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Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p < q < r \)
(B) \( p < r < q \)
(C) \( q < p < r \)
(D) \( r < p < q \)
(E) \( q < r < p \) | Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q, and r where p = 2^3009, q = 3^2006, and r = 5^1003. Hmm, comparing exponents with different bases and exponents... This might be tricky. Let me think about how to approach this.
First, I remember that comparing numbers ... | To determine which of the given statements is true, we need to compare the sizes of \( p = 2^{3009} \), \( q = 3^{2006} \), and \( r = 5^{1003} \).
First, we express each number with the same exponent by noting that:
- \( p = 2^{3009} = (2^3)^{1003} = 8^{1003} \)
- \( q = 3^{2006} = (3^2)^{1003} = 9^{1003} \)
- \( r =... | Given the values:
\[ p = 2^{3009}, \quad q = 3^{2006}, \quad r = 5^{1003} \]
1. Express \( p \) and \( q \) in terms of powers of the same base:
\[ p = 2^{3009} = 2^{3 \times 1003} = (2^3)^{1003} = 8^{1003} \]
\[ q = 3^{2006} = 3^{2 \times 1003} = (3^2)^{1003} = 9^{1003} \]
Note: \( r = 5^{1003} \) is already... | true | The solution provided correctly follows the steps to compare the values of the three numbers using both exponential form and logarithmic calculations, confirming that the order is indeed \( r < p < q \). Both the solution and the ground truth reached the same conclusion, producing the correct final boxed answer. | **Problem:** Let \( p = 2^{3009}, q = 3^{2006}, \) and \( r = 5^{1003} \). Which of the following statements is true?
(A) \( p < q < r \)
(B) \( p < r < q \)
(C) \( q < p < r \)
(D) \( r < p < q \)
(E) \( q < r < p \)
**Reasoning:** To determine which option is correct, we need to compare the sizes of \( p, ... | 4 | [
{
"from": "human",
"value": "Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements is true?\n(A) \\( p < q < r \\)\n(B) \\( p < r < q \\)\n(C) \\( q < p < r \\)\n(D) \\( r < p < q \\)\n(E) \\( q < r < p \\)"
},
{
"from": "gpt",
"value": "**Problem:**... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... |
In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \cdot DF = DA^2$. | Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem says: In triangle ABC, CA equals CB, so it's an isosceles triangle with CA = CB. Point D is the midpoint of AB. There's a line EF passing through D such that triangles ABC and EFC share the same incenter. I need to prove t... | In triangle \(ABC\), \(CA = CB\), making it isosceles with \(D\) as the midpoint of \(AB\). Line \(EF\) passes through \(D\) such that triangles \(ABC\) and \(EFC\) share the same incenter. We need to prove that \(DE \cdot DF = DA^2\).
1. **Coordinate Setup**:
- Place \(D\) at the origin \((0, 0)\), \(A\) at \((-a,... | 1. **Identify Key Elements**: Consider \( \triangle ABC \) where \( CA = CB \) and \( D \) is the midpoint of \( AB \). Line \( \mathrm{EF} \) passes through \( D \) such that \( \triangle ABC \) and \( \triangle \mathrm{EFC} \) share the same incircle (inscribed circle).
2. **Given Common Incircle**: Let the shared i... | true | The provided solution correctly follows the logical steps necessary to prove that if triangles ABC and EFC share the same incenter and D is the midpoint of AB, then DE·DF = DA². Both solutions have outlined the correct reasoning using the properties of isosceles triangles, incenters, and the Power of a Point, leading t... | Sure! Here’s a more focused version of the reasoning highlighting the key points and critical insights that lead to the final conclusion:
**Problem:** In triangle \(ABC\), \(CA = CB\), and \(D\) is the midpoint of \(AB\). Line \(EF\) passes through point \(D\) such that triangles \(ABC\) and \(EFC\) share the same inc... | 3 | [
{
"from": "human",
"value": "In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ such that triangles $ABC$ and $EFC$ share the same incenter. Prove that $DE \\cdot DF = DA^2$."
},
{
"from": "gpt",
"value": "Sure! Here’s a more focused version of the ... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... |
A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \neq 0$ and $A \neq B$. (The hundreds digit of the product is zero.) In total, how many different... | Okay, so I need to figure out how many different sets of five different odd positive integers multiply together to give a product of the form AB0AB, where A and B are digits, A isn't zero, and A isn't equal to B. Hmm, let's break this down step by step.
First, the product is a five-digit number with the pattern AB0AB.... | To determine the number of different sets of five different odd positive integers whose product is a five-digit number of the form \(AB0AB\), we analyze the prime factorization and possible combinations:
1. **Understanding the Product**:
- The product \(AB0AB\) can be written as \(1001 \times (10A + B)\), where \(1... |
1. **Observe the Structure of \( N \)**:
Let \( N = AB0AB \) and let \( t \) be the two-digit integer \( AB \).
We recognize that \( N = 1001 \cdot t \), where \( 1001 = 11 \cdot 91 = 11 \cdot 7 \cdot 13 \).
Thus,
\[
N = t \cdot 7 \cdot 11 \cdot 13
\]
2. **Formulate the Problem**:
We n... | false | The provided solution computes a total of 25 sets of five different odd positive integers whose product gives a five-digit number of the form AB0AB. However, the ground truth solution arrives at a total of 24 valid combinations after analyzing different cases, indicating a discrepancy of 1 set. The ground truth solutio... | Certainly! Here's the reasoning refined for clarity and emphasis on the critical points that led to the conclusion of the final answer:
---
**Reasoning:**
To determine how many different sets of five distinct odd positive integers greater than 2 yield a product of the form \(AB0AB\), we start by examining the struct... | 1 | [
{
"from": "human",
"value": "A set consists of five different odd positive integers, each greater than 2. When these five integers are multiplied together, their product is a five-digit integer of the form $AB0AB$, where $A$ and $B$ are digits with $A \\neq 0$ and $A \\neq B$. (The hundreds digit of the pro... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... |
Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 a^3 + 27 c - ... | Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value of the expression (2a³ + 27c - 9ab)/λ³ given some conditions on the roots of the cubic polynomial f(x) = x³ + ax² + bx + c. The roots are real, with x₂ - x₁ = λ and x₃ > (x₁ + x₂)/2.
First, I need to recall that for a ... | Given real numbers \(a, b, c\) and a positive number \(\lambda\) such that the polynomial \(f(x) = x^3 + ax^2 + bx + c\) has three real roots \(x_1, x_2, x_3\) with the conditions \(x_2 - x_1 = \lambda\) and \(x_3 > \frac{1}{2}(x_1 + x_2)\), we need to find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).
... |
We begin by analyzing the function \( f(x) = x^3 + a x^2 + b x + c \), which has three real roots \( x_1, x_2, x_3 \). We are given the following conditions:
1. \( x_2 - x_1 = \lambda \)
2. \( x_3 > \frac{1}{2} (x_1 + x_2) \)
We aim to find the maximum value of \( \frac{2a^3 + 27c - 9ab}{\lambda^3} \).
1. **Transfor... | true | Both the provided solution and the ground truth solution arrive at the same final result of \( \frac{3\sqrt{3}}{2} \). The mathematical steps taken in both solutions, including the use of Vieta's formulas, manipulation of the roots, differentiation to find maximum values, and the checks for conditions, are consistent a... | **Problem:** Given real numbers \( a, b, c \) and a positive number \( \lambda \) such that the polynomial \( f(x) = x^3 + a x^2 + b x + c \) has three real roots \( x_1, x_2, x_3 \), and the conditions \( x_2 - x_1 = \lambda \) and \( x_3 > \frac{1}{2}(x_1 + x_2) \) are satisfied, find the maximum value of \( \frac{2 ... | 2 | [
{
"from": "human",
"value": "Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial \\( f(x) = x^3 + a x^2 + b x + c \\) has three real roots \\( x_1, x_2, x_3 \\), and the conditions \\( x_2 - x_1 = \\lambda \\) and \\( x_3 > \\frac{1}{2}(x_1 + x_2) \\) are satis... | Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop w... |
"Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}(...TRUNCATED) | "Okay, so I need to prove this inequality: \\(\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}+\\frac{1}{c(1+a)} (...TRUNCATED) | "To prove the inequality \n\n\\[\n\\frac{1}{a(1+b)} + \\frac{1}{b(1+c)} + \\frac{1}{c(1+a)} \\geq \\(...TRUNCATED) | "1. Consider the given inequality:\n\n\\[\n\\frac{1}{a(1+b)}+ \\frac{1}{b(1+c)}+ \\frac{1}{c(1+a)} \(...TRUNCATED) | true | "The provided solution correctly demonstrates the inequality using appropriate mathematical techniqu(...TRUNCATED) | "Certainly! Here's a more focused and impactful version of the reasoning, emphasizing the key points(...TRUNCATED) | 0 | [{"from":"human","value":"Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) |
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