input stringlengths 112 4k | output stringlengths 36 60.3k | gts stringlengths 1 270 | score float64 0 1 | acc float64 0 1 | step int64 1 63 | uid stringlengths 36 36 | prompt_token_ids listlengths 33 1.02k | response_token_ids listlengths 18 4.1k | rollout_log_probs listlengths 18 4.1k |
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Sure, let's solve each equation step by step.
### Problem (1): \(25x^2 - 9 = 7\)
**Step 1: Isolate the term with \(x\).**
Start by adding 9 to both sides of the equation to isolate the term with \(x^2\):
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \(x^2\).**
Divide both sides by 25 to solve fo... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)**
Divide both sides by 25:
\[
x^2 = \frac{16}{25}
\]
**S... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)**
Divide both sides by 25:
\[
x^2 = \frac{16}{2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term \( 25x^2 \). To do this, add 9 to both sides of the equation:
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^2 = 16
\]
**Step 2: Solve for... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the problems step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term \( 25x^2 \). To do this, we add 9 to both sides of the equation.
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 ... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each problem step by step.
### Problem (1): \(25x^2 - 9 = 7\)
**Step 1: Isolate the term with \(x\).**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \(x\):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \(x^2\).**
Divide both sides by 25:
\[
x^2 = \frac{16}{25}
\]
**Step 3: ... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Sure, let's solve each equation step by step.
### Problem (1): \(25x^2 - 9 = 7\)
#### Step 1: Isolate the term with \(x\)
First, we need to isolate the term \(25x^2\). To do this, we add 9 to both sides of the equation:
\[25x^2 - 9 + 9 = 7 + 9\]
\[25x^2 = 16\]
#### Step 2: Solve for \(x^2\)
Next, we divide both side... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Sure, let's solve each equation step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
#### Step 1: Isolate the term with \( x \)
First, we need to isolate the term \( 25x^2 \). To do this, we add 9 to both sides of the equation:
\[ 25x^2 - 9 + 9 = 7 + 9 \]
\[ 25x^2 = 16 \]
#### Step 2: Solve for \( x^2 \)
Next, we divi... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve \( 25x^2 - 9 = 7 \)
**Step 1:** Start with the given equation:
\[
25x^2 - 9 = 7
\]
**Step 2:** Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 3:** Divide both sides by 25 to solve for \( x^2 \):
\[
\frac... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the equations step by step.
---
### **Problem (1):**
**Find the value of \( x \):**
\( 25x^2 - 9 = 7 \)
---
#### Step 1: Bring the constant term to the left side
Add \( 9 \) to both sides of the equation to isolate the term with \( x \):
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplify:
\[
25x^2 = 16
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### Problem 1: \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to get rid of the constant term on the left side of the equation. To do this, add 9 to both sides of the equation:
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplify both sides:
\[
25x^2 = 16
\]
*... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
\[
25x^2 - 9 = 7
\]
Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)**
Divide both sides by 25:
\[
x^2 = \frac{16}{2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each problem step by step.
### Problem (1): Solve for \( x \) in the equation \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term containing \( x \). Start by adding 9 to both sides of the equation:
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### Problem (1): \( 25x^2 - 9 = 7 \)
**Step 1: Isolate the term with \( x \)**
First, we need to isolate the term \( 25x^2 \). To do this, add 9 to both sides of the equation.
\[
25x^2 - 9 + 9 = 7 + 9
\]
Simplifying both sides:
\[
25x^2 = 16
\]
**Step 2: Solve for \( x^2 \)... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
### (1) Solve the equation \( 25x^2 - 9 = 7 \)
**Step 1:** Start with the given equation:
\[
25x^2 - 9 = 7
\]
**Step 2:** Add 9 to both sides to isolate the term with \( x \):
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
**Step 3:** Divide both sides by 25 to solve for \( x^2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Find the value of $x$:
(1) $25x^2-9=7$
(2) $8(x-2)^3=27$
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve each equation step by step.
---
### **Problem (1):**
Find the value of \( x \) such that
\[ 25x^2 - 9 = 7 \]
---
#### Step 1: Isolate the term with \( x \)
First, add 9 to both sides of the equation to isolate the term containing \( x \).
\[
25x^2 - 9 + 9 = 7 + 9
\]
\[
25x^2 = 16
\]
#### Step 2... | \frac{7}{2} | 1 | 1 | 1 | ddc8033e-df10-4c5f-af73-fbe8eed5a570 | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions \(\sin \alpha + \sin \beta + \sin \gamma = 0\) and \(\cos \alpha + \cos \beta + \cos \gamma = 0\).
Let's proceed step by step.
### Step 1: Understand the Given Conditions
We are given:
1. \(\sin \a... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions:
\[
\sin \alpha + \sin \beta + \sin \gamma = 0
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0
\]
Let's proceed step by step.
### Step 1: Square the Given Equations
First, square both of the giv... | \frac{3}{2} | 0 | 0 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions \(\sin \alpha + \sin \beta + \sin \gamma = 0\) and \(\cos \alpha + \cos \beta + \cos \gamma = 0\).
Let's proceed step by step.
### Step 1: Analyze the Given Equations
We are given:
\[
\sin \alpha +... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
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872,
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-0.08894,
-0.029129,
-0.45... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we are given that:
\[
\sin \alpha + \sin \beta + \sin \gamma = 0
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0
\]
We need to find the value of:
\[
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma
\]
### Step 1: Analyze the Given Equations
We have two equations involving sine and cosine of angl... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
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1124,
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9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
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897,
315,... | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we are given the equations:
\[
\sin \alpha + \sin \beta + \sin \gamma = 0
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0
\]
We need to find the value of:
\[
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma
\]
### Step 1: Analyze the Given Equations
We have two equations involving sine and cosin... | \frac{3}{2} | 0 | 0 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
15940,
1124,
32214,
28,
15,
3,
323,
57960,
9407,
1124,
7141,
41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we are given two equations:
1. \(\sin \alpha + \sin \beta + \sin \gamma = 0\)
2. \(\cos \alpha + \cos \beta + \cos \gamma = 0\)
We need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\).
### Step-by-Step Solution:
#### Step 1: Square the Given Equations
First, let's square... | \frac{3}{2} | 0 | 0 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
15940,
1124,
32214,
28,
15,
3,
323,
57960,
9407,
1124,
7141,
41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
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582,
525,
2661,
1378,
37906,
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13,
1124,
11520,
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1124,
7141,
488,
1124,
15940,
1124,
19127,
488,
1124,
15940,
1124,
32214,
284,
220,
15,
59,
340,
17,
13,
1124,
11520,
9407,
1124,
7141,
488,
11... | [
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0,
0,
0,
0... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we are given that:
\[
\sin \alpha + \sin \beta + \sin \gamma = 0
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0
\]
We need to find the value of:
\[
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma
\]
### Step 1: Use Trigonometric Identities
Recall the identity for the square of cosine:
\[
\cos... | \frac{3}{2} | 0 | 0 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
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1124,
32214,
28,
15,
3,
323,
57960,
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41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
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9407,
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... | [
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0,
0,... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we start with the given equations:
\[
\sin \alpha + \sin \beta + \sin \gamma = 0
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0
\]
We need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\).
### Step 1: Square the Given Equations
First, square both of the given equations:
... | \frac{3}{2} | 0 | 0 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
15940,
1124,
32214,
28,
15,
3,
323,
57960,
9407,
1124,
7141,
41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
1249,
11625,
279,
3491,
11,
582,
1191,
448,
279,
2661,
37906,
1447,
59,
9640,
59,
15940,
1124,
7141,
488,
1124,
15940,
1124,
19127,
488,
1124,
15940,
1124,
32214,
284,
220,
15,
198,
59,
921,
59,
9640,
59,
9407,
1124,
7141,
488,
1124,
... | [
-0.10384,
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... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions \(\sin \alpha + \sin \beta + \sin \gamma = 0\) and \(\cos \alpha + \cos \beta + \cos \gamma = 0\).
Let's proceed step by step.
### Step 1: Understand the Given Conditions
We are given:
1. \(\sin \a... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
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1124,
32214,
28,
15,
3,
323,
57960,
9407,
1124,
7141,
41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
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279,
4682,
1124,
11520,
15940,
1124,
7141... | [
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0,
0,
0,
0,
-0.000008,
-0.000003,
-0.08894,
-0.029129,
-0.45... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions \(\sin \alpha + \sin \beta + \sin \gamma = 0\) and \(\cos \alpha + \cos \beta + \cos \gamma = 0\).
Let's proceed step by step.
### Step 1: Analyze the Given Conditions
We are given:
\[
\sin \alpha ... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
15940,
1124,
32214,
28,
15,
3,
323,
57960,
9407,
1124,
7141,
41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
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3491,
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582,
1184,
311,
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897,
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488,
1124,
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61,
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1124,
19127,
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61,
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32214,
57758,
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279,
4682,
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11520,
15940,
1124,
7141... | [
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0,
0,
0,
0,
-0.000008,
-0.000003,
-0.08894,
-0.029129,
-0.45... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we are given the following equations:
1. \(\sin \alpha + \sin \beta + \sin \gamma = 0\)
2. \(\cos \alpha + \cos \beta + \cos \gamma = 0\)
We need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\).
### Step-by-Step Solution:
#### Step 1: Square the Given Equations
Start by ... | \frac{3}{2} | 0 | 0 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
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7141,
41715,
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1124,
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41715,
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32214,
28,
15,
3,
323,
57960,
9407,
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41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
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897,
315,... | [
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220,
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-0.000001,
0,
0,
0... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\) given the conditions \(\sin \alpha + \sin \beta + \sin \gamma = 0\) and \(\cos \alpha + \cos \beta + \cos \gamma = 0\).
Let's proceed step by step.
### Step 1: Analyze the Given Equations
We are given:
\[
\sin \alpha +... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
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32214,
28,
15,
3,
323,
57960,
9407,
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7141,
41715,
9407,
1124,
19127,
41715,
9407,
1124,
32214,
28,
15,
54876,
1477,
279,
897,
315,... | [
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4682,
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1124,
7141... | [
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0,
0,
0,
0,
-0.000008,
-0.000003,
-0.08894,
-0.029129,
-0.45... |
user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the problem step by step.
**Given:**
\[
\sin \alpha + \sin \beta + \sin \gamma = 0 \quad \text{(1)}
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0 \quad \text{(2)}
\]
**We need to find the value of:**
\[
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma
\]
**Step 1: Use Trigonometric Identities**
Recall th... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
151644,
872,
198,
22043,
429,
57960,
15940,
1124,
7141,
41715,
15940,
1124,
19127,
41715,
15940,
1124,
32214,
28,
15,
3,
323,
57960,
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7141,
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the problem step by step.
**Given:**
\[
\sin \alpha + \sin \beta + \sin \gamma = 0 \quad \text{(1)}
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0 \quad \text{(2)}
\]
**We need to find the value of:**
\[
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma
\]
**Step 1: Square the given equations**
First, squa... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| To solve the problem, we are given two equations:
1. \(\sin \alpha + \sin \beta + \sin \gamma = 0\)
2. \(\cos \alpha + \cos \beta + \cos \gamma = 0\)
We need to find the value of \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma\).
### Step 1: Square the Given Equations
First, let's square both of the given equations.... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
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user
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
Please reason step by step, and put your final answer within \boxed{{}}.
assistant
| Let's solve the problem step by step.
**Given:**
\[
\sin \alpha + \sin \beta + \sin \gamma = 0
\]
\[
\cos \alpha + \cos \beta + \cos \gamma = 0
\]
**We need to find the value of:**
\[
\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma
\]
**Step 1: Square the given equations**
Start by squaring both given equations:
1. \... | \frac{3}{2} | 1 | 1 | 1 | c62ea007-908d-4734-931d-d0c34b2eeb58 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the quadratic equation in terms of \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
has real roots. Let's proceed step by step.
### Step 1: Let \( z \) be a complex number
Let \( z = a + bi \), wher... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the quadratic equation in terms of \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
has real roots. Let's proceed step by step.
### Step 1: Assume the roots are real
Let the roots of the quadratic e... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the quadratic equation in \( x \) has real roots. Let's proceed step by step.
Given the quadratic equation:
\[ x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \]
where \( z \) is a complex number, and the eq... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, let's proceed step by step.
Given the quadratic equation in terms of \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
where \( z \) is a complex number. We are asked to find the minimum value of the modulus \( |z| \) such that the equation has real roots.
### Step 1: Let \( z \) be a complex numbe... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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872,
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) given that the quadratic equation in \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
has real roots. Let's proceed step by step.
### Step 1: Assume the roots
Let the roots of the quadratic equation be \( x_1... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, let's proceed step by step.
Given the quadratic equation in terms of \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
where \( z \) is a complex number, and the equation has a real root. We need to find the minimum value of the modulus \( |z| \) of the complex number \( z \).
### Step 1: Assume th... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the quadratic equation in \( x \) has real roots. Let's proceed step by step.
### Step 1: Let \( z \) be a complex number
Let \( z = a + bi \), where \( a, b \in \mathbb{R} \). Then, the equation in... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the equation \( x^2 - 2zx + \frac{3}{4} + i = 0 \) has real roots.
Let's proceed step by step.
### Step 1: Let \( z \) be a complex number
Let \( z = a + bi \), where \( a \) and \( b \) are real n... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \), given that the quadratic equation in \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
has real roots. Let's proceed step by step.
### Step 1: Understanding the Problem
We are given a quadratic equation in \( x... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \), given that the quadratic equation in \( x \) has real roots. The equation is:
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
Let's proceed step by step.
### Step 1: Assume \( z \) is a complex number
Let \( z = a +... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the quadratic equation in \( x \) has real roots. The given quadratic equation is:
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
where \( z \) is a complex number. Let's denote \( z = a + bi \), where \( a... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the quadratic equation in terms of \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
has real roots. Let's proceed step by step.
### Step 1: Assume the quadratic equation has real roots
Let the roots... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \), given that the quadratic equation in \( x \) has real roots. Let's proceed step by step.
### Given Equation:
The equation is:
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
where \( z \) is a complex number, and \( i... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) given that the quadratic equation in \( x \):
\[ x^2 - 2zx + \frac{3}{4} + i = 0 \]
has real roots. Let's proceed step by step.
### Step 1: Assume the roots are real
Let the roots of the quadratic equation be... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) given that the equation \( x^2 - 2zx + \frac{3}{4} + i = 0 \) has real roots. Let's proceed step by step.
### Step 1: Assume the roots are real
Let the roots of the quadratic equation \( x^2 - 2zx + \frac{3}{... | 1 | 0 | 0 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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user
Given \( z \in \mathbb{C} \), if the equation in terms of \( x \): \( x^2 - 2zx + \frac{3}{4} + \text{i} = 0 \) (where \( i \) is the imaginary unit) has real roots, find the minimum value of the modulus \( |z| \) of the complex number \( z \).
Please reason step by step, and put your final answer within \boxed{{}... | To solve the problem, we need to find the minimum value of the modulus \( |z| \) of the complex number \( z \) such that the equation \( x^2 - 2zx + \frac{3}{4} + i = 0 \) has real roots.
### Step 1: Let \( z \) be a complex number
Let \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the im... | 1 | 1 | 1 | 1 | 31e07f7f-23dd-46ff-8295-72284d251eb7 | [
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At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve the problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break down the problem step by step.
### Step 1: Define the Variables
- Let \( x \) be the distance from the start... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break it down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \( c... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve the problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break this down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \( ... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break it down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \( c... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the bullet hitting the target arrive simultaneously. Let's break this down step by step.
### Step 1: Define the variables
- \( d \): Length of the shooting range in meters.
- \( c \): S... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, we need to analyze the time it takes for both the bullet and the sound to travel the distance \( d \).
### Step-by-Step Solution:
1. **Define Variabl... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To determine the point on the shooting range where the sound of the gunshot and the bullet hitting the target arrive simultaneously, we need to analyze the time it takes for the bullet to reach the target and the time it takes for the sound to travel back to the starting point.
**Given:**
- Length of the shooting rang... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break this down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \(... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break down the problem step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bul... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the bullet hitting the target arrive simultaneously, we need to analyze the time it takes for the sound to travel to the target and the time it takes for the bullet to reach the target.
### Step-by-Step Solution:
1.... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on the shooting range where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break it down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \( c \) m/s
- Speed of soun... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To determine the point on a shooting range where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, we need to analyze the time it takes for both the sound and the bullet to reach the same point.
### Step-by-Step Solution:
1. **Define the Variables:**
- Let \( d \) be th... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break down the problem step by step.
### Given:
- Length of the shooting range: \( d \) meters
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on the shooting range where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break this down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \( c \) m/sec
- Speed of ... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on the shooting range where both the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break it down step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bullet: \( c \) m/s
- Speed of... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
At which point on a shooting range of length $d$ meters does the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously, if the speed of the bullet is $c \mathrm{~m}/\mathrm{sec}$, and the speed of sound is $s \mathrm{~m}/\mathrm{sec}$?
Please reason step by step, and put your fi... | To solve this problem, we need to determine the point on a shooting range of length \( d \) meters where the sound of the gunshot and the sound of the bullet hitting the target arrive simultaneously. Let's break down the problem step by step.
### Given:
- Length of the shooting range: \( d \) meters
- Speed of the bul... | \frac{}{2}(1+\frac{}{}) | 0 | 0 | 1 | b8af7727-855b-408f-95b1-6eb57a154119 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To solve the problem, we need to find the smallest natural number \( N \) greater than 2000 that is divisible by at least six of the nine natural numbers from 1 to 9. The nine natural numbers are \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \).
### Step-by-Step Solution:
1. **Identify the numbers from 1 to 9:**
The numbers are \(... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To solve this problem, we need to find the smallest natural number \( N \) that is greater than 2000 and is "six-divisible". A number \( N \) is called "six-divisible" if at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that \( N \) must be divisible by at least six of the numbers 1, ... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest natural number \( N \) greater than 2000 that is a "six-divisible number," we need to understand what it means for \( N \) to be a "six-divisible number." Specifically, \( N \) must be divisible by at least six of the nine natural numbers from 1 to 9. The nine numbers are \( 1, 2, 3, 4, 5, 6, 7, 8,... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand what it means for a number \( N \) to be "six-divisible". A number \( N \) is called "six-divisible" if at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that among the number... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand what it means for a number to be "six-divisible." A number is called "six-divisible" if at least six of the nine natural numbers from 1 to 9 can divide it. This means that at least six of the numbers 1, 2, 3, ... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To solve this problem, we need to find the smallest natural number \( N \) that is greater than 2000 and is a "six-divisible number". A "six-divisible number" is defined as a natural number \( N \) that is divisible by at least six of the nine natural numbers from 1 to 9. The nine natural numbers are 1, 2, 3, 4, 5, 6, ... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To solve the problem, we need to find the smallest natural number \( N \) greater than 2000 that is divisible by at least six of the nine natural numbers from 1 to 9. Let's break this down step by step.
### Step 1: Identify the Divisors
The natural numbers from 1 to 9 are: 1, 2, 3, 4, 5, 6, 7, 8, 9.
We need to find t... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest natural number \( N \) greater than 2000 that is a "six-divisible number," we need to understand what it means for a number to be a "six-divisible number." A number \( N \) is called a "six-divisible number" if at least six of the nine natural numbers from 1 to 9 can divide \( N \).
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" that is greater than 2000, we need to understand what it means for a number \( N \) to be a "six-divisible number". A "six-divisible number" is a natural number \( N \) such that at least six out of the nine natural numbers from 1 to 9 can divide \( N \). This means that \( N... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand what a "six-divisible number" is. A "six-divisible number" is a natural number \( N \) such that at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that among the numbers 1, 2,... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand the definition of a "six-divisible number". A "six-divisible number" is a natural number \( N \) such that at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that \( N \) must ... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand the definition of a "six-divisible number." A natural number \( N \) is called a "six-divisible number" if at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that \( N \) must ... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand what a "six-divisible number" is. A "six-divisible number" is a natural number \( N \) such that at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that \( N \) must be divisib... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest natural number \( N \) greater than 2000 that is a "six-divisible number," we need to understand what it means for a number to be a "six-divisible number." A number \( N \) is called a "six-divisible number" if at least six of the nine natural numbers from 1 to 9 can divide \( N \).
First, let's l... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To solve the problem, we need to find the smallest natural number \( N \) greater than 2000 that is divisible by at least six of the nine natural numbers from 1 to 9. Let's break this down step by step.
### Step 1: Understand the Problem
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user
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?
Please reason step by step, and put your final answer within \boxed{{}}.
ass... | To find the smallest "six-divisible number" among the natural numbers greater than 2000, we need to understand what it means for a number to be "six-divisible". A "six-divisible number" is a natural number \( N \) such that at least six of the nine natural numbers from 1 to 9 can divide \( N \). This means that at leas... | 2016 | 0 | 0 | 1 | 99bd6f55-c4f5-40c0-9e99-9d280f88c335 | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
### Problem Statement
We are given a polynomial \( P(x) \) of degree 10 with a leading coefficient of 1. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) is factored into irreducible polynomials, and it is known that at \( x ... | 243 | 1 | 1 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
We are given a polynomial \( P(x) \) of degree 10 with a leading coefficient of 1. The polynomial \( P(x) \) is of the form:
\[ P(x) = (x - r_1)(x - r_2) \cdots (x - r_{10}) \]
where \( r_1, r_2, \ldots, r_{10} \) are the roots of the polynomial.
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user
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
We are given a polynomial \( P(x) \) of degree 10 with a leading coefficient of 1. This means \( P(x) \) can be written as:
\[ P(x) = (x - r_1)(x - r_2) \cdots (x - r_{10}) \]
where \( r_1, r_2, \ldots, r_{10} \) are the roots of the polynomial.
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user
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
**Given:**
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis.
- The polynomial \( -P(x) \) is factored into irreducible polynomials.
- At \( x = 2020 \), all the resulting irred... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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user
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
### Given:
1. \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
2. The graph of \( y = P(x) \) lies entirely above the x-axis, which means \( P(x) > 0 \) for all \( x \).
3. \( -P(x) \) is factored into irreducible polynomials.
4. At \( x =... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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user
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
### Given:
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- \( P(x) \) lies entirely above the x-axis, meaning \( P(x) > 0 \) for all \( x \in \mathbb{R} \).
- \( -P(x) \) is factored into irreducible polynomials, and at \( x = 2020 \)... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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user
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
**Given:**
- A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis.
- The polynomial \( -P(x) \) is factored into irreducible polynomials.
- At \( x = 2020 \), all the resulting irreduci... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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user
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
**Given:**
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1. So, \( P(x) = x^{10} + a_9 x^9 + \dots + a_1 x + a_0 \).
- The graph of \( y = P(x) \) lies entirely above the x-axis, which means \( P(x) > 0 \) for all \( x \).
- The polynomi... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
**Given:**
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis, which means \( P(x) > 0 \) for all \( x \).
- The polynomial \( -P(x) \) is factored into irreducible polynomials.
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
### Given:
1. \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
2. The graph of \( y = P(x) \) lies entirely above the x-axis, which means \( P(x) > 0 \) for all \( x \).
3. The polynomial \( -P(x) \) is factored into irreducible polynomial... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
**Given:**
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis.
- When we factor \( -P(x) \) into irreducible factors, all the resulting irreducible polynomials take the value -3 ... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
**Given:**
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis.
- The polynomial \( -P(x) \) is factored into irreducible factors.
- At \( x = 2020 \), all the irreducible factors... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
### Given:
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis, meaning \( P(x) > 0 \) for all \( x \).
- The polynomial \( -P(x) \) is factored into irreducible polynomials.
- At... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
1. **Understanding the Polynomial \( P(x) \):**
- \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
- The graph of \( y = P(x) \) lies entirely above the x-axis, which means \( P(x) > 0 \) for all \( x \).
2. **Factoring \( -P(x) \):... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
### Given:
1. \( P(x) \) is a polynomial of degree 10 with a leading coefficient of 1.
2. The graph of \( y = P(x) \) lies entirely above the x-axis.
3. \( -P(x) \) is factored into irreducible polynomials.
4. At \( x = 2020 \), all the resulting irreducible poly... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known tha... | To solve the problem, let's break it down step by step.
We are given a polynomial \( P(x) \) of degree 10 with a leading coefficient of 1. The polynomial \( P(x) \) lies entirely above the x-axis, which means \( P(x) > 0 \) for all \( x \).
We are also given that the polynomial \( -P(x) \) is factored into irreducibl... | 243 | 0 | 0 | 1 | 446bc8cf-d27e-49bb-bd72-d5f097b2f11a | [
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Given that \(w\) and \(z\) are complex numbers such that \(|w+z|=1\) and \(\left|w^{2}+z^{2}\right|=14\), find the smallest possible value of \(\left|w^{3}+z^{3}\right|\). Here, \(|\cdot|\) denotes the absolute value of a complex number, given by \(|a+bi|=\sqrt{a^{2}+b^{2}}\) whenever \(a\) and \(b\) are real numb... | To solve the problem, we need to find the smallest possible value of \( \left| w^3 + z^3 \right| \) given the conditions \( |w + z| = 1 \) and \( \left| w^2 + z^2 \right| = 14 \).
Let's proceed step by step.
### Step 1: Express \( w^3 + z^3 \) in terms of \( w + z \) and \( wz \)
We know the identity for the sum of c... | \frac{41}{2} | 1 | 1 | 1 | 52edfb49-ec1b-4d01-a609-efb71c2cde53 | [
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Given that \(w\) and \(z\) are complex numbers such that \(|w+z|=1\) and \(\left|w^{2}+z^{2}\right|=14\), find the smallest possible value of \(\left|w^{3}+z^{3}\right|\). Here, \(|\cdot|\) denotes the absolute value of a complex number, given by \(|a+bi|=\sqrt{a^{2}+b^{2}}\) whenever \(a\) and \(b\) are real numb... | To find the smallest possible value of \( \left| w^3 + z^3 \right| \) given that \( |w + z| = 1 \) and \( \left| w^2 + z^2 \right| = 14 \), we will proceed step by step.
### Step 1: Express \( w^3 + z^3 \) in terms of \( w + z \) and \( wz \)
We know the identity for the sum of cubes:
\[
w^3 + z^3 = (w + z)(w^2 - wz +... | \frac{41}{2} | 0 | 0 | 1 | 52edfb49-ec1b-4d01-a609-efb71c2cde53 | [
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Given that \(w\) and \(z\) are complex numbers such that \(|w+z|=1\) and \(\left|w^{2}+z^{2}\right|=14\), find the smallest possible value of \(\left|w^{3}+z^{3}\right|\). Here, \(|\cdot|\) denotes the absolute value of a complex number, given by \(|a+bi|=\sqrt{a^{2}+b^{2}}\) whenever \(a\) and \(b\) are real numb... | To solve the problem, we need to find the smallest possible value of \( \left| w^3 + z^3 \right| \) given the conditions \( |w + z| = 1 \) and \( \left| w^2 + z^2 \right| = 14 \).
### Step 1: Express \( w^3 + z^3 \) in terms of \( w + z \) and \( wz \)
We know the identity for the sum of cubes:
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Given that \(w\) and \(z\) are complex numbers such that \(|w+z|=1\) and \(\left|w^{2}+z^{2}\right|=14\), find the smallest possible value of \(\left|w^{3}+z^{3}\right|\). Here, \(|\cdot|\) denotes the absolute value of a complex number, given by \(|a+bi|=\sqrt{a^{2}+b^{2}}\) whenever \(a\) and \(b\) are real numb... | To find the smallest possible value of \( \left| w^3 + z^3 \right| \) given that \( |w + z| = 1 \) and \( \left| w^2 + z^2 \right| = 14 \), we will proceed step by step.
### Step 1: Express \( w^3 + z^3 \) in terms of \( w + z \) and \( wz \)
We know the identity for the sum of cubes:
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w^3 + z^3 = (w + z)(w^2 - wz +... | \frac{41}{2} | 1 | 1 | 1 | 52edfb49-ec1b-4d01-a609-efb71c2cde53 | [
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