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so the old ratio is 160g/3£ and they add 25% more to become 200/3£ so that is 200/3 and Jane wants to reduse the price so we get the equation \(200/3=160/x\) and then we get x=0.6£ becuase the price must be 2.4£ and 3-2.4=0.6 

\(\Huge{\text{The final answer is } \quad x=0.6 \quad \text{euros}}\)

Please fix your latex you copied.

It's painful to read.

This question is an interesting AoPS question indeed.

This question has been answered multiple times already. I have personally gave hints.

Search it up in the search bar.

 \(x=\frac{2\sqrt[3]{3(270-31\sqrt{69})}}{3}\),\(y=\frac{15+\sqrt{69}}{3}\)

Acute isosceles triangle.
Sides: a = 17 b = 17 c = 16

Area: T = 120
Perimeter: p = 50
Semiperimeter: s = 25
Inradius
An incircle of a triangle is a circle which is tangent to each side. An incircle center is called incenter and has a radius named inradius. All triangles have an incenter, and it always lies inside the triangle. The incenter is the intersection of the three angle bisectors. The product of the inradius and semiperimeter (half the perimeter) of a triangle is its area.
Area =semiperimeter * inradius
Inradius =Area / semiprimeter
              = 120 / 25
              = 4.8 - the inradius of the triangle.

So you waited 6 days before asking?

Why do you expect us to do your entire AoPS homework?

If you ask us to do one, sure.

a week's? No.

Please help with these!  These are due really soon!  I need the answers!!!!!

10. Through how many different positive angles less than 360 degrees is it possible to rotate a regular 15-gon clockwise about its center such that its image coincides with the original icosagon?

11. A translation maps A to Aprime and B to Bprime. We know  AAprime = 6, AB = 7 and ABprime = 14.  Find BAprime.

12. A regular pentagon PQRST is rotated clockwise about its center by a nonzero angle. What is the smallest number of degrees it must be rotated until it coincides with its original position?

13. Roger wants to add one more dot to the diagram below so that the four dots form the four vertices of a parallelogram.  How many possible locations are there?

14.  pentagon UVWXY is translated north by 40 units to pentagon U1V1W1X1Y1. Pentagon U1V1W1X1Y1 is then translated east by 50 units to pentagon U2V2W2X2Y2. We know the perimeter of pentagon UVWXY is 30 units. Find WW2.

15. P and Q are two points on a plane. A translation maps P to Q and Q to Qprime. Another translation maps Q to P  and P to Pprime. We know PprimeQprime = 16. Find PQ.

16. OPQRSTUVWX is a regular decagon (10-gon). A rotation of theta degrees about V maps W to U. Given theta < 180 degrees, find theta.

sumfor(r, 1, 100, floor(r/100 + 1/2)) = 51

f(x)  =  x5 + 8x4 +2x2 + 5x + 15

1)  You'll need to find the anti-derivative of f(x) 

--->     F(x)  x6/6 +8x5/5 + 2x3/3 +5x2/2 +15x + c

2)  You'll need to evaluate it from -6 to 6

--->   F(6) - F(-6)

--->   [ (6)6/6 +8(6)5/5 + 2(6)3/3 +5(6)2/2 +15(6) + c ] - [ (-6)6/6 +8(-6)5/5 + 2(-6)3/3 +5(-6)2/2 +15(-6) + c ]

3)  Finish ...

Each term is of the form
\(\binom{9}{k} (2z)^{9-k}\Big(-\frac{1}{\sqrt{z}}\Big)^k = \binom{9}{k}(-1)^k 2^{9 - k} \frac{z^{9-k}}{z^{k/2}}.\)

Therefore, the constant term occurs when 9-k=k/2 , or k=6. Putting k=6 in our expression above, the constant term is

\(\binom{9}{6} (-1)^6 2^{9 - 6} = \boxed{672}.\)

I'm assuming that the function is  f(x)  =  -2x² + 32x + 5.

To divide the interval (0,16] into four rectangles, the dividing points are:  0,  4,  8,  12,  16

The left-hand endpoints are: 0, 4, 8, and 12.

f(0)  =  5

f(4)  =  101

f(8)  =  133

f(12)  =  101

To find the areas of the rectangles, multiply each height by its width (which is 4).

5x4  +  101x4  +  133x4  +  101x4  =  1360

6 et 5 ausi pour moi

x² + 6x + y²  =  4

Complete the square for x:  ( x² + 6x + 9 ) + y²  =  4 + 9

Factor:                                           (x + 3)² + y²  =  13

The problem has now become:  what two perfect squares have a sum of 13?

--  they are 4 and 9.

So, either  (x + 3)²  =  4   and   y²  = 9

--->                x + 3  =  2             y  =  3

                            x  =  -1            y  =  -3

              or    x + 3  =  -2

                           x  =  -5   

              Combining these answers to get:  (-1,3),   (-1,-3),   (-5,3),   (-5,-3)

Or:   (x + 3)²  =  9   and   y²  = 4

--->                x + 3  =  3             y  =  2

                            x  =  0            y  =  -2

              or    x + 3  =  -3

                           x  =   -6   

              Combining these answers to get:  (0,2),   (0,-2),   (-6,2),   (-6,-2)

This isn't homework. This is a test.

If you want to give us a question or two of homework, sure.

But to cheat on your test isn't good.

yea Melody was right! thanks for helping :)

I'm not sure if I have this right, but would you add 15 to 30 since that was the inital distance that the helicopter flew?

This is a bit confusing to me. I tried drawing it out, but I still don't understand. 

Please do your own homework, Silly ! 

The two 5-digit numbers are:

13225  and  24336

Your not wrong when you say this is hard...

I'll take a shot at it...

I got the values x = 2

and x = -1

They seem to work...

I first factored 4x^2 - 1 to get (2x + 1)(2x - 1). When we add fractions we multiply the denominators to get the common denominator.

When I factored 4x^2 - 1 I got (2x + 1)(2x - 1) and that is the same denominator when you multiply the other 2 denominators together.

We get 4x/(2x + 1)(2x - 1)

We multiply 1 by (2x + 1)(2x - 1) so the denominator cancels out.

We get 4x + 7 = (2x + 1)(2x - 1)

We expand (2x + 1)(2x - 1) to get 4x^2 - 1. 

4x + 7  = 4x^2 - 1

We subract 7 on both sides to receive 4x^2 - 8 = 4x

Then, we subtract 4x from both sides to get 4x^2 - 8 - 4x = 0

We can use the quadratic formula to find the solution from here on out.

a = 4, b = -4, c = -8

\(x = {-4 + \sqrt{(-4)^2-4(4(-8)} \over 2(4)}\)

Saving you time, this gives us 2 and -1.

thanks!

The derivative of the position function gives the velocity function

2 t^3 + 5 t -2    derivative   =   6 t^2 + 5

Note that 4x^2 -1  =  (2x-1)(2x+1)      multiply BOTH sides of the equation by this to get

2x+1     + 2x-1   + 7   = 4x^2 -1     simplify

4x + 7 = 4x^2 -1

4x^2-4x-8 = 0            Quadratic Formula yields   x = 2  or  -1

I dont understand either that is why i am asking 

If this question was assigned to you your learning about the pigen hole principle.

You have to look at the worst case to ensure that you will get it.

If your really unlucky, you might get 10 yellow socks in a row and 13 red socks (soxs) in a row.

Then, it is certain that you will get an either GREEN SOCK, or YELLOW SOCK so we add 1.

10 + 13 + 1 = 24

Red Sox pun not intended but definitely appreciated...

I think you have to spin the 2nd spinner if you get all the options except lose a turn in the 1st spinner. 

If so, the probability of getting lose a turn would 1/25/ The probability for the 2nd part of the question would be 6/25. For the 3rd part of the question, it would be 5/25.