SOLUTION: If (a_1,a_2,..a_17) satisfy a_1 + a_2 + a_3 = 1, a_2 + a_3 + a_4 = 2, a_3 + a_4 + a_5 = 3, a_{15} + a_{16} + a_{17} = 15, a_{16} + a_{17} + a_{1} = 16, a_{17} + a_{1}

If (,, . . . ) satisfy

 = 1, 
 = 2,
 = 3, 
. . . . . . . . .        <<<---=== I added THIS which means that there are similar equations for all intermediate indexes
 = 15, 
 = 16, 
 = 17
 
what is the value of  ?

Below is the system of equations written in full

 = 1,        (1)
 = 2,        (2)
 = 3,        (3) 
 = 4,        (4) 
 = 5,        (5)
 = 6,        (6)
 = 7,        (7) 
 = 8,       (8) 
 = 9,      (9)
 = 10,    (10) 
 = 11,    (11) 
 = 12,    (12) 
 = 13,    (13)
 = 14,    (14)
 = 15,    (15)
 = 16,     (16)
 = 17.      (17)

Add all 17 equations (1) - (17)  (both sides). You will get

 = 1 + 2 + 3 + . . . + 15 + 16 + 17 =  = 9*17 = 153.

Now divide both sides of the lest equation by 3, and you will get

 = 51.   (18)


Next subtract equation (1) from the equation (18). You will get

 = 51 - 1 = 50.   (19)


Next subtract equation (4) from the equation (19). You will get

 = 50 - 4 = 46.   (20)


Next subtract equation (7) from the equation (20). You will get

 = 46 - 7 = 39.   (21)


Next subtract equation (10) from the equation (21). You will get

 = 39 - 10 = 29.   (22)


Next subtract equation (13) from the equation (22). You will get

 = 29 - 13 = 16.   (23)


Now compare equations (23) and (16). You instantly will get 

 = 16 - 16 = 0.                  (24)


Having known  = 0, we can rewrite the equation (18) in the form

 = 51.   (25)

Next subtract equation (2) from the equation (25). You will get

 = 51 - 2 = 49.   (26)


Next subtract equation (5) from the equation (25). You will get

 = 49 - 5 = 44.   (27)


Next subtract equation (8) from the equation (27). You will get

 = 44 - 8 = 36.   (28)


Next subtract equation (11) from the equation (28). You will get

 = 36 - 11 = 25.   (29)


As the last step, subtract equation (14) from the equation (29). You will get

 = 25 - 14 = 11.   (30)


It is your answer:   = 11.