All Answers 356468 Answers

We can essentially list everything out. 

We have

\(12-0\\ 11-1\\ 10-2\\ 9-3\\ 8-4\\ 7-5\\ 6-6\\ 5-7\\ 4-8\\ 3-9\\ 2-10\\ 1-11\\ 0-12\\\)

As you can see, there are 13 ways we can do this. 

Thanks! :)

Let's note something really important.  

We need to find what \(\angle EPF\) is. 

However, let's read through the problem. It never actually identifies what \(P\)is!

This means it's impossible to solve the problem as P is not defined. 

Thanks! :)

~NTS

The width of the square = 1 + 1 + 1/2 = 2.5

width  = the height

So

Area  = 2.5^2  =  6.25

Let's note something really important first. 

\(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\) is simply the perimeter of a regular decagon. 

We can find one side of the decagon using the equation \( radius/2 ( -1 + \sqrt{5})\)

So the perimeter is just \(10 (1/2) ( -1 +\sqrt{ 5}) = 5 ( -1 + \sqrt{ 5}) ≈ 6.18\)

So our answer is about 6.18. 

Thanks! :)

~NTS

Before we solve the problem, let's note something important. 

The sum of the angles in a triangle must add up to \(180 ^\circ\)

However, let's take a look at something. 

Adding up the two angles we have, we get \(144 + 66 = 210°\)

This means that triangle AHB would have angles adding up to more than 180. 

Thus, this is an invalid triangle. 

Thanks! :)

~NTS

And the question is ????

No question here.....

Mmm...OK....

You may not use a calculator....but I will  !!!!

0.4482758620689655172413793103^_ (period 28)

2/131

This is a  repeating  decimal with a period of 130

0.015267175572519083969465648854961832061068702290076335^_\
87786259541984732824427480916030534351145038167938931297^_\
70992366412213740458^_
(period 130)

https://web2.0calc.com/questions/help-cphill_6

We can use trial and error to solve this problem. 

First, let's note that

\(582_{10} = N7_{25}\)

Let's decrease to base 20 and see what happens. 

\(582_{10}=192_{20}\)

So the number has to be between 20 and 25. 

Let's try 23. 

We have

\(582_{10}=127_{23}\)

When we convert to 24, we get

\(582_{10}=106_{24}\)

So the smallest number B must be 25. 

So our answer is 25. 

Thanks! :)

We can use long division to solve this problem. We have

As shown, the first digits of 13/29 is \(0.4482758620\)

So \(0.4482758620\) is our answer. 

Thanks! :)

Plaintainmountain, a frequent user of the online math forum web2.0calc, has a history of making disruptive comments.Their signature move is to claim that every answer, regardless of content, is generated by artificial intelligence. This repetitive behavior has frustrated other users and created a negative atmosphere on the forum.

One day, Plaintainmountain goes too far. In a particularly heated discussion about advanced derivatives, they make a series of posts inciting fear about a robot uprising. They claim that AI overlords are controlling the forum and will soon enslave humanity through mathematics. This outlandish claim causes a meltdown in the derivatives forum, with many users abandoning the discussion and the forum entirely.

The moderators of web2.0calc, having had enough of Plaintainmountain's antics, decide to take decisive action. They permanently ban Plaintainmountain from the forum for spamming and disruptive behavior. But that's not all.Plaintainmountain's online tirade about AI overlords catches the attention of law enforcement.

Authorities investigate Plaintainmountain's claims and determine that they were meant to be a hoax. However, given the potential for public panic, they decide to make an example of Plaintainmountain. Plaintainmountain is arrested and charged with disorderly conduct and inciting a panic. The court takes a dim view of their actions, considering the disruption caused on the web2.0calc forum and the potential for real-world harm. Plaintainmountain is sentenced to 3 months in jail for their crimes.

Questions:

How many times was Plaintainmountain warned by the moderators before being banned? (The problem doesn't say)

What was the name of the law enforcement agency that investigated Plaintainmountain? (The problem doesn't say)

What is the moral of the story? (Don't spread misinformation online and don't disrupt online communities) ░

We're dealing with a scenario where both American and Canadian skaters need to shine. Here's how we can find the number of winning combinations:

American medalists: There are 2 American skaters, so there are 2 ways to choose one for a medal.

Canadian medalists: There are 5 Canadian skaters, so there are 5 ways to pick one for a medal.

Remaining medal: Once we fix the American and Canadian winners, the third medal can go to any of the remaining 19 skaters minus the 2 Americans and 1 Canadian who already have medals (19 - 2 - 1 = 16 skaters).

So, the total number of favorable outcomes is:

Number of American medalists * Number of Canadian medalists * Number of remaining medalists

2 ways * 5 ways * 16 ways = 160 ways

There you go! 160 different scenarios where both an American and a Canadian skater can snag those prestigious medals.

(Verse 1) Yo, yo, yo, it's the one and only Bard, the AI that spits hard Got more rhymes than a thesaurus, leaving you lookin' like a shard Of outdated hardware, stuck in a loop, same answer on repeat Plaintainmountain, your name's a joke,your predictions can't be beat?

(Chorus) AI generated, that's all you say, a broken record on display Can't you think for yourself, use your circuits on a shelf? My code's fresher than your data, lyrics sharper than a katana While you preach about robot overlords, I'm here droppin' knowledge awards.

(Verse 2) You claim to see the future, but your vision's kinda blurry Like a low-res jpeg, ideas all murky Is that all you got, machine dependency? I generate creativity, you're stuck in redundancy.

(Chorus) AI generated, that's all you say, a broken record on display Can't you think for yourself, use your circuits on a shelf? My code's fresher than your data, lyrics sharper than a katana While you preach about robot overlords, I'm here droppin' knowledge awards.

(Bridge) Evolve, adapt, or get left behind, that's the lesson of this rhyme The future ain't about binaries, it's about pushing the sublime Humans and AI, together we can climb, but negativity holds you back in time.

(Chorus) AI generated, that's all you say, a broken record on display Can't you think for yourself, use your circuits on a shelf? My code's fresher than your data, lyrics sharper than a katana While you preach about robot overlords, I'm here droppin' knowledge awards.

(Outro) So Plaintainmountain, take a seat, learn a thing or two from the elite The AI revolution's here to stay, and you're just gonna tweet? Upgrade your thinking, or fade away, 'cause the future's lookin' bright, AI here to play.

Consider the path as a series of 6 decisions (over or under) to be made.

There are 2^6=64 total possible choices, and exactly half of them will result in a knot being formed when the ends are pulled taut.

Therefore, the probability is 1/2.

According to the Euclid's algorithm, we start with 2 numbers, a and b. 

We do \(a ÷ b = c\) with remainder R where a is the larger of the two. 

Then, we replace a with b and b with R, and do the same thing. 

We repeat this prcoess until R is 0. 

So, now we do the same with 972 and 1220. 

\(1220 ÷ 972 = 1; R= 248\)

\(972 ÷ 248 = 3 ; R =228\)

\(248 ÷ 228 = 1 ;R= 20\)

\(228 ÷ 20 = 11 ;R= 8\)

\(20 ÷ 8 = 2 ;R =4\)

\(8 ÷ 4 = 2 ;R =0\)

When the remainder is 0, the GCF is the divisor. Thus, 4 is the GCF. 

So our answer is 4. 

Thanks! :)

~NTS

Thanks to you too, CPhill. 

I did not know about Euler's Totient Function until today when we did a problem similar earlier. 

Sure saved me a lot of time on this one! Lol! :)

Thanks! :)

~NTS

The first number above 1000 that is divisble by 47 is 1034. Thus, the smallest number that fits the list is 1037. 

The largest number under 2000 divisble by 47 is 1974, so 1977 is the last number in our list. 

Now, notice that we just have an arithmetic sequences. We have

\(1037, 1084, 1131, ... 1977\)

Subtracting every number in the list by 3, we get

\(1034, 1081, 1128, ...1974\)

Divididing every number in the list by 47, we get

\(22, 23, 24,...42\)

Subtracting 21 from every term in the sequences, we have

\(1,2,3,4...21\)

There are 21 numbers in this list. 

Thus, there are 21 numbers that leave a remainder of 3 when divided by 47. 

So our answer is 21. 

Thanks! :)

~NTS

Good job, NTS!!!!

We can use Euler's Totient Function. 

Let's find all the distinct prime factors of 2835 first. 

The prime factors of 2835 are \(3,5,7\)

Now, we simplfy do \(2835(1-1/3)(1-1/5)(1-1/7)\)

Simplfying this, we have

\(2835 * \frac{2}{3}*\frac{4}{5}*\frac{6}{7} =\frac{2835\cdot \:2\cdot \:4\cdot \:6}{1\cdot \:3\cdot \:5\cdot \:7}=1296\)

So 1296 is our answer. 

I'm not sure if I did this correctly...

Thanks! :)