instruction_seed stringlengths 16 5.67k | response_seed stringlengths 16 5.67k ⌀ | reasoning stringlengths 0 107k | deepseek_solution stringlengths 0 13.3k | __original_row_idx int64 1 40k ⌀ | source stringclasses 2
values | final_reasoning_trace stringlengths 19 108k | conversations listlengths 2 2 |
|---|---|---|---|---|---|---|---|
** Give an example of a polynomial $f(x)$ with integer coefficients, such that for $m$ different positive integers $x$, it yields $m$ different prime numbers. | null | Okay, so I need to come up with a polynomial f(x) with integer coefficients. The condition is that for m different positive integers x, the polynomial f(x) should produce m different prime numbers. Hmm, let me think about how to approach this.
First, I remember that there's a theorem related to primes and polynomials.... | To find a polynomial \( f(x) \) with integer coefficients such that for \( m \) different positive integers \( x \), it yields \( m \) different prime numbers, we can consider the simplest polynomial: the identity function.
1. **Polynomial Choice**: The polynomial \( f(x) = x \) has integer coefficients.
2. **Prime Ge... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I need to come up with a polynomial f(x) with integer coefficients. The condition is that for m different positive integers x, the polynomial f(x) should produce m different prime numbers. Hmm, let me think about how to approach this.
First, I remember that there's a theorem related to primes and poly... | [
{
"from": "human",
"value": "** Give an example of a polynomial $f(x)$ with integer coefficients, such that for $m$ different positive integers $x$, it yields $m$ different prime numbers."
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to come up with a polynomial f(x) with integer coeff... |
Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 24.$ Find the largest possible value of $b.$ | Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 24.$ Find the largest possible value of $b.$ | Okay, so I need to find the largest possible value of \( b \) in a three-term geometric series where all terms are positive, and their sum is 24. Let me start by recalling what a geometric series is. In a geometric series, each term is multiplied by a common ratio to get the next term. So if the three terms are \( a \)... | Given a three-term geometric series \(a, b, c\) with all terms positive and their sum equal to 24, we need to find the largest possible value of \(b\).
Since \(a, b, c\) form a geometric series, we can write \(b = ar\) and \(c = ar^2\) where \(r\) is the common ratio. The sum of the terms is:
\[
a + ar + ar^2 = 24
\]
... | 19,277 | ai2-adapt-dev/openmath-2-math | <think>
Okay, so I need to find the largest possible value of \( b \) in a three-term geometric series where all terms are positive, and their sum is 24. Let me start by recalling what a geometric series is. In a geometric series, each term is multiplied by a common ratio to get the next term. So if the three terms are... | [
{
"from": "human",
"value": "Let $a,$ $b,$ $c$ be a three-term geometric series where all the terms are positive, such that $a+b+c = 24.$ Find the largest possible value of $b.$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I need to find the largest possible value of \\( b \\) in a three-te... |
q ' = 3 q - 3 , what is the value of ( 3 ' ) ' ? | null | Okay, let me try to figure out this problem. The question is: given that q' = 3q - 3, what is the value of (3')'? Hmm, primes can sometimes be confusing because they might mean different things in different contexts. But here, since it's defined as q' = 3q - 3, I think the prime symbol (') is being used to denote a fun... | Given the operation defined by \( q' = 3q - 3 \), we need to find the value of \( (3')' \).
First, we compute the inner prime operation on 3:
\[
3' = 3 \cdot 3 - 3 = 9 - 3 = 6
\]
Next, we apply the prime operation again to the result:
\[
6' = 3 \cdot 6 - 3 = 18 - 3 = 15
\]
Thus, the value of \( (3')' \) is \(\boxed{... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, let me try to figure out this problem. The question is: given that q' = 3q - 3, what is the value of (3')'? Hmm, primes can sometimes be confusing because they might mean different things in different contexts. But here, since it's defined as q' = 3q - 3, I think the prime symbol (') is being used to deno... | [
{
"from": "human",
"value": "q ' = 3 q - 3 , what is the value of ( 3 ' ) ' ?"
},
{
"from": "gpt",
"value": "<think>\nOkay, let me try to figure out this problem. The question is: given that q' = 3q - 3, what is the value of (3')'? Hmm, primes can sometimes be confusing because they might mean d... |
"Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\\circ}}.\\]Enter (...TRUNCATED) | "Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\\circ}}.\\]Enter (...TRUNCATED) | "Okay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(cos10° + c(...TRUNCATED) | "To simplify the expression \\(\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\cos{20^\(...TRUNCATED) | 33,695 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, let's see. I need to simplify this trigonometric expression: (sin10° + sin20°)/(co(...TRUNCATED) | [{"from":"human","value":"Simplify\n\\[\\frac{\\sin{10^\\circ}+\\sin{20^\\circ}}{\\cos{10^\\circ}+\\(...TRUNCATED) |
"Shyne can grow some eggplants in every seed packet and 10 sunflowers in every seed packet. She boug(...TRUNCATED) | null | "Okay, let's see. The problem says Shyne can grow some eggplants in every seed packet and 10 sunflow(...TRUNCATED) | "Let \\( x \\) be the number of eggplants in each seed packet. Shyne bought 4 seed packets of eggpla(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let's see. The problem says Shyne can grow some eggplants in every seed packet and 1(...TRUNCATED) | [{"from":"human","value":"Shyne can grow some eggplants in every seed packet and 10 sunflowers in ev(...TRUNCATED) |
"A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshe(...TRUNCATED) | "A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the same size. If the bookshe(...TRUNCATED) | "Alright, so I need to figure out the least number of shelves required to store 24 books of differen(...TRUNCATED) | "The problem requires determining the least number of shelves needed to store 24 books of different (...TRUNCATED) | 3,003 | ai2-adapt-dev/openmath-2-math | "<think>\nAlright, so I need to figure out the least number of shelves required to store 24 books of(...TRUNCATED) | [{"from":"human","value":"A bookshelf has 5 shelves, and each shelf can hold at most 8 books of the (...TRUNCATED) |
"A sector with an obtuse central angle $\\phi$ is cut from a circle of radius 8. Find the radius of (...TRUNCATED) | null | "Okay, so I have this problem here: A sector with an obtuse central angle φ is cut from a circle of(...TRUNCATED) | "To find the radius of the circle circumscribed about a sector with an obtuse central angle \\(\\phi(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I have this problem here: A sector with an obtuse central angle φ is cut from a (...TRUNCATED) | [{"from":"human","value":"A sector with an obtuse central angle $\\phi$ is cut from a circle of radi(...TRUNCATED) |
"Solve the following system of equations:\n\n\\begin{align*}\nx+y+z&=6,\\\\\nx+2y+3z&=12,\\\\\nx+3y+(...TRUNCATED) | "Solve the following system of equations:\n\n\\begin{align*}\nx+y+z&=6,\\\\\nx+2y+3z&=12,\\\\\nx+3y+(...TRUNCATED) | "Okay, so I need to solve this system of equations. Let me write them down again to make sure I have(...TRUNCATED) | "To solve the system of equations:\n\n\\[\n\\begin{align*}\nx + y + z &= 6, \\\\\nx + 2y + 3z &= 12,(...TRUNCATED) | 6,462 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to solve this system of equations. Let me write them down again to make su(...TRUNCATED) | [{"from":"human","value":"Solve the following system of equations:\n\n\\begin{align*}\nx+y+z&=6,\\\\(...TRUNCATED) |
Let's prove that
$$
183^{n}-126^{n}
$$
is divisible by 1957 if $n$ is an even number. | null | "Alright, so I need to prove that \\(183^{n} - 126^{n}\\) is divisible by 1957 when \\(n\\) is even.(...TRUNCATED) | "To prove that \\(183^{n} - 126^{n}\\) is divisible by 1957 if \\(n\\) is an even number, we start b(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nAlright, so I need to prove that \\(183^{n} - 126^{n}\\) is divisible by 1957 when \\(n\\)(...TRUNCATED) | [{"from":"human","value":"Let's prove that\n\n$$\n183^{n}-126^{n}\n$$\n\nis divisible by 1957 if $n$(...TRUNCATED) |
"1.17. Let $x_{1}$ and $x_{2}$ be the roots of the quadratic equation $x^{2} + a x + b = 0$, and let(...TRUNCATED) | null | "Okay, so I have this problem here about quadratic equations and something called Waring's formula. (...TRUNCATED) | "To prove Waring's formula for the sum \\( \\frac{s_n}{n} \\) where \\( s_n = x_1^n + x_2^n \\) and (...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I have this problem here about quadratic equations and something called Waring's (...TRUNCATED) | [{"from":"human","value":"1.17. Let $x_{1}$ and $x_{2}$ be the roots of the quadratic equation $x^{2(...TRUNCATED) |
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