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lol.

just add all equations. You will get 2A+2B+2C=60

so A+B+C=60/2 = 30

it is easy

I was just tricking you

Hello, I realize that problems from AoPS may be challenging. Please use the message board to ask for help. Thank you!

This should be easy. Multiply the expression by 100 then group then divide. Steps:

30+31+32+...+49+50

80 x 10

800

800/100=8

In the left-hand side, the y-coordinate is -2t - 2, which can be any real number.  If k = 0, then in the right-hand side, the y-coordinate can only be 0, so the answer is k = 0.

sumfor(n, 30, 50, n/100) = 8.4

The probability is 1/16*1/12*1/8*1/4 = 1/6144.

10 to 630 = 8 1/2 hours

8 1/2 /24   = 17/48  probability   (?)   

Angle QPR works out to 90 - (41 + 63)/2 = 38 degrees.

Note: I do not know if this is correct

35 x 10^-6  =  k/ (180^4)      k= 36741.6

Power = 36741.6 / 540⁴ =  432 .09   nanowatts       or         4.32 x 10^-7 watts

Wait, I'm pretty sure doing it that way overcounts some things. For instance, doing \(\binom{4}{2}\) for 2-2-0-0 will count each way twice since you choosing two balls or you choosing the other two will result in the same balls in boxes. But building off of your idea, I think I can get the answer. Thanks!

By similar triangles, YZ = 2*15^2/25 = 18.

1. You can write the expression as cos(2x) + cos(4x) + cos(6x) + cos(8x), so a + b + c + d = 2 + 4 + 6 + 8 = 20.

(20 p *24 m) / 6000 ft^2   =  .08  p-m / ft²

.08  p-m / ft2  * 8000 ft² / 20 p 

see how things 'cancel out'   and yo are left with minutes?    .08 * 8000 / 20  = 32 minutes

You try the second one......

Equilateral triangle height  h = sin( 60°) = 0.866

Small square side     a = 1

Large square area   A = ? 

You posted it TWICE!!. Look at the question below this one!

Well, you should have: 50C2 =1,225 ways of pairing those 50 numbers. But only 12 pairs sum up to 25. Technically, there should be 12 x 2 = 24 pairs if the cards are DISTINCT such as different-colored cards, or cards of different shapes and sizes....etc. To my understanding, then you should have at most 24 pairs that sum up to 25:

Such as: 24 + 1, 23 +2, 22 + 3, 21 + 4, 20 +5......etc. And if they are DISTINCT, then: 1 + 24, 2 + 23, 3 + 22, 4 + 21, 5 + 20........and so on.

Note: Somebody should check this. Thanks.

what first post?

See your first post.

12  x  50 = 600 m^2 - the area of the wall.

1 square meter =~10.76 square feet.

600  x  10.76 =6,456 square feet - the area of the wall.

6,456 / 200 =32.28 gallons, or round it to 33 whole gallons of paint.