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We can solve for n  thusly

Sum =   (first term + last term)  (number of terms) /2

Note.....number of terms  = n - 20 + 1  =  n -19

So  we  have that

(20 + n)  ( n - 19)/2  =  2021    multiply through by 2    and simplify

n^2 + n - 380   =  4042          subtract  4042 from both sides

n^2  + n - 4422   =   0  

We can factor this as

(n + 67)  ( n - 66)    =  0

The second factor set  =   0   gives us the positive answer we  need  for n

n - 66    =  0

n  =  66

See the graph here  :  https://www.desmos.com/calculator/bjpubu0a6p

Boundary points  =  5  +  5    =10

Interior  points  = 24 

So....34  lattice points

Incidentally.....area of  bounded region  =   24  + 10/2  - 1   =  28

\(\sum_{20}^{66}\) n   = 2021

When Caroline walks 20 seconds, she walked 30 meters.

Each hour has 3600 seconds, which is 180 repetitions of 20 seconds. So she walked 180 repetitions of 30 meters, which is 5400 meters

log _2.62074 (18) = 3

m - n = 17.

Ok thanks melody! I will sign my name from now on!

-Grace 

600 - 3 = 597

597 x 1/3 =199 - which is your prime number p.

199 x 2 = 398

398 =[1, 2, 199, 398] - Number of its divisors.

Sum[1 + 2 + 199 + 398]== 600

See 2 solutions here: https://web2.0calc.com/questions/help-me_17340

a=1; b=1; if(gcd(24 , b)==(a + 4) and lcm(24,b)==a*(a+4), goto loop, goto next);loop:printa, b;print gcd(24, b);print lcm(24,b);next: a++;if(a<1000, goto2, 0);a=1;b++;if(b<1000, goto2, discard=0; 

OUTPUT:

x = 20   and   GCD =20 +4 =24   and   LCM =20*(20 + 4) = 480

The smallest possible value of 2nd number = 480, so that:

GCD[24, 480] == 24   and   LCM[24, 480] ==480, which agrees with the value of x ==20.

a=0;p=1; b=0;c=0;n=(a^3+b^3+c^3);if(a!=b and a!=c and b!=c and n<100, goto loop, goto next);loop:printp," - ", a, b, c," = ",n;p=p+1;next:a++;if(a<100, goto4,0);a=0;b++;if(b<100, goto4, 0);a=0;b=0;c++;if(c<100, goto4,0)

OUTPUT:

n     x y z       sum

1  -  2 1 0  =  9
2  -  3 1 0  =  28
3  -  4 1 0  =  65
4  -  1 2 0  =  9
5  -  3 2 0  =  35
6  -  4 2 0  =  72
7  -  1 3 0  =  28
8  -  2 3 0  =  35
9  -  4 3 0  =  91
10  -  1 4 0  =  65
11  -  2 4 0  =  72
12  -  3 4 0  =  91
13  -  2 0 1  =  9
14  -  3 0 1  =  28
15  -  4 0 1  =  65
16  -  0 2 1  =  9
17  -  3 2 1  =  36
18  -  4 2 1  =  73
19  -  0 3 1  =  28
20  -  2 3 1  =  36
21  -  4 3 1  =  92
22  -  0 4 1  =  65
23  -  2 4 1  =  73
24  -  3 4 1  =  92
25  -  1 0 2  =  9
26  -  3 0 2  =  35
27  -  4 0 2  =  72
28  -  0 1 2  =  9
29  -  3 1 2  =  36
30  -  4 1 2  =  73
31  -  0 3 2  =  35
32  -  1 3 2  =  36
33  -  4 3 2  =  99
34  -  0 4 2  =  72
35  -  1 4 2  =  73
36  -  3 4 2  =  99
37  -  1 0 3  =  28
38  -  2 0 3  =  35
39  -  4 0 3  =  91
40  -  0 1 3  =  28
41  -  2 1 3  =  36
42  -  4 1 3  =  92
43  -  0 2 3  =  35
44  -  1 2 3  =  36
45  -  4 2 3  =  99
46  -  0 4 3  =  91
47  -  1 4 3  =  92
48  -  2 4 3  =  99
49  -  1 0 4  =  65
50  -  2 0 4  =  72
51  -  3 0 4  =  91
52  -  0 1 4  =  65
53  -  2 1 4  =  73
54  -  3 1 4  =  92
55  -  0 2 4  =  72
56  -  1 2 4  =  73
57  -  3 2 4  =  99
58  -  0 3 4  =  91
59  -  1 3 4  =  92
60  -  2 3 4  =  99
 

I don't know! 

I don't know! 

Thanks Grace, 

I have let admin know.

In the mean time could you please sign your name 'Grace' at the end of your posts so we know who you are.

And don't worry, you haven't done anything wrong.

I was really only upset with the asker.   

What is the largest number less than 1000 such that the remainder is 1 when is divided by each of 3, 4, 5, 6, and 7?  

My submission is 841.  I multiplied 4 x 5 x 6 x 7 and got 840 then added 1. 

I didn't multiply by 3 because that ends up over a 1000 and anyway the 6 took care of the 3. 

The area of the triangle AOD is 1/4 of [ABCD].

To find a radius of a circle we ought to multiply [AOD] by 2 and then divide the product by 25.

r = 300 / 25  

I am only on as a guest because i tried to register for an account at the end it said to check my email for account conformation or something and i did but there was nothing in my email and when i tried to log in it said invalid username or password......

​​​​​​​​​​​​​​​​​​​​

-Grace

I dont understand what you are trying to find.....

Do you want to find the value of x and y?

If so you should have put that in the question.....

I actually do not know who answered that.  That is the problem when people post as guests.

I cannot comment because I do not know which answerer I am talking to.

No do I know which answer/s you are referring to.  

What values of x satisfy |x - 4| + |x + 4| <= 20?

x-4 is positive  when x>4

x+4 is positive when x>-4

So we have three regions to consider.

x<=-4,    x>=4    and  -4

If x <= -4 then

|x - 4| + |x + 4| <= 20

 -(x-4) +  -(x+4) <=20

-x+4 -x-4  <= 20

-2x<= 20

x>=-10

So the statement is true for   -10 <=x <= -4

You can do the other 2 regions.

Let X, Y, and Z be points such that XZ/XY = ZY/XY = 1/2. If Y = (1, 7), Z = (-1, -7), then what is the sum of the coordinates of X?

Coordinates of X are:      X ( -2/3, -7/3)

The sum of the coordinates of X is -3

I'm not Guest

Thank you, CPhill ! 

By Cauchy-Schwarz, to get the minimum value, let a = b = c = 1/3.  So the minimum value is (1/3)^2 + 2*(1/3)^2 + (1/3)^2 = 4/9.

A

B = 1.5 A

C+D  = 2 (A+ 1.5A)

E = 10  

Totaled = 100

A + 1.5A + 2(A + 1.5A) + 10 = 100

7.5 A + 10 = 100

A = 12

   Then B = 1.5 A = 1.5 (12) = 18 miles

Hi! It's a pretty long process, so I won't post how i got the answer, but I got 7. Hope it's correct!!