reasoning, which could further assist a student in understanding how such problems can be approached and solved, even if the complete proof is not provided.\n\nGiven that the text directly addresses the theorem used in the solution and explains it in a manner that is relevant to solving the problem, it is clear that 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 Ramsey Number: The text provides a definition and explanation of Ramsey's Theorem and the concept of Ramsey numbers, which is directly related to the solution of the problem statement. The problem involves coloring the edges of a complete graph with two colors and finding the smallest n that guarantees either a red triangle or a blue 6-clique. This is a direct application of Ramsey's Theorem, which deals with finding monochromatic complete subgraphs in a sufficiently large complete graph colored with a given number of colors. The text explains that for any given number of colors and given integers (in this case, the sizes of the monochromatic subgraphs we are looking for), there is a Ramsey number that guarantees the existence of a complete subgraph of the required size in one of the colors. This explanation is essential for understanding the solution to the problem, as the solution references the Ramsey number R(3,6) = 18, which is derived from the theorem's principles. Therefore, the text can help a student who is not familiar with Ramsey's Theorem to understand and ultimately solve the given problem by applying the theorem in a related way as the solution does.\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 party and you have a game that involves connecting every pair of guests with either a red or a blue string. Each guest is connected to every other guest. You want to find out the minimum number of guests needed so that you're guaranteed to have either a group of three guests all connected by red strings or a group of six guests all connected by blue strings. What is the smallest number of guests that will ensure this happens?"}


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

{"reasoning": "Reasoning for Finite Semigroup Equal Elements for Different Powers: The text presents a theorem and its proof related to finite semigroups, specifically stating that for any element \\(x\\) in a finite semigroup, there exist natural numbers \\(m\\) and \\(n\\), where \\(m \\ne n\\), such that \\(x^m = x^n\\). The proof utilizes the Pigeonhole Principle by arguing that because the semigroup has a finite number of elements, listing the positive powers of any element must eventually lead to repetitions, demonstrating an application of the Pigeonhole Principle.\n\nWhile the domain of application (semigroups and their elements' powers) in the text is quite different from the problem statement concerning people and their birthday months, the underlying mathematical concept used in both the solution to the problem and the text is the Pigeonhole Principle. Both applications involve a scenario where there are more items (people or element powers) than containers (months or unique element powers), leading to the conclusion that there must be at least one container with more than one item.\n\nTherefore, even though the specific details and areas of application are different, the text can be helpful to a student not familiar with the Pigeonhole Principle. It provides an alternate example of how this principle can be applied to prove the existence of repetitions or duplicates in a finite set, which is a closely related reasoning step to solving the given problem. Thus, it can assist in understanding the application of the Pigeonhole Principle in a broader sense, beyond the specific context of birthdays and months.\n\nAnswer: True\n\nReasoning for Pigeonhole Principle: The provided text presents a theorem and its proof that directly correlates with the application of the Pigeonhole Principle mentioned in both the problem solution and the theorem definition. The theorem in the text states that if you divide a finite set into k subsets, at least one of these subsets must contain at least the ceiling of n/k elements. This aligns with the problem's approach, which leverages the Pigeonhole Principle to deduce that with 13 people (n+1, where n = 12 months), there must be at least one month (a \"subset\") where two people share a birthday month. The solution to the problem explicitly uses the Pigeonhole Principle by illustrating a scenario where more entities (people) are placed into a fixed number of categories (months) than the categories can hold without overlap, guaranteeing repetition in at least one category. The text explains a generalized form of this principle through a mathematical proof that reinforces the understanding and reasoning behind the principle's application in the problem's solution. Therefore, the text can be helpful for a student not familiar with the theorem to understand and solve the problem since it illustrates very similar reasoning steps and applies the theorem in a related way as the solution applies it.\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 unique social media challenge where participants must submit photos of themselves born in different months. To ensure the challenge's uniqueness, you aim to have participants representing each month of the year. What is the minimum number of participants you need to guarantee that at least two participants were born in the same month?"}