ThatOnePerson

That means that there are 12 numbers that are not multiples of 5. Also, 8 integers that are multiples have to be even. 12+8 = 20 integers.

No wait, Guest's answer may be wrong if my calculations are correct. Since AT and AS trisect angle DAB, that means that angle SAD = 30 degrees and so triangle SAD is a 30-60-90 triangle. Since it's a 30-60-90 triangle, AD = sqrt3. Looking at TAD now, we get that it's a 30-60-90 triangle as well. That means that TD = sqrt3 * sqrt3 = 3. Adding CT and TD gives us that CD = 2sqrt3. Multiplying that by sqrt3, we get that the area of ABCD = 2sqrt3 * sqrt3 = 6. Hence, the area of ABCD is 6.

I am confused on how you got the 3 out of nowhere. Could you please explain more?

2. I don't see the image, could you please attach? Thanks!

1. You know that triangles EFC and BDE are similar by AA property. We also know that FC = 12-5 = 7. Since DE = 5 and FC = 7, you know that the similar ratio is 5/7. Since EF is a side of a square, it has a length of 5 as well. 5 * 5/7 = 25/7. Hence, BD has a length of 25/7.

This problem is simply asking how many numbers from 2 - 100 have 1 or more square factors. We first list out the multiples of the square factors and then eliminate common multiples.

4             9            We don't need to do 16.          25            We don't need to do 36.          49

8            18                                                           50                                                            98

12          27                                                           75

16                                                                               

20          45         

24          54         

28          63         

32                   

36          81         

40          90         

44          99         

48

52

56

60

64

68

72

76

80

84

88

92

96

100

25+9+3+2 = 39 integers that can be simplified.

We rewrite the equation as a + (a+1) + (a+2) + ... + (a+10583) = x^3. Adding them all up and factoring, we get (5292)(2a+10583) = x^3. We can factor 5292 as 3^3 * 14^2. That means that 2a+10583 = 14y^3, which has no solution because of the parity mismatch.

Quite similar to what milkcloud said, I would subtract 25 from 137 to get 112. Then, I would divide by 2 to get 56. After that, I would add 25 back to 56 to get a final answer of 81 pounds, like milkcloud. 

I believe that all of your guesses are correct. I checked over them, and I am pretty sure that they're correct.

I think it's supposed to be a troll... or maybe latex just didn't work out well.