CPhill

40^13 mod 81  = 22

https://web2.0calc.com/questions/rectangle_44

https://web2.0calc.com/questions/rectangle_37

The bags  must have been 3/5 of the total items

Let's call the number of total items  = T

So

Number of hats bought * 12.80  +  Number of bags bought * 23.30  =  191

(2/5)T* 12.80  + (3/5)T * 23.30  = 191

5.12T   + 13.98T = 191

T ( 5.12 + 13.98)  = 191 

T (19.1)  = 192

T = 191 / 19.1  =   10   items were bought

The number of bags =  (3/5) * 10   =   6

At the end, the number of each color sticker was  94080 / 4  =  23520

At first there must have been  23520 / 1.60 =  14700 red stickers

And there must have been  23520 / .70  =  33600 blue stickers

If 2/7  of the white stickers were colored green, then

Original White - (2/7)  original white = 23520

(5/7) Original White = 23520

Original White  = (23520)(7/5)  = 32928  

Then  32928(2/7) = 9408 original white stickers were coloured green

Then the number of green stickers at first must have been 23520 - 9408  = 14112

So....the total number of stickers at first must have been

14700 + 33600 + 32928 + 14112  =   95340 

Actually..we didn't need to worry about the green or white stickers....their total number (combined) didn't change 

We gained (23520 - 14700) = 8820 red stickers  and lost  (33600 - 23520)  = 10080 blue stickers

The net loss  =  10080 - 8820  = 1260

So  95340 beginning  - 1260 lost  =  94080  at the end

If 5 is the units digit of a perfect square.....the original squared integer must end in 5

Let the other part of the original integer be some non-negative  multiple of 10...call this part "a"

Then

(a + 5) (a + 5)  = 

a^2 + 10a + 25

a^2 will either = 0  or it will end in at least two 0's

10a  will = 0  or it will end in at least two 0's

Then the tens digit of the perfect square wil always = 2

Slope =  [ b^2 - a^2 ] / [ b - a]  = [ (b + a) (b -a) ] / (b - a)  = 2

So

a + b  =  2

Very nice, NTS !!!!

I like your solution....

https://web2.0calc.com/questions/geometry_35384

CA = 19

BA = 13

BC = 3

This isn't possible

ABC must form a triangle but  BC + BA < CA  which violates the triangle inequality