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Let \( a, b, c \) be positive real numbers. Prove that
$$
\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if \( a = b = c = 1 \). | Okay, so I need to prove this inequality: \(\frac{1}{a(1+b)}+\frac{1}{b(1+c)}+\frac{1}{c(1+a)} \geq \frac{3}{1+abc}\), where \(a, b, c\) are positive real numbers, and equality holds if and only if \(a = b = c = 1\). Hmm, let me try to think through this step by step.
First, I remember that inequalities often involve ... | To prove the inequality
\[
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{1+abc}
\]
for positive real numbers \(a, b, c\) with equality if and only if \(a = b = c = 1\), we proceed as follows:
### Key Steps and Proof:
1. **Substitution for \(abc = 1\):**
- Assume \(abc = 1\). Let \(a = \fr... | 1. Consider the given inequality:
\[
\frac{1}{a(1+b)}+ \frac{1}{b(1+c)}+ \frac{1}{c(1+a)} \geq \frac{3}{1 + abc}
\]
To simplify, we add \( \frac{3}{1 + abc} \) to both sides. The new inequality becomes:
\[
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} + \frac{3}{1 + abc} \geq \frac{6}{1 + abc}
\]
2. Let's ... | true | The provided solution correctly demonstrates the inequality using appropriate mathematical techniques such as substitution, analysis of cases, and the AM-GM inequality. Furthermore, it establishes the equality condition accurately, matching the claims of the ground truth solution. | 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... | [
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},
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"A set consists of five different odd positive integers, each greater than 2. When these five intege(...TRUNCATED) | "Okay, so I need to figure out how many different sets of five different odd positive integers multi(...TRUNCATED) | "To determine the number of different sets of five different odd positive integers whose product is (...TRUNCATED) | "\n1. **Observe the Structure of \\( N \\)**:\n Let \\( N = AB0AB \\) and let \\( t \\) be the tw(...TRUNCATED) | false | "The provided solution computes a total of 25 sets of five different odd positive integers whose pro(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. A set consists of five differe(...TRUNCATED) |
"Given real numbers \\( a, b, c \\) and a positive number \\( \\lambda \\) such that the polynomial (...TRUNCATED) | "Okay, let's try to tackle this problem step by step. The problem is about finding the maximum value(...TRUNCATED) | "Given real numbers \\(a, b, c\\) and a positive number \\(\\lambda\\) such that the polynomial \\(f(...TRUNCATED) | "\nWe begin by analyzing the function \\( f(x) = x^3 + a x^2 + b x + c \\), which has three real roo(...TRUNCATED) | true | "Both the provided solution and the ground truth solution arrive at the same final result of \\( \\f(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Given real numbers \\( a, b, c(...TRUNCATED) |
"In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ passes through point $D$ s(...TRUNCATED) | "Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem s(...TRUNCATED) | "In triangle \\(ABC\\), \\(CA = CB\\), making it isosceles with \\(D\\) as the midpoint of \\(AB\\).(...TRUNCATED) | "1. **Identify Key Elements**: Consider \\( \\triangle ABC \\) where \\( CA = CB \\) and \\( D \\) i(...TRUNCATED) | true | "The provided solution correctly follows the logical steps necessary to prove that if triangles ABC (...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. In triangle $ABC$, $CA = CB$, (...TRUNCATED) |
"Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of the following statements(...TRUNCATED) | "Okay, so I need to figure out which of the given options is correct by comparing the sizes of p, q,(...TRUNCATED) | "To determine which of the given statements is true, we need to compare the sizes of \\( p = 2^{3009(...TRUNCATED) | "Given the values:\n\\[ p = 2^{3009}, \\quad q = 3^{2006}, \\quad r = 5^{1003} \\]\n\n1. Express \\((...TRUNCATED) | true | "The solution provided correctly follows the steps to compare the values of the three numbers using (...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Let \\( p = 2^{3009}, q = 3^{2(...TRUNCATED) |
"When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left(a^{3}+1\\right) x^{2}+\\l(...TRUNCATED) | "Okay, so I need to figure out the nature of the roots for the quadratic equation \\((a^3 + 1)x^2 + (...TRUNCATED) | "To determine the nature of the roots for the quadratic equation \\((a^3 + 1)x^2 + (a^2 + 1)x - (a +(...TRUNCATED) | "\nGiven the quadratic equation:\n\n\\[\n(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0\n\\]\n\nwhere \\((...TRUNCATED) | true | "Both solutions correctly analyze the discriminant, the sum and product of the roots, and arrive at (...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. When \\( a < -1 \\), the natur(...TRUNCATED) |
"Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respectiv(...TRUNCATED) | "Okay, let's see. So, we have four points A, B, C, D in a plane. Then there are midpoints of the seg(...TRUNCATED) | "Given four points \\(A, B, C, D\\) in the plane, we consider the midpoints \\(Q, R, S, T, U, V\\) o(...TRUNCATED) | "To find the distance between the midpoints of $Q U$ and $R V$, let's break down the given informati(...TRUNCATED) | true | "The solution accurately applies the properties of midpoints and parallelograms to determine the dis(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Suppose that $A, B, C, D$ are (...TRUNCATED) |
"Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in which the product of natural (...TRUNCATED) | "Okay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where each of (...TRUNCATED) | "To determine the number of ordered triples \\((x, y, z)\\) where \\(1 \\leq x, y, z \\leq 6\\) and (...TRUNCATED) | "Given the constraints \\(1 \\leq x, y, z \\leq 6\\), we are to find the number of natural number co(...TRUNCATED) | true | "The provided solution correctly uses the principle of inclusion-exclusion to count the number of or(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Given that \\(1 \\leq x, y, z (...TRUNCATED) |
"Prove that for each \\( n \\geq 2 \\), there is a set \\( S \\) of \\( n \\) integers such that \\((...TRUNCATED) | "Okay, so I need to prove that for each integer n ≥ 2, there exists a set S of n integers such tha(...TRUNCATED) | "To prove that for each \\( n \\geq 2 \\), there exists a set \\( S \\) of \\( n \\) integers such t(...TRUNCATED) | "\nGiven the problem:\n> Prove that for each \\( n \\geq 2 \\), there is a set \\( S \\) of \\( n \\(...TRUNCATED) | false | "While the provided solution attempts to construct a valid set S, there are critical missteps in rea(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Prove that for each \\( n \\ge(...TRUNCATED) |
"Find all prime numbers \\( p \\) such that for any prime number \\( q < p \\), if \\( p = kq + r \\(...TRUNCATED) | "Okay, let's try to tackle this problem. So, we need to find all prime numbers \\( p \\) such that f(...TRUNCATED) | "To find all prime numbers \\( p \\) such that for any prime number \\( q < p \\), if \\( p = kq + r(...TRUNCATED) | "\n**Step 1:** Identify the prime numbers \\( p \\) that meet the given conditions: \n\nFor every pr(...TRUNCATED) | true | "The solution provided correctly identifies all prime numbers that satisfy the condition laid out in(...TRUNCATED) | "Your role as an assistant involves thoroughly exploring questions through a systematic long thinkin(...TRUNCATED) | [{"from":"user","value":"Return your final response within \\boxed{}. Find all prime numbers \\( p \(...TRUNCATED) |
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