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1001_n is a prime number in the following bases:

base 5, base 22, base 24, base 25, base 26, base 28, base 33,  and base 36.

You are awesome thank you!!!!

Please try again

52 weeks  in a year    (actually 365/7 =52.14   so we will jus settle for 52 weeks in a year)

280 recipes/52 weeks = 5.38 recipes per week....approx 5 per week...every third week do 6  .

f(x) =9x^2-6x+3      'complete the square'    to do this , leading coeff has to be 1

divide by 9  (we'll put it back in shortly)

f(x)/9   =  x^2 - 2/3 x +1/3     now complete the square

              x^2 - 2/3 x + 1/9       -1/9  +1/3

f(x) / 9 =  (x-1/3)^2     - 1/9 + 3/9

f(x) / 9   =  (x- 1/3)^2   + 2/9                   Now remember that division by 9?   Put the 9 back in by multiplying by 9

f(x) = 9 (x-1/3)^2  + 2          END !

Thank you for your solution.

A perfect sphere is inscribed inside a right circular cone which has a vertex angle of 60 degrees. Find the ratio of the volume of the sphere to volume of the cone.

Acceleration is change in velocity divided by change in time.

vf = vo + at       velocity final = velocity original + acceleration * time

0 = 20 m/s + (-2 m/s^2) t       solve for t = 10 sec

xf = x0 + vo t + 1/2 a t^2  

xf = 0 + 20 (10) + 1/2 (-2)(10^2)     

xf = 200 + (-100) 

xf = final position = 100 m                     Correct !

Units must match !      (this is fast 254 km/hr = 159 miles per hour !)

   254 km/hr  x 1000m/km  x 1 hr/3600s = 70.556 m/s  = vo

vf = 0 = vo + (a) t =

0 = 70.556 m/s    + (-5 m/s^2)* t      solve for t = 14.11 sec

xf = x0 + vo t + 1/2 (a) t^2

xf = 0 + 70.556  (14.11) + 1/2 (-5) (14.11)^2  =  497.9 m      It cannot stop in 23 m (unless it hits a wall !)

change in velocity / change in time =   remember to keep you 'units' matched !

    change in velocity   is    126- 54  km/hr   Thi s equals   72 km/hr x 1000m/km x 1hr/3600 s =  20 m/s

       now     change in velocity / change in time =  20 m/s / 8 s = 2.5 m/s^2

Since the altitudes are inside the triangle, the triangle is acute.  This gives us the inequalities 12^2 + 14^2 > h^2, h^2 + 12^2 > 14^2, and h^2 + 14^2 > 12^2.  Then the largest h can be is 18.

Let r be the radius of the sphere.  Then the volume of the sphere is 4/3*pi*r^3 and the voluem of the cone is 1/3*pi*(2r)^2*3r = 4*pi*r^3, so the ratio is 1/3.

Solve for x:
x^(1/4) + sqrt(x) = 12

Subtract 12 from both sides:
-12 + x^(1/4) + sqrt(x) = 0

Simplify and substitute y = x^(1/4).
-12 + x^(1/4) + sqrt(x) = -12 + x^(1/4) + (x^(1/4))^2
 = y^2 + y - 12:
y^2 + y - 12 = 0

The left hand side factors into a product with two terms:
(y - 3) (y + 4) = 0

Split into two equations:
y - 3 = 0 or y + 4 = 0

Add 3 to both sides:
y = 3 or y + 4 = 0

Substitute back for y = x^(1/4):
x^(1/4) = 3 or y + 4 = 0

Raise both sides to the power of four:
x = 81 or y + 4 = 0

Subtract 4 from both sides:
x = 81 or y = -4

Substitute back for y = x^(1/4):
x = 81 or x^(1/4) = -4

Raise both sides to the power of four:
x = 81 or x = 256

x^(1/4) + sqrt(x) ⇒ 81^(1/4) + sqrt(81) = 12:
So this solution is correct

x^(1/4) + sqrt(x) ⇒ 256^(1/4) + sqrt(256) = 20:
So this solution is incorrect

The solution is:

x = 81

The answer is 54. Try drawing it out.

That's incorrect, "the edges of the pyramid" includes its hexagonal base.

The points (x,y) are (-1,7) and (3,3).

n = 22.

The red figure at the right is made of two hemispheres of equal radius. The blue cylinder at the left has the same radius and height as the red figure.

Which shape has a larger surface area?

I already tried this value before and it is incorrect

Straight line 1 is:

5x-8y+5=0 

well solve for y 

-8y=-5x-5

8y=5x+5 

divide by 8

\(y=\frac{5}{8}x+\frac{5}{8}\)

Straight line 2 is:

3x-4y+10=0

4y=3x+10 

divide by 4

\(y=\frac{3}{4}x+\frac{10}{4}\)

Well 2 sides of square are based upon two straight lines, so both sides of square must be equal (Actually 4 sides of square are equal but let's see the context here) , so the 2 equations must be equal I.e.

\(\frac{5}{8}x+\frac{5}{8}\)=\(\frac{3}{4}x+\frac{10}{4}\)

solve for x 

x=-15   (I.e. the length of square can't be negative but this square is in a negative quadrant that's why the x value is -15 but we write it just as 15)

So, area of square = s^2 , (-15)^2=225

A=abs((30*12) - (5.5*10)), solve for A
A = 55 Degrees - the angle at 10 minutes past noon.
55=abs((30*12) - (5.5*M)), solve for M
M =610 / 11 =55.4545-minutes after 12 noon when the angle will be 55 degrees again.

I guess this is meant to be the following:

\(\text{For how many positive integers }n\geq2\text{ is }1001_n\text{ a prime number?}\)
 

I presume the subscript n indicates a number base. 

If you want to prepare for a math competition, then you should use math competition books (Artofproblemsolving books are good).

We know  that  we can find the slope as

[ f (2016)  - f ( 2004) ]                              5

__________________  =                     ____

           2016  - 2004                                12

So....this implies  that

[ f (2012 )   - f (1184) ]                             5

__________________     =                 _____

2012  - 1184                                           12

[ f(2012)  -  f(1184)                                5

________________  =                        ___               multiply both sides by 28

           28                                              12

                                                    5 (28)                  35

f (2012)  - f (1184)     =                  ______  =       ______   

                                                       12                     3