I wrote a mathematical Olympiad the other day and there was one question that I could not work out.

The question was If ab = 1, bc =2, cd =3, de =4 and ea = 6, what does a + b + c + d +e =

It was a multiple choice and the answers were : 43/6; 47/6; 49/6; 53/6; 61/6 The correct answer was 61/6, but how can do you get to that answer?

Aurora (student) (grd9)
middle

From ab = 1 we have b = ¹/_a.

From bc = 2 we have c = ²/_b, substituting b = ¹/_a we have that c = 2a.

From cd = 3 we have d = ³/_c, substituting c = 2a we have that d =  ³/_2a.

From de = 4 we have e = ⁴/_d, substituting d = ³/_2a we have e = ^8a/₃.

Since ea = 6 and e = ^8a/₃ we find  ^{8a 2}/₃ = 6, solving for a, we have a = ³/₂.

From this b = ¹/_a = ²/₂, c = 2a = 3, d = ³/_2a = 1 and e = ^8a/₃ = 4.

Add these numbers.