Pangolin14

You are off to a good start. We know that because BDI is a 3-4-5 triangle, it is right, and because an angle bisector is a perpendicular in triangle ABC, ABC is isosceles. As ABC is isosceles, a perpendicular from A is a median of BC! Therefore, BC = 2*BD = 8. 

I'm going to coordinate bash. Let B be (-4,0) and C be (4,0). Then, reverse-using the formula for the incenter's coordinates, or (0,3), A = (0, 96/7).

We have that the height is 96/7, so the area is 96/7 * 4 = 54.8571428571 or 54 6/7.

jugoslav, nice to see we agree :)

Keep in mind that we only need 1 example to disprove something.

A. As one angle determines all the other ones in a parallelogram, A is true, because all the other angles are 180-90 = 90, so it is a rectangle.

B. True, same reason as E

C. True, I think same reason as E

D. True, opposite angles are equal, so all the angles are auto-completed, making the equal-sided rhombus a square.

E.  True, as the diagonals must bisect each other (definition of a parallelogram) and so all the small triangles formed by the diagonals are congruent by SSS and have largest angle 360/4=90˚. Because a parallelogram's angle bisectors are diagonals, and all the triangles are isosceles right, each angle is 45*2 = 90, and as at least two sides of a parallelogram are equal, and each angle is 90 from part A, we have that all the sides and angles are equal, so it is a square.

Remember DIRT - distance is rate * time

Happy new year!

Perhaps we can simplify the cases. We know that the probability the leftmost lamp is red is 1/2, so we just multiply our final answer by that.

Case 1: leftmost lamp is on (and obviously red)

5 choose 2/ 6 choose 3 = 1/2

Case 2: the second lamp is the first on and is red 

1/2 * 3/5 *2/5 = 3/25

Case 3: the third lamp is the first on and is red 

1/2 *2/5 *3/4 * 2/5 = 3/50

Case 4: the fourth lamp is the first on and is red  

1/2 * 2/5 *1/4* 1 *2/5 = 1/50

Not sure where you went wrong... though.

d = rt: 20 = 40t --> t = 1/2 hr = 30 mins

d = rt: 15 = 60t --> t = 1/4 hr = 15 mins + 3 + 3 = 21 mins

30-21 = 9 mins

Or for all the complicated equation enthusiasts. Solve this equation to get the answer: $(\sqrt{6} + \sqrt{6}-x)* x = \frac{(\sqrt{6}-x)^2}{2}$ for $(\sqrt{6} + \sqrt{6}-x)* x$

It would just be that 2(black area) + white area = 6 cm^2, so 3 * white area = 6 cm^2 so white = black = 2 cm^2

That does not seem to be one of the answer choices. I forgot to include the answer choices with my post, here they are:

A) 8 B)10 C) 3√7 D)9 E)√65

1 =  2 + 0 - 2 + 1

2  = (2 + 0) / 2  + 1

3  =  2^2 - 1 - 0

4  =  2^2  - 0/1

5 =  2 + 0 + 2 + 1

6 =  (2 + 1)2  + 0

7  =  10/2 + 2

8  =  (2 + 1)! + 2   - 0

9 =  (2 + 1)^2  + 0/2

10  = (20/2)  / 1

11  = 20 / 2 + 1

12  =  12  + 0/2

13  =   12 + (0/2)!

14  =  12 + 2 + 0

15  =  C(10,2) / ( 2! + 0! )

16  =   2^ ( 0! + 1!  + 2!)

17 = 20 - 2 -1 

18  =  20  - (2/1)!

19  = 20 - 2! + 1!

20 =   20 / (2 -1)

21  =  20 + 2  - 1

22  =  20 + 2/1

23  = 20 + 2 + 1

24  = ( 12) ( 2 + 0)

25 =   122  + 0!

26  =  20 + (1 + 2)!

27 = (2 ! + 0! ) ^( 2 + 1)

28 = C(10-2, 2)

29 = [(2+0!)! / .2] -1

30 = (2 + 0!)! * (1/.2)

31 = 2^(1/.2) - 0!

32 = 10  + 22

33 = 2^(1/.2) + 0!

34 = (2^2)! + 10

35 = [ (2 + 1) 1] ^2  - 0!

36 = [ (2 + 1)!  ] ^2  - 0

37 = 2! + 0!)! ^( 2)  +  1

38 = (20-1) * 2

39 =  20*2 - 1

40 = 10 (2 + 2)

41 =  20 * 2 + 1

42 = 21*2  - 0

43 = 21 *2  + 0!

44 = 22(1 + 0!)

45 = p(10, 2)/2

46 = 10.2(bar)/0.2(bar)

47 = C(10,2) + 2

48 = (10/.2)  -  2

49 =((2+1)! + 0!)^2

50 = 10^2/2

51 = 102/2

52 = (10/.2) + 2

53 = ( 12/.2 bar) - 0!

54 = ( 12/.2 bar) + 0

55 = (12 - 0!)/.2

56 = P(10-2,2)

57 = 

58 =[ ( 2 + 0! )! / .1] -  2 

59 = 12/.2  - 0!

60 = 20 ( 2 + 1)

61 = 12/.2  + 0!

62 = [( 2 + 0! )!/ .1] + 2

63 = 2^( [ 2 + 1]!)  - 0!

64 = 2^[(2 + 1)! ] + 0

65 = C(12,2) - 0!

66 = C(12,2) - 0 

67 = C(12,2)  + 0!

68 = 

69 = 

70 = floor (sqrt (2)  /.2  )  *  10

71 = ceil(sqrt(2)/.2 * 10)

72 =  (12)  ( 2 + 0!)!

73 = 

74 = 

75 = 

76 =

77 =

78 = C(12+0!, 2)

79 = [( 0! / .1 bar ) ^2] -2

80 = [2^( 2 + 0!)] / .1

81 =[( 2^0/.1 bar)^2]

82 =

83 = [ (0!/.1 bar)^2] +2

84 =

85 =

86 =

87 =

88 = P(10,2) - 2

89 = (20/.2 bar) - 1

90 = C(10,2) * 2

91 = ( 20/.2 bar) + 1

92 = P (10,2) +2

93 =

94 =

95 = (20 -1)/.2

96 = floor [(sqrt (21))^(2 + 0!)]   =  floor [ (21)^(3/2)]    

97 = ceiling [ (sqrt (21))^(2 + 0!)]

98 = 10^2  - 2

99 = 20/.2 - 1

100 = 102  - 2

Above is edited. Total count = 85

Adding on to catmg's post:

If you know 2048, you know 2^6 = 64.

And also 19^2 is memorizable, it's 361.

But note that 64 = 8^2, so it's really just (8 * 19)^2, which is 152^2.

Then 2 * 3^3 * 5^3 = 125 * 2 * 27, or 250 * 27, and you can use the x25 rule to get 6750.

152^2 = 152 * 152  = 04 _ (15 * 2 + 15 * 2) + 15^2 = 04 _ 60 + 225 = 23104, by LOIFing.

Thus, easy subtraction to get 16354. Divide it by 2 (long division) and reach 8177, which is easily divisible by 13, = 629, which is easily divisible by 17, 37, which is a prime.

And as none of the other factors 152 or 98 are divisible by 37, the answer is 37!!!