CPhill

https://web2.0calc.com/questions/algebra_45144

A7^2 + 7B + C  = C5^2 + 5B + A

49A + 7B + C = 25C + 5B + A

48A + 2B  = 24C

24A + B  = 12C

Let B = 0     C =4   and A  = 2

A = 2    B = 0  C = 4

204₇  = 402₅  ???

2(7)^2 + 0 + 4  = 4(5)^2 + 0 + 2

98 + 4  =  100 + 2

102  = 102

N  = 102

250₁₀  =   fa₁₆ →  B = 16

f = 15

a =10

15 (16)  + 10   =  240 + 10 =  250

https://web2.0calc.com/questions/number-theory_75476

Simplify as

x^2 -4x + y^2 - 12y  =  0       complete the square on x,y

x^2 -4x + 4 + y^2 - 12y + 36  = 4 + 36

(x -2)^2 + (y -6)^2  = 40

This is a circle centered at (2,6)  with a radius of sqrt (40)

max x + y  can be found by the intersection of the line  y =(x -2) + 6  → y = x + 4  with the circle

The intersection point is

x =  2 + sqrt 40 * cos 45°      y =  6 + sqrt 40 * sin 45°  

max [ x ] + [ y ]  =  [  2 + sqrt (40) / sqrt (2)]  + [ 6 + sqrt (40)/sqrt (2)]   =

8 + 2sqrt (20)  =

8 + 4sqrt (5)

Simplify as

(7 + n)x < -36

x <  -36 / ( 7 + n)

This will have no solution if n =  -7

The radius will be the circumradius of the triangle ABC

semi-perimeter  =  [ 8 + 12 + 15 ] /  2 = 17.5

Area  =  sqrt [17.5  (9.5) (5.5) (2.5) ]   

Radius =   product of the sides  / 4 * area  =  

[8 * 12 * 15 ]  / [ 4 * sqrt [17.5  (9.5) (5.5) (2.5) ] ] ≈  7.53 

https://web2.0calc.com/questions/incenter_1

We want to  solve this

500 (1/20)^(n -1)  > 1       divide both sides by 500

(1/20)^(n -1)  >  1/500       take the log of both sides and  we  can wtite

(n -1) log (1/20) > log (1/500)

{divide both sides by log (1/20)...since thisis negative,  reverse the inequality sign }

n -1 <  log (1/500) / log (1/20)

n <  [ 1 + log (1/500) /log (1/20) ]  

n <  3.07

The first three terms  will be > 1

y =  (x + 1) (x^2 + 1)^-1

y' = (x^2 + 1)^-1  - (x +1) (x^2 + 1)^-2 * 2x  = 0

(x^2 + 1)^-2  [  (x^2 + 1) - 2x ( x  + 1) ]  = 0

-x^2  - 2x + 1   = 0

x^2 + 2x - 1  =  0

x^2 + 2x  = 1

x^2 + 2x + 1  =  2

(x + 1)^2   =  2       take both roots

x + 1 = sqrt 2            x + 1  = -sqrt 2

x = sqrt (2)  - 1         x = -sqrt (2)  - 1

y  =  sqrt 2 / [ (sqrt (2) -1)^2 + 1]  = sqrt 2 / [ 4 - 2sqrt 2]  =  max

y = -sqrt (2) / [ (-sqrt (2) - 1)^2 + 1 ]  = -sqrt (2) / [ 4 + 2sqrt 2] = min

Sum of  max and  min  = [ sqrt 2][ 4 + sqrt 8] / 8  - [sqrt 2][ 4 -sqrt 8] /8  =

2sqrt (16) / 8   =

[2 * 4] / 8  = 

8  / 8    =   

1