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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 ... | math | numina_math | null | null | <think>
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 ... | [
{
"from": "human",
"value": "Let \\( a, b, c \\) be positive real numbers. Prove that\n\n$$\n\\frac{1}{a(1+b)}+\\frac{1}{b(1+c)}+\\frac{1}{c(1+a)} \\geq \\frac{3}{1+abc},\n$$\n\nand that equality occurs if and only if \\( a = b = c = 1 \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need... |
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... | math | numina_math | null | null | <think>
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 tha... | [
{
"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... |
"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) | math | numina_math | null | null | "<think>\nOkay, so I have this geometry problem here. Let me try to understand it step by step. The (...TRUNCATED) | [{"from":"human","value":"In triangle $ABC$, $CA = CB$, and $D$ is the midpoint of $AB$. Line $EF$ p(...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) | math | numina_math | null | null | "<think>\nOkay, so I need to figure out which of the given options is correct by comparing the sizes(...TRUNCATED) | [{"from":"human","value":"Let \\( p = 2^{3009}, q = 3^{2006}, \\) and \\( r = 5^{1003} \\). Which of(...TRUNCATED) |
"When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left(a^{3}+1\\right) x^{2}+\\l(...TRUNCATED) | "Okay, so I need to figure out the nature of the roots for the quadratic equation \\((a^3 + 1)x^2 + (...TRUNCATED) | "To determine the nature of the roots for the quadratic equation \\((a^3 + 1)x^2 + (a^2 + 1)x - (a +(...TRUNCATED) | "\nGiven the quadratic equation:\n\n\\[\n(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0\n\\]\n\nwhere \\((...TRUNCATED) | math | numina_math | null | null | "<think>\nOkay, so I need to figure out the nature of the roots for the quadratic equation \\((a^3 +(...TRUNCATED) | [{"from":"human","value":"When \\( a < -1 \\), the nature of the roots for the equation\n$$\n\\left((...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) | math | numina_math | null | null | "<think>\nOkay, let's see. So, we have four points A, B, C, D in a plane. Then there are midpoints o(...TRUNCATED) | [{"from":"human","value":"Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, (...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) | math | numina_math | null | null | "<think>\nOkay, let's see. The problem is asking how many ordered triples (x, y, z) there are, where(...TRUNCATED) | [{"from":"human","value":"Given that \\(1 \\leq x, y, z \\leq 6\\), how many cases are there in whic(...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) | math | numina_math | null | null | "<think>\nOkay, let's try to tackle this problem. So, we need to find all prime numbers \\( p \\) su(...TRUNCATED) | [{"from":"human","value":"Find all prime numbers \\( p \\) such that for any prime number \\( q < p (...TRUNCATED) |
"Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of triangle $ABC$. It turns ou(...TRUNCATED) | "Okay, so I need to prove that triangle ABC is isosceles given some conditions about the perimeters (...TRUNCATED) | "Given triangle \\(ABC\\) with points \\(M\\) on \\(AB\\) and \\(N\\) on \\(BC\\), we know the perim(...TRUNCATED) | "\n1. Denote the perimeter with the letter $P$. According to the conditions of the problem, we know (...TRUNCATED) | math | numina_math | null | null | "<think>\nOkay, so I need to prove that triangle ABC is isosceles given some conditions about the pe(...TRUNCATED) | [{"from":"human","value":"Points $M$ and $N$ are taken on the sides $AB$ and $BC$ respectively of tr(...TRUNCATED) |
"Let \\( a, b, c \\) be given complex numbers, with \\( |a+b| = m \\) and \\( |a-b| = n \\). Given t(...TRUNCATED) | "Okay, so I need to prove that the maximum of |ac + b| and |a + bc| is at least mn over the square r(...TRUNCATED) | "Given complex numbers \\(a, b, c\\) with \\(|a + b| = m\\) and \\(|a - b| = n\\) where \\(m \\cdot (...TRUNCATED) | "Given that \\( a, b, c \\) are complex numbers with \\( |a+b|=m \\) and \\( |a-b|=n \\), and \\( mn(...TRUNCATED) | math | numina_math | null | null | "<think>\nOkay, so I need to prove that the maximum of |ac + b| and |a + bc| is at least mn over the(...TRUNCATED) | [{"from":"human","value":"Let \\( a, b, c \\) be given complex numbers, with \\( |a+b| = m \\) and \(...TRUNCATED) |
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