Imcool

So first, lets understand this problem with an equation. This is what i got:

$\frac{x}{x+1}\cdot\frac{x+1}{x+2} ...\frac{x+19}{x+20}=3$

now we cancel. You see by cross canceling, the x+1's dissapear as all of them do and you are left with

$\frac{x}{x+20} = 3$

well, you might think that since it has to be a positive fraction and x has to be negative, this is wrong, BUT keep in mind, if both x in the numerator and denominator or negative, the fraction is a postive. so simplify this equation.

$x = 3x+60$  multiply both sides by $x+20$

$2x=-60$   subtracting x on both sides and moving the 60 over.

you will get x = -30, so the first fraction will be$\frac{30}{31}$,which will be your answer!

Im very confused. You say it is a triangle, but it is not. Do you mean connecting Q and R to form a triangle? But then the answer would be pretty straight forward. $\frac{12*14}{2}$ which will be 84. But i assume you mean the area of the figure shown here. So i will solve that too

Here is what i hope to be the solution: 

First lets find $\overline{PQN}$ which will be 42 (use the triangle calculation formula $\frac{base*height}{2}$)

Heres where things get trickier, but I want you to figure it out, but i will give you some hints

1. draw a line from P to the midpoint of $\overline{QR}$

2. try to see if you can combine triangles in some way to single out $\overline{NOR}$ to find it

Hint: since each one of them is less than 3, the only possible way is the stick being broken into length 2,2, and 2.

find the total ways the stick can be broken up into. then it is 1/(total ways) <---- that will be your answer.

Hope this helps!

can anybody online check my work? My brain is damaged.

Where do i go? to the left, where nothing is right, or to the right, where nothing is left?

Find all real numbers x such that

$(3^x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3$

well, this is going to be a pain to expand, but i see no other way... Oh god.

Okay after 30 minutes of backbending expanding, i got this:

$27^x-9^{2+x}+3^{x+7}-19683 + 19683x^3-6561x^2+729x-27 = 27000x^3-81000x^2+81000x-27000$

yeah.... thats, quite a bit of math. Now lets simplify! YAY! *sigh..*

$27^x-9^{2+x}+3^{x+7} - 7317x^3 + 74439x^2 + 80271x = 0$

Im acually having fun doing this! No sarcasm like acually!

Okay lets acually do it another way.

$[ (3x - 27 + 27x - 3) ] [ (3x - 27)^2 - (3x - 27)(27x - 3) + ( 27x - 3)^2 ] = 27000x^3 - 81000x^2 + 81000x - 27000$

$( 30x - 30) [ (3(x - 9))^2 - ( 3 (x - 9) * 3(9x - 1) ) + (3 (9x - 1))^2 ] = the thing idon't wanna write$

$(30x - 30) [ 9 ( x^2 - 18x + 81) - ( 9(9x^2 - 82x + 9)) + (9 ( 81x^2 - 18x + 1)) = againidon't wannawriteit$

$9 (30x - 30) [ x^2 - 18x + 81 - 9x^2 + 82x - 9 + 81x^2 - 18x + 1 ] = yeahyeahyeah$

$270 ( x - 1) [ 73x^2 + 46x + 73 ] = ohgod$

the discriminant of $[ 73x^2 + 46x + 73 ]$is negative, so it has no real roots, just saying, you know.

so basically $270 ( x - 1) = (30x-30)^3$

more simplifying....

$270 ( x - 1) = 270(100x^3 - 300x^2 + 300x - 100)$

$x-1=100x^3 - 300x^2+300x-100$

$100x^3 - 300x^2 + 299x - 99 = 0$

i think ill let you take it from here

Note: I've spent more than an hour on this prob, but i haven't solved it yet, anybody that can give the solution i will appreciate it

Basically, I graphed it and made the triangle with the equations

$y\ -4\ =\ x\ -\ 3$

$-\frac{x}{2}-1=y+4$

$3x+1\ =\ y+6$

try to graph what i just said and form a 90 degree line and you should be able to get some equation.

I got this

$3x\ =\ \frac{3y}{2}+3$

I get it now. Okay so first, look at the constant term 18. we need some 2 numbers to be in a 3:1 ratio and multiply to get 18 so

$xy=18$

$4y=x$

from this system of equations we get

$y = \sqrt{4.5}$

$x = \sqrt{72}$

these are the 2 solutions the this quadratic.

we can express this as

$(x - \sqrt{4.5})(x-\sqrt{72}) = 0$

so b is equal to $\sqrt{4.5} + \sqrt{72}$

hope this helped(i hope this is not wrong)

I don't see the picture, try to put the picture in?

Name the total amount of problems in the November math meet x

Name the total amount of problems in the March math meet y

Now according to the information: we get the following equations:

$x+y=28$

$\frac{1}{3}x + \frac{2}{7}y = 15$

2 variables, 2 equations, we can solve for x and y, but we just need y, so

substitue $x$ for $28 - y$ into the 2nd equation we get

$\frac{28-y}{3} + \frac{2y}{7}= 15$

then we simplify to get

$\frac{196-7y}{21} + \frac{6y}{21} = 15$

which is

$\frac{196-y}{21} = 15$

multiply both sides by 21, we get

$196 - y = 315$

so simplify to get

$y = -119$

wait a sec, that can't be right, can someone fix what i did wrong? hmm....