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1 inch = 4 feet

1 inch = 4x12 = 48 inches

scale factor 1:48    

Between 1 and 99 =900 and 

Between 100 and 999 =12,600

12,600 + 900 =13,500 between 1 and 999 inclusive.

.94s = 1.14 v      where s and v are the priginal populations

s/v = 1.14/.94

     = 114 /94    = 57/47  

As this is an AoPS question I will disclose a hint.

use the center. Use approximation to guess. It might take a try or two.

I think that you have to add them all up sequentially, that is: 1+2; 1+3; 1+5.....etc. When you finish with 1 then you start with 2: 2+3; 2+5; 2+8.......and so on. When you do that, this is what you will get:

(3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 21, 22, 23, 24, 26, 29, 34, 35, 36, 37, 39, 42, 47, 55, 56, 57, 58, 60, 63, 68, 76, 89). From this list, you can easily tell that you cannot make 12 or 17 or 19 or 20.......and so on. Just go through the entire list and write down the numbers that are not on the list, all the way up to 89. The list has 36 numbers, which means you have to write down: 89 - 36 =53 numbers that are not on the list. Make sure to count them to ensure that you have written down all 53 numbers.

Oh thank do you have any work on the second one yet?

The answer to the first is 2889. I'm stuck on the second one though.

What is the answer?

Here is a hint.

Using angle equalities what type can your prove that TSR and PQR are?

As this is an AoPS question please do not post full solutions, but hints are ok.

Good job violetzhang07 !. That is exactly right.

Hold up, I accidentally did the wrong perimeter. If you plug in 56 instead of 18.4, you will get the answer! Sorry about the inconvinience!

What is wrong with these numbers?
x        y        z
1        3        8
1         4       5
2         2       4
z =8 + 5 + 4 = 17

If they can be arranged into 24, different ways, there are 6 ways each number gets. So, since 3142 starts with 3, it would be past the 12 mark. If you think about it there is only one smaller number than 3142, and that is 3124. So, 3142 would be after 3124, and 3124 is number 13. So therefore, 3142 is in the 14th position. 

LMK if that is correct! I didn't write it down or anything so I might've got something wrong but you can think about it this way!

thank you so much you are the best

Were any of these correct?

Thank you!

P = 2L + 2w

18.4 = 2(3w+4) + 2w

18.4 = 6w + 8 + 2w

18.4 = 8w + 8

10.4 = 8w

w = 10.4/8 = 1.3

L = 3(1.3)+4 = 3.9 + 4 = 7.9

I hope that answers what you are looking for! 

a+b+c =6.

There are these ways to make 6:

3+2+1

2+2+2,

We can then sort this list into 3 dfferent, 2 different, and no different.

Lets start easy.

3 same gets us only one way to permutate the numbers.         1

2 same gives us 2! ways to permutate the numbers(But we don't have any 2s)     0

and three different gives us 3! ways to permute          6

We have 7 ways to make 6 out of non zero positive whole numbers.

The product of  a·cis(A)  and  b·cis(B)  is  (a·b)·cis(A + B).

For your problem, find the values of a, A, b, and B.

Then place them into the formula (and simplify).

Take out a sheet of graph paper.

1)  Plot the point (1,0); call this point A.

2)  Plot the point (2,2); call this point B.

3)  Plot the point (4,5); call this point C.

4)  Draw the line segments AB and BC; these will be the bottom and the right side of the parallelogram.

5)  Use this diagram to find point D.

6)  Find the length BC using the distance formula:  distance  =  sqrt[  (x₂ - x₁)² + (y₂ - y₁)² ] 

7)  Find the distance between BC and AD.

8)  Use the results of step 6 and step 7 to find the area of the parallelogram.

The first equation:  y  =  |x + 2| - 2  has its vertex at (-2,-2) and opens upward. The equation of its line of symmetry

is y  =  -2.

The second equation:  x  =  |y + 4|  has its vertex at (0,-4) and opens to the right. Can you find its line of symmetry?

If the original length is x and the original width is y, then the area will be:  A  =  xy.

If you increase the length by 100%, you are doubling the length to 2x.

If you increase the width by 100%, you are doubling the width to 2y.

The new area will be:  A = 2x·2y = 4xy.

The increase in area will be:  4xy - xy  =  3xy.

The percentage increase will be:  3xy / xy = 3, or 300%.

The seven consecutive integers can be written as:

\(x+(x+1)+(x+2)+(x+3)+(x+4)+(x+5)+(x+6)\)= \(7x+21\)= \(77\)

\(7x+21=77\)

\(7x=56\)

\(x=8\)

Since x is the smallest integer and (x+6) is the largest answer, 8+6 = 14 which is the largest number.