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[
{
"role": "user",
"content": "Return your final response within \\boxed{}. Given two moving points \\(A(x_1, y_1)\\) and \\(B(x_2, y_2)\\) on the parabola \\(y^2 = 6x\\), where \\(x_1 \\neq x_2\\) and \\(x_1 + x_2 = 4\\). If the perpendicular bisector of segment \\(AB\\) intersects the \\(x\\)-axis at point... |
[
{
"role": "user",
"content": "Return your final response within \\boxed{}. Prove that in the sequence \\( a_n = 6^n \\), the numbers formed by the last two digits of this power form a periodic sequence. Find the period of this sequence."
},
{
"role": "assistant",
"content": "<|begin_of_thought|>... |
[
{
"role": "user",
"content": "What are some famous formulas or principles associated with the number \"0\"?"
},
{
"role": "assistant",
"content": "What a great question! The number 0 has played a significant role in various mathematical and scientific concepts, leading to some fundamental formul... |
[
{
"role": "user",
"content": "Write a function to find the sum of the largest contiguous sublist in the given list. Additionally, return the start and end indices of this sublist. The list can contain negative and positive integers."
},
{
"role": "assistant",
"content": "Here is a Python solutio... |
[
{
"role": "user",
"content": "Return your final response within \\boxed{}. A function \\(\\psi: \\mathbb{Z} \\rightarrow \\mathbb{Z}\\) is said to be zero-requiem if for any positive integer \\(n\\) and any integers \\(a_{1}, \\ldots, a_{n}\\) (not necessarily distinct), the sums \\(a_{1}+a_{2}+\\cdots+a_{n... |
[
{
"role": "user",
"content": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.It is well known that Berland has n cities, which form the Silver ring β cities i and i + 1 (1 β€ i... |
[
{
"role": "user",
"content": "Return your final response within \\boxed{}. Let $\\left(a_{n}\\right)_{n \\geq 0}$ be the sequence of rational numbers with $a_{0}=2016$ and\n\n$$\na_{n+1}=a_{n}+\\frac{2}{a_{n}}\n$$\n\nfor all $n \\geq 0$.\n\nShow that this sequence does not contain any square of a rational n... |
[
{
"role": "user",
"content": "In a fictional country where 80% of the land is owned by 10% of the population, what might be some potential social and economic impacts of this level of land inequality?"
},
{
"role": "assistant",
"content": "What a thought-provoking question! In a country where 80... |
[
{
"role": "user",
"content": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.There are n stairs, a person standing at the bottom wants to reach the top. The person can climb e... |
[{"role":"user","content":"Return your final response within \\boxed{}. Petya wrote all the positive(...TRUNCATED) |
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