prompt stringlengths 21 10.2k ⌀ | response stringlengths 134 181k | conversations listlengths 2 2 | problem stringlengths 17 8.21k ⌀ | deepseek_reasoning stringlengths 837 114k ⌀ | deepseek_solution stringlengths 0 13.1k ⌀ | ground_truth_solution stringlengths 1 8.33k ⌀ | domain stringclasses 6
values | source stringclasses 9
values | test_cases stringlengths 33 79.2M ⌀ | starter_code stringlengths 0 1.47k ⌀ |
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null | <think>
Okay, so I need to prove that both polynomials P and Q are constants if the equation P(x² - x + 1) = Q(x² + x + 1) holds for all real x. Hmm, let's start by understanding what the problem is asking.
First, both P and Q are polynomials, and they're being composed with different quadratic expressions. The key h... | [
{
"from": "human",
"value": "Polynomials \\( P(x) \\) and \\( Q(x) \\) are such that for any real \\( x \\), the equality \\( P\\left(x^{2} - x + 1\\right) = Q\\left(x^{2} + x + 1\\right) \\) holds. Prove that both polynomials are constants."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I nee... | Polynomials \( P(x) \) and \( Q(x) \) are such that for any real \( x \), the equality \( P\left(x^{2} - x + 1\right) = Q\left(x^{2} + x + 1\right) \) holds. Prove that both polynomials are constants. | Okay, so I need to prove that both polynomials P and Q are constants if the equation P(x² - x + 1) = Q(x² + x + 1) holds for all real x. Hmm, let's start by understanding what the problem is asking.
First, both P and Q are polynomials, and they're being composed with different quadratic expressions. The key here is t... | To prove that polynomials \( P(x) \) and \( Q(x) \) must be constants given the identity \( P(x^2 - x + 1) = Q(x^2 + x + 1) \) for all real \( x \), we proceed as follows:
### Key Steps and Reasoning:
1. **Quadratic Compositions**:
- Let \( A(x) = x^2 - x + 1 \) and \( B(x) = x^2 + x + 1 \). The given condition be... | 1. Consider the given equation for the polynomials \( P(x) \) and \( Q(x) \) where for any real \( x \),
\[
P(x^2 - x + 1) = Q(x^2 + x + 1).
\]
2. We substitute the argument \(-x\) in place of \(x\) in the given equation to obtain:
\[
P((-x)^2 - (-x) + 1) = Q((-x)^2 + (-x) + 1).
\]
3. Simplifying both... | math | numina_math | null | null |
null | <think>
Okay, so I need to show that for all positive real numbers \( a, b, \) and \( c \), the inequality \( a^{2} + b^{2} + c^{2} \geq ab + bc + ca \) holds. Then, using that result, I have to deduce that \( (a + b + c)^{2} \geq 3(ab + bc + ca) \). Hmm, let's start with the first part.
I remember that inequalities ... | [
{
"from": "human",
"value": "Show that for all positive real numbers \\( a, b, \\) and \\( c \\), we have \\( a^{2} + b^{2} + c^{2} \\geq ab + bc + ca \\). Deduce from this that \\( (a+b+c)^{2} \\geq 3(ab + bc + ca) \\)."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to show that for al... | Show that for all positive real numbers \( a, b, \) and \( c \), we have \( a^{2} + b^{2} + c^{2} \geq ab + bc + ca \). Deduce from this that \( (a+b+c)^{2} \geq 3(ab + bc + ca) \). | Okay, so I need to show that for all positive real numbers \( a, b, \) and \( c \), the inequality \( a^{2} + b^{2} + c^{2} \geq ab + bc + ca \) holds. Then, using that result, I have to deduce that \( (a + b + c)^{2} \geq 3(ab + bc + ca) \). Hmm, let's start with the first part.
I remember that inequalities involvin... | To show that for all positive real numbers \( a, b, \) and \( c \), the inequality \( a^{2} + b^{2} + c^{2} \geq ab + bc + ca \) holds, we can use the sum of squared differences:
\[
(a - b)^2 + (b - c)^2 + (c - a)^2 \geq 0
\]
Expanding each term:
\[
(a - b)^2 = a^2 - 2ab + b^2
\]
\[
(b - c)^2 = b^2 - 2bc + c^2
\]
\[... |
1. **Step 1: Initial Inequality and Reformulation**:
We begin with the inequality $a^2 + b^2 + c^2 \geq ab + bc + ca$. To demonstrate this, we can reformulate the inequality by moving all terms to one side:
\[
a^2 + b^2 + c^2 - ab - bc - ca \geq 0
\]
2. **Step 2: Multiplication and Grouping**:
... | math | numina_math | null | null |
null | "<think>\nOkay, let's see. I need to prove that for any prime number p, p divides a_p, where the seq(...TRUNCATED) | [{"from":"human","value":"For the sequence $\\{a_n\\}$ defined by $a_1 = 0$, $a_2 = 2$, $a_3 = 3$, $(...TRUNCATED) | "For the sequence $\\{a_n\\}$ defined by $a_1 = 0$, $a_2 = 2$, $a_3 = 3$, $a_{n+3} = a_{n+1} + a_n$,(...TRUNCATED) | "Okay, let's see. I need to prove that for any prime number p, p divides a_p, where the sequence {a_(...TRUNCATED) | "To prove that for any prime number \\( p \\), \\( p \\) divides \\( a_p \\) for the sequence \\( \\(...TRUNCATED) | "1. Let's define the sequence $\\{a_n\\}$ with initial conditions and recurrence relation:\n - $a(...TRUNCATED) | math | numina_math | null | null |
null | "<think>\nOkay, let's try to figure out this problem step by step. So, the situation is that Sri's c(...TRUNCATED) | [{"from":"human","value":"The clock in Sri's car, which is not accurate, gains time at a constant ra(...TRUNCATED) | "The clock in Sri's car, which is not accurate, gains time at a constant rate. One day as he begins (...TRUNCATED) | "Okay, let's try to figure out this problem step by step. So, the situation is that Sri's car clock (...TRUNCATED) | "Sri's car clock gains time at a constant rate. When he starts shopping, both his car clock and his (...TRUNCATED) | "1. **Identify the rate of gain of the car clock**: \n The car clock shows 12:35 when the accurate(...TRUNCATED) | math | numina_math | null | null |
"How can I effectively convey the subtle curvature of a circular staircase in a one-point perspectiv(...TRUNCATED) | "<think>\nOkay, so I want to draw a circular staircase using one-point perspective. I've heard that (...TRUNCATED) | [{"from":"human","value":"How can I effectively convey the subtle curvature of a circular staircase (...TRUNCATED) | null | null | null | null | null | null | null | null |
null | "<think>\nOkay, let's try to figure out how many points the graphs of y = 2 log x and y = log 2x int(...TRUNCATED) | [{"from":"human","value":"Consider the graphs of $y=2\\log{x}$ and $y=\\log{2x}$. We may say that:\n(...TRUNCATED) | "Consider the graphs of $y=2\\log{x}$ and $y=\\log{2x}$. We may say that:\n$\\textbf{(A)}\\ \\text{T(...TRUNCATED) | "Okay, let's try to figure out how many points the graphs of y = 2 log x and y = log 2x intersect. T(...TRUNCATED) | "To determine how many points the graphs of \\( y = 2\\log{x} \\) and \\( y = \\log{2x} \\) intersec(...TRUNCATED) | "1. **Set the equations equal to each other**: Given the equations $y = 2\\log{x}$ and $y = \\log{2x(...TRUNCATED) | math | numina_math | null | null |
null | "<think>\nOkay, so I need to find the number of pairs of integers x and y that have different pariti(...TRUNCATED) | [{"from":"human","value":"Find the number of pairs of integers $x, y$ with different parities such(...TRUNCATED) | "Find the number of pairs of integers $x, y$ with different parities such that $\\frac{1}{x}+\\fr(...TRUNCATED) | "Okay, so I need to find the number of pairs of integers x and y that have different parities (meani(...TRUNCATED) | "To find the number of pairs of integers \\( x, y \\) with different parities such that \\( \\frac{1(...TRUNCATED) | "1. Start with the given equation:\n \\[\n \\frac{1}{x} + \\frac{1}{y} = \\frac{1}{2520}\n \\](...TRUNCATED) | math | numina_math | null | null |
null | "<think>\nOkay, so I need to prove that for all positive real numbers \\( a, b, c \\), the inequalit(...TRUNCATED) | [{"from":"human","value":"For all positive real numbers \\(a, b, c\\), show that\n\n$$\n\\left(a^{2}(...TRUNCATED) | "For all positive real numbers \\(a, b, c\\), show that\n\n$$\n\\left(a^{2} b+b^{2} c+c^{2} a\\right(...TRUNCATED) | "Okay, so I need to prove that for all positive real numbers \\( a, b, c \\), the inequality \n\n\\[(...TRUNCATED) | "To prove the inequality for all positive real numbers \\(a, b, c\\):\n\n\\[\n\\left(a^{2} b + b^{2}(...TRUNCATED) | "\n1. To prove the inequality \\((a^2 b + b^2 c + c^2 a)(a b^2 + b c^2 + c a^2) \\geqslant 9 a^2 b^2(...TRUNCATED) | math | numina_math | null | null |
null | "<think>\nOkay, let's try to figure out if Ivan can reach the summit of Stromboli volcano and return(...TRUNCATED) | [{"from":"human","value":"From the volcano station to the summit of Stromboli volcano, it takes 4 ho(...TRUNCATED) | "From the volcano station to the summit of Stromboli volcano, it takes 4 hours by road and then 4 ho(...TRUNCATED) | "Okay, let's try to figure out if Ivan can reach the summit of Stromboli volcano and return safely w(...TRUNCATED) | "Ivan will be able to reach the summit and return safely by starting his journey 38 hours after the (...TRUNCATED) | "\nWe are given the following information:\n1. The total time to walk to the volcano's summit and ba(...TRUNCATED) | math | numina_math | null | null |
null | "<think>\nOkay, let's try to figure out how many 10-digit numbers there are where each digit is 1, 2(...TRUNCATED) | [{"from":"human","value":"How many 10-digit numbers are there whose digits are all 1, 2, or 3 and in(...TRUNCATED) | "How many 10-digit numbers are there whose digits are all 1, 2, or 3 and in which adjacent digits di(...TRUNCATED) | "Okay, let's try to figure out how many 10-digit numbers there are where each digit is 1, 2, or 3, a(...TRUNCATED) | "To determine the number of 10-digit numbers composed of digits 1, 2, or 3 where adjacent digits dif(...TRUNCATED) | "\nTo determine the number of 10-digit numbers whose digits are all 1, 2, or 3 and in which adjacent(...TRUNCATED) | math | numina_math | null | null |
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