problem stringlengths 11 5.47k | reasoning stringlengths 708 124k | deepseek_solution stringlengths 0 10.3k | ground_truth_solution stringlengths 97 9.16k |
|---|---|---|---|
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 ... |
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... |
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... |
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... |
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... |
When \( a < -1 \), the nature of the roots for the equation
$$
\left(a^{3}+1\right) x^{2}+\left(a^{2}+1\right) x-(a+1)=0
$$
is:
(A) Two negative roots.
(B) One positive root and one negative root, with the absolute value of the negative root being larger.
(C) One positive root and one negative root, with the absolute v... | Okay, so I need to figure out the nature of the roots for the quadratic equation \((a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0\) when \(a < -1\). The options are given as (A) two negative roots, (B) one positive and one negative root with the negative root having a larger absolute value, (C) similar to B but the negative r... | To determine the nature of the roots for the quadratic equation \((a^3 + 1)x^2 + (a^2 + 1)x - (a + 1) = 0\) when \(a < -1\), we analyze the discriminant, sum, and product of the roots.
1. **Discriminant Analysis**:
- The discriminant \(D\) is given by:
\[
D = (a^2 + 1)^2 - 4(a^3 + 1)(-(a + 1))
\]
... |
Given the quadratic equation:
\[
(a^3 + 1) x^2 + (a^2 + 1) x - (a + 1) = 0
\]
where \( a < -1 \).
1. **Identify the coefficients and analyze their signs:**
- \( a^3 + 1 \)
- \( a^2 + 1 \)
- \( -(a+1) \)
2. **Signs of the coefficients under the condition \( a < -1 \):**
- Since \( a < -1 \), we ha... |
"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) |
"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) |
"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) |
"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) |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 5