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Remove the number  6  because it can't be combined with any other number to create a sum of  12.

The centroid of a triangle is found 2/3^rds of the way from the vertex to the opposite side.

Since D and E are midpoints of two sides of a triangle, they create a line parallel to the base of the triangle (BC) and cut all transversals at their midpoints. Therefore, AM is 1/2 the way from the vertex to the opposite side.

This means that MG is (2/3 - 1/2)  =  1/6^th the distance.

GM/GA  =  (1/6) / (2/3)  =  1/4

where did you get the 90?

um the question is missing vital information!

You are lucky I use AoPS and was able to render your image :/

Angle DCB + Angle DAB = 180

Lets call ACD \(x\)

and lets call CAB \(y\)

\(x+y+70=180\)

Now.

Can you figure out the answer? (simple algebra!)

s = 1/7 and t = -4/7, so s + t = -3/7.

1) Exactly 4 of 9:  ₉C₄ · (0.50)⁴ · (0.50)⁵

2) At most 3 of 9:  ₉C₀ · (0.50)⁰ · (0.50)⁹  +  ₉C₁ · (0.50)¹ · (0.50)⁸  · ₉C₂ · (0.50)² · (0.50)⁷ · ₉C₃ · (0.50)³ · (0.50)⁶

3)  At least 8:  ₉C₈ · (0.50)⁸ · (0.50)¹  +  ₉C₉ · (0.50)⁹ · (0.50)¹

Angle CAB + angle ACD = 30 + 40 + 90 = 160 degrees.

I think you mis-typed 2/7 as 27

arctan  5/9

Let the ten's digit of the original be x and the one'e digit be y.

The value of the original number is:    10x + y

The value of the reversed number is:  10y + x

If you double the reversed number and add the result to the original number, you get 222: 

     This equation is:  2(10y + x) + (10x + y)  =  222

                                       20y + 2x + 10x + y  =  222

                                                    12x + 21y  =  222

                                                        4x + 7y  =  74

Since the sum of the digits of the number is 14:     x + y =  14

Combining these two equations:     4x + 7y  =  74                      --->      4x + 7y  =  74 

                                                          x  +  y  =  14     --->   x -4   --->     -4x - 4y  =  -56

Adding down the columns;                                                                              3y  =  18

                                                                                                                          y  =  6

Since  x + y  =  14:     x  =  8

Points  A  and  B  divide the circle into two arcs:  major arc(AB)  and  minor arc(AB).

The size of angle(AXO)  =  26°.

This means that the size of angle(BXO) is also 26°.

The total size of angle(AXB)  =  52°.

Since we want to find the size of minor arc(AB), call minor arc(AB) = x.

Since the sum of the major arc(AB) and minor arc(AB) = 360°, major arc(AB)  =  360° - x.

The appropriate formula is:  [ major arc(AB) - minor arc(AB) ] / 2  =  angle(AXB).

                                                                        [  (360° - x) - x ] / 2  =  52°

                                                                                [ 360° - 2x] / 2  =  52°

                                                                                       360° - 2x  =  104°

                                                                                               256°  =  2x

                                                                                                    x  =  128°

By the property of telescoping sum, we have quite easily that \dfrac{1}{a_n} = 2-\sum_{k=0}^{n-1}\dfrac{1}{n+a_k}.

First we show that a_n<1 which is analogous to proving that \sum_{k=0}^{n-1}\dfrac{1}{n+a_k}<1. Now, we note that \{a_k\} is an increasing sequence, hence a_k\geq a_0 for all k\geq0. This gives \sum_{k=0}^{n-1}\dfrac{1}{n+a_k}\leq\dfrac{n}{n+a_0}=\dfrac{2n}{2n+1}<1 and thus we are done.

Now we prove the other part of the inequality. We note that \sum_{k=0}^{n-1}\dfrac{1}{n+a_k}\geq\dfrac{n}{n+a_n} because of increasing property of the sequence a_n. Now using the fact that \dfrac{1}{a_n}=2-\sum_{k=0}^{n-1}\dfrac{1}{n+a_k} we have, after some algebra that 2a_n^2+(n-1)a_n-n\geq0 implying, and keeping in mind that \{a_k\} is a positive sequence, a_n\geq\dfrac{-(n-1)+\sqrt{(n-1)^2+8n}}{4}. It remains to show that this huge quantity is greater than 1-\dfrac{1}{n}. By squaring both sides, and cancelling out terms, we come to 3n>2 which is true for any n. Hence, 1/(a_1 + 1) + 1/(a_2 + 1) + ... + 1/(a_n + 1) < 2.

∑[(2^2^n) /(4^2^n - 1), n, 0, ∞] = 1

x = adults

256-x = children

$ 2.75 x     +    $ 1.75 *  (256-x)    =  $ 551.50     solve for x = adults,   then    256-x = children

He gets a semicircle of radius   10    

then when the rope hits the corner of the building it will only be 4 m ...he will get two quarter circles of this radius

pi r^2 = area       semicircle =  1/2  pi  (10)^2

         and two quarter circles = one semicircle of radius 4     1/2  pi (4)^2        Add the reds together for total area.

After I put the answer in, this was the answer it gave me. Turns out it was actually 15. Thank you still for trying to help! Much appreciated!

Case 4-0-0-0: There is only one way.
Case 3-1-0-0: There are four ways to choose the ball that is by itself. Then the other three balls must go in a different box.
Case 2-2-0-0: There C(4,2) = 6 ways to choose two balls to go in one box. Then the other two balls must go in a different box. But since order doesn't matter (there is no first box or second box), we divide by 2.
Case 2-1-1-0: There are C(4,2) = 6 ways to choose two balls to go in one box. Then the other two balls go in separate boxes.
Case 1-1-1-1: There is only one way.
This gives a total of 15 ways.

15 was correct, thanks!

I remember that website from AOPS

About q2:

x^2 - 3x + 1 = x^2 - 2 * (3/2) * x + (9/4) - (9/4) + 1 = (x - 3/2)^2 - 5/4.

=> b = -3/2 ; c = -5/4

"the polynomial x^2+bx+c has exactly one real root" => Discriminant : b^2 - 4*a*c = 0

but a = 1 => b^2-4c = 0 (1)

we also have : b = c+1 (2)

Solve for the set of equation of (1) and (2) => c = 1.

right triangle      ladder is hypotenuse (110 ft)    one leg is 28 ft   the other  is the height

28^2 + x^2 = 110^2      x = height from truck = 106.38 ft

    you must add the eight feet off of the ground to this       114.38 ft

First: list all the possible lengths of the sides:

   2,9,9     3,8,9     4,8,8     4,7,9     5,6,9     5,7,8     6,6,8     6,7,7

and then, find all the ways that these lengths can be written:

as:  for  2,9,9  they can be interchanged as:  2,9,9     9,2,9     9,9,2

and for  5,7,8  they can be interchanged as:  5,7,8     5,8,7     7,5,8     7,8,5     8,5,7     8,7,5

The axis of symmetry will be at the vertex  ....where  x = -3     this is also =  -b/2a

-b/2a  = -3       multiply both sides by -2

b/a = 6