All Answers 358090 Answers

Let's tackle each of these questions one by one:

1. Find a set of three different whole numbers whose sum is equal to their total when multiplied.

   One such set of three different whole numbers that satisfy this condition is {2, 2, 3}.

   - The sum of the numbers: 2 + 2 + 3 = 6
   - The product of the numbers: 2 x 2 x 3 = 6

   As you can see, the sum (6) is equal to the product (6).

2. What is the smallest whole number that is equal to seven times the sum of its digits?

   Let's call the two digits of this number 'a' and 'b'. The number can be represented as 10a + b. According to the given condition, this number should be equal to seven times the sum of its digits, which is 7(a + b).

   Therefore, we have the equation:
   10a + b = 7(a + b)

   Let's simplify this equation:
   10a + b = 7a + 7b

   Now, subtract 7a from both sides of the equation:
   3a + b = 7b

   Next, subtract b from both sides:
   3a = 6b

   Divide both sides by 3:
   a = 3b

   So, the smallest whole number that satisfies this condition has two digits: one digit is 3 times the other digit. The smallest such number is 31.

3. What is the smallest number that increases by 12 when it is flipped and turned upside down?

   A number that remains the same when flipped and turned upside down is a "palindromic number." To find the smallest such number that increases by 12 when flipped, let's start by considering two-digit palindromic numbers. We can write them in the form of 'ABA,' where A and B represent digits.

   The smallest two-digit palindromic number is 11. When we flip and turn it upside down, it remains 11, which is not increased by 12.

   Let's consider the next two-digit palindromic number, 22. When flipped and turned upside down, it remains 22, which is also not increased by 12.

   Let's continue this pattern until we find a palindromic number that increases by 12 when flipped. 

   The next two-digit palindromic number is 33, and when we flip and turn it upside down, it becomes 33 + 12 = 45. So, the smallest number that increases by 12 when flipped and turned upside down is 33.

Please don't post AoPS Beast Academy questions.

Guess Who?  .....

Ok I’ll guess. It’s not a difficult guess. You’re TPM (The Puppet Master –aka The Blarney Master).

You decided to join in the fun and games when WAT started posting his (AI generated) bombastic prose toward me and then in my name.  You recognized this as a great opportunity to troll me and the forum as the Blarney Master that you are. We’ve had several public debates; all of them ending with you on the losing side.

Another way to find the lowest point would have  

been to recognize that the equation is a parabola  

opening upward, and solving for the vertex.  

_.

let P=x^2-6x-13. If x is real, what is the smallest value of P? what value of x produces this minimum value of P?   

When x = 3, then P = –22     I took the equation to Desmos graphing calculator.    

_.

1.   1, 2, 3    

2.   21   

3.   86    

_.

(a) To simplify (n/k)/(n/(k-1)), we can simplify the expression inside the parentheses first:

n/(k - 1) = n * 1/(k - 1) = n/(k - 1)

Now, we can rewrite the entire expression as:

(n/k)/(n/(k - 1)) = (n/k) * (k - 1)/n

Simplifying further:

= n * (k - 1)/(k * n)

= (k - 1)/k

Therefore, (n/k)/(n/(k - 1)) simplifies to (k - 1)/k.

(b) In the expansion of (1 + x)^n, the coefficients follow the pattern of Pascal's triangle. The pattern for the nth row of Pascal's triangle is given by the binomial coefficients for (n C 0), (n C 1), (n C 2), ..., (n C n).

In this case, we are given three consecutive coefficients that satisfy the ratio 1:8:28. This means that (n C 0) = 1, (n C 1) = 8, and (n C 2) = 28.

Using the formula for binomial coefficients, we can write this as equations:

(n C 0) = 1 --> n! / [(0!)(n - 0)!] = 1 --> 1 = n! / n! --> 1 = 1

(n C 1) = 8 --> n! / [(1!)(n - 1)!] = 8 --> n = 8(n - 1)

(n C 2) = 28 --> n! / [(2!)(n - 2)!] = 28 --> n(n - 1) = 56

From the first equation, we know that n = 1 is a solution. However, n must be a positive integer, so n = 1 is not valid.

Now, we can use the second and third equations to solve for n. 

From (n C 1) = 8, we have n = 8(n - 1). Expanding this:

n = 8n - 8
8n - n = 8
7n = 8
n = 8/7

From (n C 2) = 28, we have n(n - 1) = 56. Expanding this:

n^2 - n = 56
n^2 - n - 56 = 0

This quadratic equation can be factored as:

(n - 8/7)(n + 7) = 0

Setting each factor equal to 0, we have:

n - 8/7 = 0 --> n = 8/7

n + 7 = 0 --> n = -7

However, n must be a positive integer, so n = -7 is not valid.

Therefore, the only valid solution is n = 8/7.

Answer: n = 8/7.

N = 26.14

(-2x + 5)  / 7    + ( 3x - 1) / 2    simplify

[ 2 (-2x + 5)  + 7 ( 3x - 1) ] / [ 7 * 2 ]   =

[ -4x + 10  + 21x  - 7 ]  / 14

[17x + 3 ]  / 14

So

[17x + 3 ] / 14  = 0           and               [ 17x + 3 ] / 14   =1

x = -3/17                                                17x + 3 =  14

                                                               17x  =  11

                                                                  x =  11/17

The real numbers fall into  the  interval  [ -3/17 , 11/17 ]

3 meters =  3 x 100 centimeters = 300 centimeters

70 centimeters  = 70 centimeters

3 decimeters =  3 x 10 centimeters  = 30 centimeters

(300 + 70 + 30)  =    400 centimeters

Nope

For them to trigger again after 360 seconds,  56 must be a divisor of 180  (but , it isn't)

They will trigger again in  

56 =  2^3 * 7

180  = 2^2 * 3^2 * 5

2^3 * 7 * 3^2 * 5  =    2520 seconds  = 42 minutes

a + b + c  = 60

(a - 1) = N    →  a  = N + 1

(b + 2)  = N  →  b =  N - 2

3c  = N        →   c =  N /  3

(N + 1) + (N - 2) + N / 3  = 60

(7/3)N  =  61

N = (61 * 3)  / 7 =   183 / 7

a^2b^3  = 33/27     (1)

ab^3  = 27/4           (2)

Divide (1) by (2)

a = (33/27) / (27/4)  =   44 / 243

Using (2)

(44/243) * b^3  = 27/4

b^3 =  (27/4) / ( 44/243)   = 6561  /176

b = ∛ [ 6561 / 176 ] 

The vertex of the parabola described by the equation $y=-3x^2-30x-81+15x-9$ is $(m,n)$. What is $n$?   

                                   y  =  –3x² – 30x – 81 + 15x – 9    

                                   y  =  –3x² – 15x – 90    

The vertex is where the curve turns around and goes the other way.   

At that point, the slope of the curve is zero. 

The first derivative of an equation is the slope.   

So, set the first derivative equal to zero and solve for x.   

                                   y'  =  –6x – 15     

                                   –6x – 15  =  0          

                                           –6x  =  15         

                                               x  =  –2.5    

Plug x back into   

original equation           y  =  –3(–2.5²) – 15(–2.5) – 90   

                                      y  =  –3(6.25) + 37.5 – 90     

                                      y  =  –18.75 + 37.5 – 90   

                                      y  =  –71.25     (the "n" asked for in the problem is the y-coordinate)  

_.

a - b = -2 (c + d)    (1)

b = a - 5                 (2)

d = c + 3                 (3)

Sub (2) , (3) into (1)

a - (a -5)  = -2 (c + c + 3)

5 = -2 ( 2c + 3)

5 = -4c - 6

11 =  -4c

c = -11 / 4

"In your opinion after spending a few hundred thoudsand $ in updating and improving the website, the website will attract more visitors? And then the revenue will increase?"

Yes.

"Lacking the funds to rent a server with a powerful NVIDIA graphics card, I can't run a high-quality AI model"

Then don't run an AI model at all.

"Well, 99% of posts are spam or inappropriate content. You expecting that the volutary moderators of this forum should moderate this all manually in their free time is a little bit strange.."

That's a bit of an exaggeration. And yes, they are called moderators for a reason, they have powers - moderation powers that normal users dont. Everyone can answer questions, not everyone can moderate. Also, wouldn't deploying a senseless AI just increase the amount of work they have to do in terms of unhiding posts...?

"Sorry for the webite problems. A few years ago this website ran on more than a dozen high performance dedicated servers. But now it runs on a few cheap virtual low power servers"

It's not just server problems, the formulary is incomplete, for example.

"Especially if one of the many AI companies scrape the forum to train their models."

I doubt they'd want to.

"Students have no time to spend on solving math problems, I guess they are to busy watching TikTok all day."

Damn right.

6x+99=7x-189x solve this equation    

                                    6x + 99  =  7x – 189x       

Combine like terms.   

We could do several at once  

but I'll take them singly.  

                                    6x + 99  =  7x – 189x       

                                    6x + 99  =  –182x       

                                            99  =  –188x       

                                            99  

                                          ––––  =  x    

                                         –188           

                                                x  =  –0.5266   

_.

I am a three digit number. My tens digit is five more than my ones digit. My hundreds digit is eight less than my tens digit. What number am I?  

                       H  =  hundreds digit  

                       T  =  tens digit   

                       U  =  ones digit       

                       T – H  =  8  

Only two ways to do this.  If T=8 or if T=9   

But if T=8 then H=0 and we can't have that.   

So T=9 and, knowing that, the rest is a cinch.  

                       H  =  1    

                       T  =  9     

                       U  =  4        

The number in question is 194.    

_. 

194  fits the bill

Hi "GW", thanks for your feedback!

It's about time you started putting education before profits.

Lol. What profits?

If you truly wanted this site to become a useful repository for future students and a welcoming community for those who decide to post a question, you wouldn't have to worry about making enough money to keep this piece of junk afloat.

Now the forum generates $0 income. In your opinion after spending a few hundred thoudsand $ in updating and improving the website, the website will attract more visitors? And then the revenue will increase? Like what, 50%? Let's see, $0 + 50%, Wow, nice.

deploying your hopeless AI moderator

Lacking the funds to rent a server with a powerful NVIDIA graphics card, I can't run a high-quality AI model. Renting a Google Cloud "a2-highgpu-1g" would be too costly. If you're open to donating several thousand dollars monthly, please inform me. Currently, I'm using a smaller 7 billion parameter model. Further reducing its capability with 4-bit quantization to fit a budget server inevitably limits its performance.

We do not need your attempts at moderation

Well, 99% of posts are spam or inappropriate content. You expecting that the volutary moderators of this forum should moderate this all manually in their free time is a little bit strange..

A place where users struggle daily with errors and malfunctions while

Sorry for the webite problems. A few years ago this website ran on more than a dozen high performance dedicated servers. But now it runs on a few cheap virtual low power servers. Of course there are very often peformance problems resulting in all kinds of timeout errors. Especially if one of the many AI companies scrape the forum to train their models.

the only questions being posted are still sham AoPS class material.

Learning math has no future, at least in the western world. The traffic to all math related websites clearly show that. Many students have no time to spend on solving math problems, I guess they are to busy watching TikTok all day.