instruction_seed stringlengths 17 5.02k | response_seed stringlengths 25 4.3k ⌀ | reasoning stringlengths 698 82.8k | deepseek_solution stringlengths 42 2.97k | __original_row_idx int64 1 40k ⌀ | source stringclasses 2
values | final_reasoning_trace stringlengths 1.01k 84.8k | conversations listlengths 2 2 |
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The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | The function $f(x)$ satisfies
\[f(x - y) = f(x) f(y)\]for all real numbers $x$ and $y,$ and $f(x) \neq 0$ for all real numbers $x.$ Find $f(3).$ | Okay, let's see. I need to find f(3) given that the function f(x) satisfies the equation f(x - y) = f(x)f(y) for all real numbers x and y, and also that f(x) is never zero. Hmm, functional equations can be tricky, but maybe I can figure this out by plugging in some specific values for x and y. Let me start by understan... | Given the function \( f(x) \) satisfies the equation \( f(x - y) = f(x) f(y) \) for all real numbers \( x \) and \( y \), and \( f(x) \neq 0 \) for all real numbers \( x \), we need to find \( f(3) \).
First, by setting \( x = y \), we get:
\[
f(0) = f(x - x) = f(x) f(x) = [f(x)]^2
\]
This implies \( [f(x)]^2 = f(0) \... | 11,083 | ai2-adapt-dev/openmath-2-math | <think>
Okay, let's see. I need to find f(3) given that the function f(x) satisfies the equation f(x - y) = f(x)f(y) for all real numbers x and y, and also that f(x) is never zero. Hmm, functional equations can be tricky, but maybe I can figure this out by plugging in some specific values for x and y. Let me start by u... | [
{
"from": "human",
"value": "The function $f(x)$ satisfies\n\\[f(x - y) = f(x) f(y)\\]for all real numbers $x$ and $y,$ and $f(x) \\neq 0$ for all real numbers $x.$ Find $f(3).$"
},
{
"from": "gpt",
"value": "<think>\nOkay, let's see. I need to find f(3) given that the function f(x) satisfies t... |
1. Let the complex number $z$ satisfy $|z|=1$. Then the maximum value of $\mid z^{3}+3 z+2 \mathrm{i}$ is ( ).
(A) $4 \sqrt{2}$
(B) $3 \sqrt{3}$
(C) $2 \sqrt{5}$
(D) $4 \sqrt{3}$ | null | Okay, so I have this complex number problem here. Let me read it again: Let the complex number z satisfy |z| = 1. Then the maximum value of |z³ + 3z + 2i| is one of the options given. The options are A) 4√2, B) 3√3, C) 2√5, D) 4√3. Hmm, I need to figure out which one is the maximum value.
First, since |z| = 1, z is a ... | Given the complex number \( z \) satisfying \( |z| = 1 \), we need to find the maximum value of \( |z^3 + 3z + 2i| \).
1. **Parametrization**: Since \( |z| = 1 \), we can write \( z = e^{i\theta} \) where \( \theta \) is the argument of \( z \). This gives \( z^3 = e^{i3\theta} \) and \( 3z = 3e^{i\theta} \).
2. **Ex... | null | AI-MO/NuminaMath-1.5 | <think>
Okay, so I have this complex number problem here. Let me read it again: Let the complex number z satisfy |z| = 1. Then the maximum value of |z³ + 3z + 2i| is one of the options given. The options are A) 4√2, B) 3√3, C) 2√5, D) 4√3. Hmm, I need to figure out which one is the maximum value.
First, since |z| = 1,... | [
{
"from": "human",
"value": "1. Let the complex number $z$ satisfy $|z|=1$. Then the maximum value of $\\mid z^{3}+3 z+2 \\mathrm{i}$ is ( ).\n(A) $4 \\sqrt{2}$\n(B) $3 \\sqrt{3}$\n(C) $2 \\sqrt{5}$\n(D) $4 \\sqrt{3}$"
},
{
"from": "gpt",
"value": "<think>\nOkay, so I have this complex number pr... |
"Let $m$ and $n$ be positive integers. A $2 m \\times 2 n$ grid of squares is coloured in the usual (...TRUNCATED) | null | "Okay, so I need to find the number of ways to place mn counters on the white squares of a 2m x 2n c(...TRUNCATED) | "To solve the problem of finding the number of ways to place \\( mn \\) counters on the white square(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I need to find the number of ways to place mn counters on the white squares of a (...TRUNCATED) | [{"from":"human","value":"Let $m$ and $n$ be positive integers. A $2 m \\times 2 n$ grid of squares (...TRUNCATED) |
"Given that $\\sin \\theta + \\cos \\theta = \\frac{4}{3}$, with $\\theta \\in \\left(0, \\frac{\\pi(...TRUNCATED) | null | "Okay, let's see. I have this problem where sinθ + cosθ = 4/3, and θ is between 0 and π/4. I nee(...TRUNCATED) | "Given that \\(\\sin \\theta + \\cos \\theta = \\frac{4}{3}\\) with \\(\\theta \\in \\left(0, \\frac(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let's see. I have this problem where sinθ + cosθ = 4/3, and θ is between 0 and π(...TRUNCATED) | [{"from":"human","value":"Given that $\\sin \\theta + \\cos \\theta = \\frac{4}{3}$, with $\\theta \(...TRUNCATED) |
"3. As shown in Figure 1, in $\\triangle A B C$, $D$ and $E$ are points on $A C$ and $A B$ respectiv(...TRUNCATED) | null | "Alright, so I've got this geometry problem here. Let me try to visualize it first. There's triangle(...TRUNCATED) | "Given triangle \\( \\triangle ABC \\) with points \\( D \\) on \\( AC \\) and \\( E \\) on \\( AB \(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nAlright, so I've got this geometry problem here. Let me try to visualize it first. There's(...TRUNCATED) | [{"from":"human","value":"3. As shown in Figure 1, in $\\triangle A B C$, $D$ and $E$ are points on (...TRUNCATED) |
"One endpoint of a line segment is (7, 2) and its midpoint is (5, 6). What is the difference between(...TRUNCATED) | "One endpoint of a line segment is (7, 2) and its midpoint is (5, 6). What is the difference between(...TRUNCATED) | "Okay, so I need to find the other endpoint of a line segment when one endpoint is (7, 2) and the mi(...TRUNCATED) | "Given one endpoint of a line segment as \\((7, 2)\\) and its midpoint as \\((5, 6)\\), we need to f(...TRUNCATED) | 27,515 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the other endpoint of a line segment when one endpoint is (7, 2) a(...TRUNCATED) | [{"from":"human","value":"One endpoint of a line segment is (7, 2) and its midpoint is (5, 6). What (...TRUNCATED) |
"Given the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0).$ The four vertices (...TRUNCATED) | null | "Okay, let's try to tackle this hyperbola problem step by step. The question is about finding the ec(...TRUNCATED) | "Given the hyperbola \\( E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\) with \\( a > 0 \\) and \\( b(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, let's try to tackle this hyperbola problem step by step. The question is about findi(...TRUNCATED) | [{"from":"human","value":"Given the hyperbola $E: \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b (...TRUNCATED) |
"Among the following judgments: \n① $x^2 \\neq y^2 \\Leftrightarrow x \\neq y$ or $x \\neq -y$; (...TRUNCATED) | null | "Okay, so I need to check which of these five statements are correct. Let's take them one by one. \n(...TRUNCATED) | "Let's analyze each statement one by one:\n\n1. **Statement ①**: \\( x^2 \\neq y^2 \\Leftrightarro(...TRUNCATED) | null | AI-MO/NuminaMath-1.5 | "<think>\nOkay, so I need to check which of these five statements are correct. Let's take them one b(...TRUNCATED) | [{"from":"human","value":"Among the following judgments: \n① $x^2 \\neq y^2 \\Leftrightarrow x \\(...TRUNCATED) |
What is the base-5 equivalent of the decimal number 219? | What is the base-5 equivalent of the decimal number 219? | "Okay, so I need to figure out what the base-5 equivalent of the decimal number 219 is. Hmm, base co(...TRUNCATED) | "To convert the decimal number 219 to base-5, we use the division method by keeping track of the rem(...TRUNCATED) | 6,281 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to figure out what the base-5 equivalent of the decimal number 219 is. Hmm(...TRUNCATED) | [{"from":"human","value":"What is the base-5 equivalent of the decimal number 219?"},{"from":"gpt","(...TRUNCATED) |
"What is the largest power of 5 that divides the product of all positive integers from 1 to 25, incl(...TRUNCATED) | "What is the largest power of 5 that divides the product of all positive integers from 1 to 25, incl(...TRUNCATED) | "Okay, so I need to find the largest power of 5 that divides the product of all positive integers fr(...TRUNCATED) | "To find the largest power of 5 that divides the product of all positive integers from 1 to 25 (i.e.(...TRUNCATED) | 17,073 | ai2-adapt-dev/openmath-2-math | "<think>\nOkay, so I need to find the largest power of 5 that divides the product of all positive in(...TRUNCATED) | [{"from":"human","value":"What is the largest power of 5 that divides the product of all positive in(...TRUNCATED) |
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