category large_string | source large_string | input large_string | output large_string |
|---|---|---|---|
math | OpenMathReasoning | Find all prime numbers \( p \) such that \( p^3 + 1 = k^2 \). | <think>
Okay, let's see. I need to find all prime numbers p where p cubed plus one equals a perfect square k squared. So the equation is p³ + 1 = k². Hmm, primes are numbers greater than 1 that have no divisors other than 1 and themselves. So possible primes to check could be 2, 3, 5, 7, etc. Maybe start with small pri... |
math | OpenMathReasoning | Determine the locus of the complex number \( z \) that satisfies the equation \( \left| z + 2 - 6i \right| + \left| z - 2 - 3i \right| = 4 \). | <think>
Okay, so I need to find the locus of the complex number z that satisfies the equation |z + 2 - 6i| + |z - 2 - 3i| = 4. Hmm, let's break this down.
First, I remember that in the complex plane, a complex number z can be represented as x + yi, where x and y are real numbers. So maybe I can rewrite the equation i... |
math | NuminaMath-CoT | A train 18 m long passes a pole in 9 seconds. How long will it take to pass a bridge 36 m long? | <think>
Okay, so I need to figure out how long it takes for a train that's 18 meters long to pass a bridge that's 36 meters long. The problem already says that it takes 9 seconds for the train to pass a pole. Let me start by understanding what that means.
When the train passes a pole, the entire length of the train ha... |
math | OpenMathReasoning | Given a polynomial of degree \( n \) with roots \( x_1, x_2, \ldots, x_n \), how can one compute the sum of the roots raised to a power \( m \) when \( m > n \)? Newton sums do not apply since there are no coefficients for negative degree terms, and Newton-Girard formulas are only for symmetric polynomials. | <think>
Okay, let's see. I need to figure out how to compute the sum of the roots of a polynomial raised to a power m when m is greater than the degree n of the polynomial. The problem mentions that Newton sums don't apply here because there are no coefficients for negative degrees, and Newton-Girard formulas are for s... |
math | OpenMathReasoning | Compute the largest squarefree positive integer $n$ such that $Q(-n)$ is a unique factorization domain. | <think>
Okay, let's see. I need to find the largest squarefree positive integer n such that Q(-n) is a unique factorization domain. Hmm, right, quadratic fields. So, first, I remember that for a quadratic field Q(√d), where d is a square-free integer, the ring of integers has unique factorization if and only if the cla... |
math | NuminaMath-CoT | Compute the definite integral:
$$
\int_{-5 / 3}^{1} \frac{\sqrt[3]{3x+5} + 2}{1 + \sqrt[3]{3x+5}} \, dx
$$ | <think>
Okay, so I need to compute the definite integral from -5/3 to 1 of [cube root(3x + 5) + 2] divided by [1 + cube root(3x + 5)] dx. Hmm, that looks a bit complicated, but maybe substitution can help here.
First, let me write down the integral again to get a clear view:
∫ from -5/3 to 1 of [ (³√(3x + 5) + 2 ) / ... |
math | OpenMathReasoning | For what complex numbers $z$ does the limit $\lim_{n\to \infty} \left( \frac{z}{\bar{z}} \right)^n$ exist? | <think>
Okay, let's try to figure out for which complex numbers \( z \) the limit \( \lim_{n\to \infty} \left( \frac{z}{\overline{z}} \right)^n \) exists. Hmm, first, I need to recall that for a complex number \( z \), its complex conjugate is \( \overline{z} \). So, the expression inside the limit is \( \left( \frac{z... |
math | OpenMathReasoning | Define the new nested factorial function recursively by \( f_1(x) = x! \) and \( f_n(x) = (f_{n-1}(x))! \) for \( n \geq 2 \). Suppose you have 5 objects. You find all the possible ways to order these 5 objects in a list, where order matters. Then, with all of these possible lists, you create new lists with all the pos... | <think>
Okay, let me try to work through this problem step by step. So, the question is about the nested factorial function and figuring out the values of j and k such that the total number of new, distinct lists can be represented by f_j(k). Then, we need to find j + k.
First, let me make sure I understand the probl... |
math | OpenMathReasoning | Let ABCDEF be a regular hexagon. The distance between the midpoint of AB and DE is 12. What is the side length of the hexagon? | <think>
Okay, so I need to find the side length of a regular hexagon where the distance between the midpoint of AB and DE is 12. Hmm, let's start by recalling some properties of a regular hexagon.
First, a regular hexagon can be divided into six equilateral triangles, all with side length equal to the hexagon's side.... |
math | NuminaMath-CoT | There is a quadratic function \( f(x) \) that satisfies the conditions \( f\left(x^{2}\right) = f(f(x)) = (f(x))^{2} \). How many such quadratic functions exist?
A. 0
B. 1
C. 2
D. Infinitely many | <think>
Okay, so I need to figure out how many quadratic functions satisfy the conditions f(x²) = f(f(x)) = (f(x))². Let me start by understanding the problem.
First, a quadratic function is of the form f(x) = ax² + bx + c, where a ≠ 0. The problem states that two equations must hold for all x: f(x²) = f(f(x)) and f(f... |
math | NuminaMath-CoT | In a certain country, there are 30 cities, and each city is connected to every other city by a road.
What is the maximum number of roads that can be closed for repairs while still allowing travel between any two cities? | <think>
Okay, so I need to figure out the maximum number of roads that can be closed for repairs in a country with 30 cities, where each city is connected to every other city by a road. The condition is that even after closing those roads, it should still be possible to travel between any two cities. Hmm, let's start b... |
math | OpenMathReasoning | Find all positive integer solutions to the equation $x^4 - 5y^4 = -1$. | <think>
Okay, so I need to solve the equation \( x^4 - 5y^4 = -1 \) for positive integers x and y. Hmm, let's see. Let me start by understanding what the equation is saying. It’s a Diophantine equation because we’re looking for integer solutions. The left side is x to the fourth power minus five times y to the fourth p... |
math | OpenMathReasoning | Solve the equation $\lg^4(x-1)^2 + \lg^2(x-1)^3 = 25$. | <think>
Okay, so I need to solve the equation lg^4(x-1)^2 + lg^2(x-1)^3 = 25. Hmm, let me start by making sure I understand the notation here. The "lg" usually stands for logarithm base 10, right? So lg(x) is log base 10 of x. Now, the equation has lg^4(x-1)^2. Wait, does that mean [lg((x-1)^2)]^4 or lg[(x-1)^2]^4? I t... |
math | NuminaMath-CoT | In a company of 10 people, 14 pairs of quarrels have occurred. Prove that it is still possible to form a group of three friends. | <think>
Okay, let's see. The problem says that in a company of 10 people, there have been 14 pairs of quarrels. We need to prove that it's still possible to form a group of three friends. Hmm, so friends are people who haven't quarreled, right? So we need to show that even with 14 quarrel pairs, there's still a trio of... |
math | NuminaMath-CoT | If \(3 \leq p \leq 10\) and \(12 \leq q \leq 21\), then the difference between the largest and smallest possible values of \(\frac{p}{q}\) is
(A) \(\frac{29}{42}\)
(B) \(\frac{29}{5}\)
(C) \(\frac{19}{70}\)
(D) \(\frac{19}{12}\)
(E) \(\frac{19}{84}\) | <think>
Okay, so I need to solve this problem where they give me ranges for p and q, and I have to find the difference between the largest and smallest possible values of p/q. Let's see, p is between 3 and 10 inclusive, and q is between 12 and 21 inclusive. The options are given, so maybe I can figure it out step by st... |
math | OpenMathReasoning | Let \( S \) be a set of \( n \) elements. Find the number of ways to choose \( n \) distinct non-empty subsets \( X_1, X_2, \ldots, X_n \) of \( S \) such that \( X_1 \subset X_2 \subset \cdots \subset X_n \). | <think>
Okay, so I have this problem where I need to find the number of ways to choose n distinct non-empty subsets X₁, X₂, ..., Xₙ of a set S with n elements, such that each subset is properly contained in the next, like X₁ ⊂ X₂ ⊂ ... ⊂ Xₙ. Hmm, let's see. Let me try to break this down.
First, S has n elements. Each ... |
math | OpenMathReasoning | Let \( A = \{1, 2, 3, \ldots, n\} \) be a set. Find the number of \( k \)-element subsets of \( A \) such that each subset has no consecutive elements. | <think>
Okay, so I need to find the number of k-element subsets of the set A = {1, 2, 3, ..., n} such that no two elements in the subset are consecutive. Hmm, let me think about how to approach this.
First, I remember that when dealing with subsets that have no consecutive elements, there's a combinatorial approach in... |
math | OpenMathReasoning | On a hot day, Megan likes to eat a Popsicle every 15 minutes. Assuming she keeps up that rate of consumption, how many Popsicles can Megan finish in 4 hours and 30 minutes? | <think>
Okay, so the problem is about Megan eating popsicles every 15 minutes, and we need to figure out how many she can finish in 4 hours and 30 minutes. Let me start by breaking this down.
First, I need to convert the total time she's eating into minutes because the rate is given in minutes. The time given is 4 hou... |
math | OpenMathReasoning | How large should $n$ be so that the Trapezoidal Estimate for $\int_1^3 \dfrac{1}{x} \, dx$ is accurate within $0.001$? | <think>
Okay, so I need to figure out how large n should be for the Trapezoidal Estimate to approximate the integral from 1 to 3 of 1/x dx with an error less than 0.001. Hmm, trapezoidal rule error bound. Let me remember. The formula for the error in the trapezoidal rule is given by something like (b - a)^3 divided by ... |
math | OpenMathReasoning | Find the value of \(76 \cot {(\cot^{-1}2+\cot^{-1}4+\cot^{-1}6+\cot^{-1}8)}\). | <think>
Okay, so I need to find the value of 76 times the cotangent of the sum of four inverse cotangents: cot⁻¹2 + cot⁻¹4 + cot⁻¹6 + cot⁻¹8. Hmm, that seems a bit complex. Let me try to break it down step by step.
First, I remember that the cotangent of the sum of angles can be tackled using the cotangent addition fo... |
math | NuminaMath-CoT | Is it possible to place the numbers from 1 to 12 on the edges of a cube so that the sums at all vertices (summing the numbers on the edges originating from the given vertex) are equal? | <think>
Okay, so I need to figure out if I can place the numbers 1 to 12 on the edges of a cube such that the sum at each vertex is the same. Hmm, let me start by understanding the problem better.
A cube has 12 edges and 8 vertices. Each vertex is where three edges meet. So, if I label each edge with a unique number f... |
math | OpenMathReasoning | Evaluate the sum
\[ \sum_{n=2}^\infty \frac{1}{2^n}\tan \frac{\pi}{2^n}. \] | <think>
Okay, so I need to evaluate the infinite sum ∑_{n=2}^∞ (1/2^n) tan(π/2^n). Hmm, infinite series involving trigonometric functions can be tricky. Let me start by recalling some related formulas or series expansions. Maybe there's a telescoping series or a known identity that can help here.
First, let's write ou... |
math | OpenMathReasoning | For each nonnegative integer $n$, define $A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}$. Find the greatest common divisor of the numbers $A_0, A_1, \ldots, A_{1999}$. | <think>
Okay, let's try to solve this problem. So, we have these numbers A_n defined as 2^{3n} + 3^{6n+2} + 5^{6n+2} for each nonnegative integer n. We need to find the greatest common divisor (GCD) of all these A_n from n=0 to n=1999. Hmm, okay. The problem is asking for the GCD of A_0, A_1, ..., A_{1999}.
First, ma... |
math | OpenMathReasoning | There are positive integers \( m \) and \( n \) such that \( m^2 - n = 32 \) and \( \sqrt[5]{m + \sqrt{n}} + \sqrt[5]{m - \sqrt{n}} \) is a real root of the polynomial \( x^5 - 10x^3 + 20x - 40 \). Find \( m + n \). | <think>
Okay, let's try to solve this problem. We have two positive integers m and n such that m² - n = 32. Also, the expression ∛(m + √n) + ∛(m - √n) is a real root of the polynomial x⁵ - 10x³ + 20x - 40. We need to find m + n. Wait, wait, actually the problem says fifth roots, not cube roots. Let me check again. It's... |
math | OpenMathReasoning | Find the number of integral solutions for the equation \(x(3x+1)+2y(y-2x)-2 = 0\). | <think>
Okay, let's see. I need to find the number of integral solutions for the equation x(3x + 1) + 2y(y - 2x) - 2 = 0. Hmm, where do I start? Maybe I should try to rearrange the equation to make it easier to handle. Let's expand the terms first.
Starting with x(3x + 1): that's 3x² + x. Then 2y(y - 2x) would be 2y² ... |
math | NuminaMath-CoT | Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{3+6+9+\ldots+3n}{n^{2}+4}$$ | <think>
Okay, so I need to find the limit as n approaches infinity of the sequence (3 + 6 + 9 + ... + 3n) divided by (n² + 4). Let me think step by step about how to approach this.
First, the numerator is the sum of an arithmetic series. The terms are 3, 6, 9, ..., 3n. Let me recall the formula for the sum of an arith... |
math | NuminaMath-CoT | Prove that if $\sqrt{1+x}+\sqrt{1+y}=2 \sqrt{1+\alpha}$, then $x+y \geqslant 2 \alpha$. | <think>
Okay, so I need to prove that if √(1 + x) + √(1 + y) equals 2√(1 + α), then x + y is at least 2α. Hmm, let me think about how to approach this.
First, maybe I can start by squaring both sides of the equation to eliminate the square roots. Let me try that. If I square both sides, I get:
[√(1 + x) + √(1 + y)]²... |
math | OpenMathReasoning | Given that \(1 < a < b\) and the equations \(\frac{3a}{\log_{a}b} = \frac{b}{2\log_{b}a} = \frac{3a+b}{3}\), find the value of \(5\log_{a}b\). | <think>
Okay, let's tackle this problem step by step. So, we have two variables, a and b, with the condition that 1 < a < b. The problem gives us two equations:
First, (3a)/(log_a b) = (b)/(2 log_b a), and both of these are equal to (3a + b)/3. We need to find the value of 5 log_a b. Hmm, that's a bit to unpack. Let m... |
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