CPhill

Only the odd terms are positive

So

1000*(1/3)^(n -1 )  > 1      

(1/3)^(n-1)  >  1/1000  take the log of both sides

(n -1) log (1/3)  > log (1/1000)                   (log 1/3  is  negative....reverse the sign )

n -1 <  log (1/1000) /(log (1/3)

n < 1  + log (1/1000) /log (1/3)

n < 7.28

The 1st , 3rd, 5th and  7th terms will all be  > 1  ........4 terms

1/4  + 12d = 2/5

12d = 2/5 -1/4

12d =  3/20

d =  3/20 * 1/12  =  1/80

2/5 + 13d =  48th term

2/5 + 13 ( 1/80)  = 

[ 32]/80 + [13]/80  = 

45 / 80  = 

9 / 16

The numerator and  denominator are  defined  for all real  numbers.....so...

domain  =  ( -inf,  inf)

We need to solve this

C (n ,14)  >  100

By brute force

C (15,14)  =  15

C ( 16 ,14)    =  120

So......n =  16

1.  Since we have the condition "at least once," then "24" could  also appear twice, as well ...so... I think we could also have

2424x     =    10

24x24     =    10

x2424   =        9

= 19  more added to   blueorange's answer

Simplify

x^2 - mx + 14  = 0

Factors of 14                                                m

(1 , 14)                   (x + 1) (x + 14)              -14

(-1, -14)                  (x - 1)  (x - 14)               14

(2,7)                       (x + 2) ( x + 7)                -9

(-2, -7)                   (x - 2) ( x - 7)                   9

First one   C(18,2)

Second one :  Choose Mark and  1 of 17 other people =17 ways  

Lst one :  C(17,2)

A

      5

            X

               5

B                 C

AX =CX  . ..so ....BX  is also an angle  bisector .......so  AB = BC

Triangle AXB ≈  Triangle ABC

AX / AB  = AB / AC

5 / AB = AB / 10

AB^2  = 50

AB = 5sqrt (2) = BC

BX / AB  = BC / AC

BX / AB = AB  / AC

BX / AB = AB / 10

10BX = AB^2

10 BX = 50

BX = 50 / 10    =  5

For the first function x can  be any real number except 0

For the second function , we  have to  have that

5x + 6 - 11 ≥ 0

5x - 5 ≥  0

5x ≥  5

x ≥  1    [this is the domain ]

[ y  - - 2] / [ 6 -1] =  -7/4

[y + 2 ] = -(7/4) (5)

y + 2 = - 35/4

y =  -35/4 - 2

y = -43/4