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It's correct :) thanks so much!

Square both sides to get

x+5 = 1 + x + 2 sqrtx  

4 = 2 sqrt x

2 = sqrt x

x = 4

It's not Tuesday :(

For 1 see https://web2.0calc.com/questions/let-f-x-be-a-polynomial-such-that-f-0-4-f-1-5-and_1#r2

0⁵ -5⁴ + 4^{0   =}   0 -625+1 = -624

216 possible rolls

6 rolls will fit the criteria   666 555 444 333 222 111

6 /216 = 1/36 chance

Perpindicular slope will be   -1/2

  y intercept (put 0 in for x)  - 4

y = -1/2 x  -4

2y+x +  8=0

Thank You!

Huh, thats a nice way to do it.

Thanks Alan!

The smallest positive n =98, 98+1, 98+2=98, 99,100

100 mod 2^2 =0

100 mod 5^2 =0

99 mod 3^2   =0

98 mod 7^2   =0

I'm gonna attempt to answer Guest's question: What is the angle APM?

∠BON = ∠AOT = 60º                  ∠APT = 30º

∠BOT = ∠AON= 120º

Radius  TO = 1 

Diameter TN = 2

PT = TO / tan(30º)

∠NPT = tanarc( TN / PT ) ≈ 49.107º

∠AMP ≈ 19.107º

(1st question is for 5th graders and the second one for college students.) 

Every multiple 3x4 will be marked false   but they are true    so that will be 8 marked incorrectly  (12  24  36 48 60 72 84 96

  all of the other multiples of 4  (25 of them) will be marked as true minus the 8 above

    17 /25   will be correct 

edited:     sorry...i misread the question

17 of 25 will be correct for the multiples of 4

there are 75 other questions that have 'false' as an answer

      33 multiple of 3 will be marked false which is a correct answer EXCEPT for the 8 above

            33 - 8 = 25 will be correctly marked

25 + 17  out of 100 will be correct   42/100     ( I think !    Confusing  )

First die =any number =6/6

2nd die - The probability that it is different from the first die = 5/6

3rd die -  The probability that it is different from the first two= 4/6

Then, the probability that ALL three are different from each other =6/6  x  5/6  x  4/6 =120 / 216

Then, the probability that ALL three show the same number  = 1 - 120 / 216 = 96 / 216 = 4 / 9

The 1000th day of Lucy's life fell on: Tuesday, August 28, 2007

This question is mistaken...

The question should be: what is the angle APM? 

The area of the shaded region is    A = 2 [ 2 / cos(30º)]

3 years  is  12 quarters

interest per quarter = 15%/4 = .15/4 = .0375

FV = Future Value = 7000 (1+.0375)¹²  =  $ ............

Extend line from O to N  angle BON = AOT = 60 

APT is in a triangle witht wo other angles 90 and 60 degrees

APT = 180 - 60- 90  = ......

common differece, d = 8         a1 = 11        a20 = -11 + 8(19) = 141

Sum = n/2 (a₁ + a₂₀)

        = 20/2 (-11 + 141) = ..........

(x-2)(x-1) <0        ONE of these must be less than zero   if EITHER is = 0   or if both are negative then the equation is not true

(x-2) < 0      (x-1)<0

x <2               x < 1              so between 1 and 2  (non-inclusive)   this is true    :    (1,2)      

The smallest number that works is 1223 (1224 is divisible by 4 and 9, 1225 is divisible by 25 and 49).

Thank you so so much! :)

Let's put it into this way.

If a player wins game 1: (1/3)*(1/2)=1/6

If the player wins before the 4th turn: P(wins game 1)+P(lose game 1, wins game 2)+P(loses game 1 and 2, but wins game 3)

Player loses game 1 but wins game 2: (5/6)*(1/6)=5/36

Player loses game 1 and 2, but wins game 3: [(5/6)^2](1/6)=25/216 (very confusing, I'm sorry)

Player wins before 4th turn: 1/5+5/36+25/216=91/216

Yay!

Have you tried using the AoPS message board yet though? :)

My computer says there are 16 such numbers as follows:

(13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231) = 16 such numbers.