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"content": "Given the sequence $(a_n)$ with $a_1=1$ and $|a_n| = |a_{n-1}+2|$ for every positive integer $n \\geq 2$, what is the minimum possible value of $\\sum_{i=1}^{2000} a_i$?",
"role": "user"
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"content": "<think>\nOkay, let's try to figure out this problem. So, we have a sequence where ... |
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"content": "Find all functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that \\( f(x + y) = f(\\ln xy) \\) for all \\( x, y > 0 \\).",
"role": "user"
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"content": "<think>\nOkay, let's see. I need to find all functions f from the real numbers to the real numbers such that for any p... |
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{
"content": "In how many ways can five numbers be chosen from the numbers $1, 2, 3, \\ldots, 12$ arranged in a circle so that no two adjacent numbers are selected?",
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"content": "<think>\nOkay, so I need to figure out how many ways I can choose five numbers from the numbers 1 to ... |
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"content": "Let \\( x, y, z > 0 \\) such that:\n\\[\n\\begin{cases}\nx^2 + xy + y^2 = 9 \\\\\ny^2 + yz + z^2 = 16 \\\\\nz^2 + zx + x^2 = 25\n\\end{cases}\n\\]\nEvaluate \\( P = xy + yz + zx \\).",
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"content": "<think>\nOkay, let's see. I have this problem where I need to find th... |
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"content": "Determine the value of\n\\[2002 + \\frac{1}{2} \\left( 2001 + \\frac{1}{2} \\left( 2000 + \\dots + \\frac{1}{2} \\left( 3 + \\frac{1}{2} \\cdot 2 \\right) \\right) \\dotsb \\right).\\]",
"role": "user"
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"content": "<think>\nOkay, let's see. I need to solve this nested fraction expr... |
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{
"content": "In a 2-player game played on $n \\geq 3$ points (where no 3 points are collinear), each move consists of selecting 2 points and drawing a new line segment connecting them. The first player to draw a line segment that creates an odd cycle loses. An odd cycle must have all its vertices among the $n$... |
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{
"content": "Let \\( a = \\underbrace{19191919 \\ldots 1919}_{19 \\text{ is repeated } 3838 \\text{ times}} \\). What is the remainder when \\( a \\) is divided by 13?",
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"content": "<think>\nOkay, so I need to find the remainder when this huge number a is divided by 13. The numb... |
[
{
"content": "What is the density function of the x-coordinate when a point is picked randomly and uniformly from the perimeter of a unit circle?",
"role": "user"
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"content": "<think>\nOkay, so the problem is asking for the density function of the x-coordinate when a point is picked randomly and... |
[
{
"content": "For a given positive integer $n$, evaluate the expression $ \\sum_{d|n} d \\sum_{k|d} \\frac{ \\varphi(k) \\varphi(d/k)}{k}$.",
"role": "user"
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"content": "<think>\nOkay, so I need to evaluate this sum: for a given positive integer n, it's the sum over all divisors d of n of [d mul... |
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{
"content": "Let \\( A \\), \\( B \\), \\( C \\), and \\( D \\) be points on a circle such that \\( AB = 11 \\) and \\( CD = 19 \\). Point \\( P \\) is on segment \\( AB \\) with \\( AP = 6 \\), and point \\( Q \\) is on segment \\( CD \\) with \\( CQ = 7 \\). The line through \\( P \\) and \\( Q \\) intersect... |
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{
"content": "The idea of moral injury involves\nA. the natural law idea of the doctrine of double effect.\nB. the Rossian idea of prima facie duties.\nC. the utilitarian idea of greatest good for the greatest number.\nD. the Kantian idea of the equal dignity of all persons.\nPlease think step by step and provi... |
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{
"content": "In triangle \\(ABC\\), point \\(M\\) is on segment \\(AB\\) such that \\(BM = 2AM\\). Given that \\(\\angle ACM = 15^\\circ\\) and \\(\\angle CMB = 60^\\circ\\), find \\(\\angle ACB\\).",
"role": "user"
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"content": "<think>\nOkay, so I need to find the measure of angle ACB in trian... |
[
{
"content": "What is the probability that a number chosen at random between 0 and 60 is less than a number chosen at random between 10 and 70?",
"role": "user"
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"content": "<think>\nOkay, let me try to figure out this probability problem. So the question is: What is the probability that a numbe... |
[
{
"content": "Scientists analyzed the cells of an animal to determine the cause of an infection. The scientists did not agree on the results of the analysis. What does this disagreement indicate about the nature of science?\nA. Scientists can present data in different formats.\nB. Scientific results can be inte... |
[
{
"content": "Write a persuasive article that highlights the luxurious experience and culinary artistry of non-vegan high-end restaurants. The article should be structured into three sections, each clearly highlighted with a subheading. Conclude the article with the exact sentence: \"In every bite, a story of p... |
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{
"content": "A 3-digit number \\(abc\\) in base 6 is equal to \\(cba\\) in base 9. Find the digits \\(a\\), \\(b\\), and \\(c\\).",
"role": "user"
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"content": "<think>\nOkay, let's tackle this problem. So, we have a 3-digit number in base 6, which is written as abc (base 6), and this same numbe... |
[
{
"content": "The number \\(n\\) can be written in base \\(14\\) as \\(\\underline{a}\\) \\(\\underline{b}\\) \\(\\underline{c}\\), in base \\(15\\) as \\(\\underline{a}\\) \\(\\underline{c}\\) \\(\\underline{b}\\), and in base \\(6\\) as \\(\\underline{a}\\) \\(\\underline{c}\\) \\(\\underline{a}\\) \\(\\under... |
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