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1. We start with the given conditions: \(a_1 = 1\) and \((m+n)a_{m+n} \le a_m + a_n\) for all positive integers \(m\) and \(n\).
2. We need to show that \(\frac{1}{a_{200}} > 4 \cdot 10^7\).
3. First, we calculate \(a_2\):
\[
2a_2 \le a_1 + a_1 = 2 \implies a_2 \le 1
\]
4. Next, we calculate \(a_3\):
\[
... | The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$ . Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ .
. | math | AI-MO/NuminaMath-CoT/aops_forum | {'source': 'aops_forum', 'messages': [{'content': 'The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\\le a_m +a_n$ for all positive integers $m$ and $n$ . Show that $\\frac{1}{a_{200}} > 4 \\cdot 10^7$ .\n.', 'role': 'user'}, {'content': '1. We start with the given conditions: \\(a_1... | null | [
"The problem gives a condition on the sequence of positive numbers $a_1, a_2, ...$: $(m+n)a_{m+n} \\le a_m + a_n$ for all positive integers $m$ and $n$. We are given $a_1 = 1$. We need to show that $\\frac{1}{a_{200}} > 4 \\cdot 10^7$.\n\nLet's explore the given inequality with small values of m and n.\nFor m=1, n=... | Solution:
We are given the condition $(m+n)a_{m+n} \le a_m + a_n$ for all positive integers $m$ and $n$.
Let $m=n$. Then $2n a_{2n} \le a_n + a_n = 2a_n$, so $n a_{2n} \le a_n$.
This implies $a_{2n} \le \frac{a_n}{n}$.
By repeatedly applying this inequality, we have:
$a_{2^k} \le \frac{a_{2^{k-1}}}{2^{k-1}} \le \frac{... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\le a_m +a_n$ for all positive integers $m$ and $n$ . Show that $\frac{1}{a_{200}} > 4 \cdot 10^7$ .
.<|im_end|>
<|im_st... |
null | null | null | null | null | null | null | null | General Question
MilkyWay's avatar
What exactly is classed as an "assault" ?
Asked by MilkyWay (13162 points ) January 26th, 2012
Does the violent action have to leave a mark? What acts are classed as assault?
Observing members: 0 Composing members: 0
8 Answers
WestRiverrat's avatar
Assault legally is the threa... |
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To JHK Please take a look at Gdingen and Gdynia... |
null | null | null | null | null | null | null | null | Concerning the death of Mr. Hummell when the police shot him, he had the sword. Does the Las Cruces Police Department have tazers? I don't recall ever reading anything about how they tazed somebody.
If the county is in such bad shape financially, why did they allow the Sheriffs Department to promote eight people to ca... |
null | null | null | null | null | null | null | null | Short of destroying a whole world with planet-breaking weapons, the most action-filled moments in science fiction come when opposing spacecraft clash. As phasers fire and missiles launch, ships frantically maneuver, attack, or spectacularly explode. But sometimes the aim is to capture a spaceship intact, or if she has ... |
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January 15
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When you sign a two year contract with your wireless carrier, you probably know there's a termination fee if you want to get out early. You may think that fee could be waived if you could demonstrate a real need to end your contract, but this s... |
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Celebrity Phobia’s
Yep! They are just as goofy as we are. Being a famous actor or actress requires a huge amount of travel, so you wouldn’t think a jet setting celebrity would have a fear of flying […]
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80. Re: Diablo III Hotfixes May 25, 2012, 13:12 xXBatmanXx
nin wrote on May 25, 2012, 09:19:
Elim wrote on May 25, 2012, 05:17:
I feel like a fucking retard for buying it.
Then the disconnects, lags etc.
60 Euros was... |
null | null | null | null | null | null | null | null | Every single one of you that is reading this has a birthday. A given fact.
Accepting that fact is step number six of my 11 steps on aging gracefully. It reads "Accept some realities." My reality is I plan on having just as many birthdays as I can.
I had a birthday a couple months ago and the cards I received are a cl... |
262 | The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$ . There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$ . Find $c_k$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-9', 'Year': 2016, 'Problem Number': 9, 'Part': 'II'} | null | [
"\nThe problem asks for the value of $c_k$, given information about an arithmetic sequence and a geometric sequence whose terms are summed to form a third sequence.\n\nLet the arithmetic sequence be $a_n = 1 + (n-1)d$, where $d$ is a positive integer (since the sequence is increasing and terms are integers).\nThe f... | Solution:
Let the arithmetic sequence be $a_n = 1 + (n-1)d$, where $d$ is a positive integer.
Let the geometric sequence be $b_n = r^{n-1}$, where $r$ is an integer greater than 1.
The sequence $c_n = a_n + b_n = (1 + (n-1)d) + r^{n-1}$.
We are given $c_{k-1} = 100$ and $c_{k+1} = 1000$.
$c_{k-1} = 1 + (k-2)d + r^{k-2... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$ . There is an integer $k... |
728 | Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,$ and $61$ . | math | qq8933/AIME_1983_2024 | {'ID': '2016-II-8', 'Year': 2016, 'Problem Number': 8, 'Part': 'II'} | null | [
"\nThe problem asks for the number of sets $\\{a,b,c\\}$ of three distinct positive integers such that $abc = 11 \\times 21 \\times 31 \\times 41 \\times 51 \\times 61$.\n\nFirst, let's find the prime factorization of the product $P = 11 \\times 21 \\times 31 \\times 41 \\times 51 \\times 61$.\nThe prime factorizat... | Solution:
Let the product be $P = 11 \times 21 \times 31 \times 41 \times 51 \times 61$.
The prime factorization of $P$ is $3^2 \times 7^1 \times 11^1 \times 17^1 \times 31^1 \times 41^1 \times 61^1$.
We are looking for sets $\{a,b,c\}$ of three distinct positive integers such that $abc = P$.
Consider the number of or... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,$ and $61$ .<|im_end|>
<|im_start|>as... |
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null | null | null | null | null | null | null | null | Two Door Cinema Club returned with Beacon after a couple of years touring in support of their debut album, Tourist History. While that set of songs was already pretty sleek thanks to the production skills of Eliot James, the band opted to polish things further with the help of Jacknife Lee, who has worked with R.E.M., ... |
To determine how many integers from $1$ to $2023$ inclusive are in the range of the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$, we need to analyze the behavior of the function and find the range of values it can produce.
1. **Analyze the function components:**
-... | Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$ . How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$ ? Note that $\lfloor x\rfloor$ is known as the $\textit{floor}$ function, which returns th... | math | AI-MO/NuminaMath-CoT/aops_forum | {'source': 'aops_forum', 'messages': [{'content': 'Define the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}} \\rfloor$ for all positive real numbers $x$ . How many integers from $1$ to $2023$ inclusive are in the range of $f(x)$ ? Note that $\\lfloor x\\rfloor$ ... | null | [
"\nThe problem asks for the number of integers from $1$ to $2023$ inclusive that are in the range of the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}} \\rfloor$, where $x > 0$.\n\nLet $y = f(x)$. We are looking for the number of distinct integer values of $y$ such t... | Solution:
The function is defined as $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for $x > 0$.
The range of $f(x)$ consists of the values taken by $f(n)$ for positive integers $n$, since the jumps in the function occur at integer values of $x$.
Let $f(n) = n + \lfloor \sqrt{n}... | <|im_start|>system
You are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>
<|im_start|>user
Define the function $f(x) = \lfloor x \rfloor + \lfloor \sqrt{x} \rfloor + \lfloor \sqrt{\sqrt{x}} \rfloor$ for all positive real numbers $x$ . How many integers from $1$ to $2023$ inclusive are in ... |
null | null | null | null | null | null | null | null | Lightning 1, Sabres 1, OT
Via: AP
We battled to the bitter end, Ruff said. We could have easily won the game. We skated better than they did in the third period, although they might have been just trying to preserve the one point, which was important to them. Sometimes you look over and they were not taking a lot of ... |
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