Road to Emerald City

Number Theory Level 3

Dorothy began her journey along the golden road to the Emerald city. After walking for some distance, she found the Scarecrow, who joined the quest with her.

After walking together for some time, both of them found the Tinman, and at this point, the Scarecrow had walked 2 5 \dfrac{2}{5} of Dorothy's distance. The Tinman then joined the two in this trip.

Afterwards, the three companions met the Cowardly Lion, and at this point, the Tinman had walked 2 5 \dfrac{2}{5} of Scarecrow's distance.

If all distances travelled between any meeting points were integer kilometers and Dorothy's total distance to the Lion's point was prime, how long was the total distance in kilometers?


The answer is 19.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

Let A , B , C , D A, B, C, D be the starting points for Dorothy, Scarecrow, Tinman, & Lion. And let A B = a AB = a , B C = b BC=b , C D = c CD=c for some integers a , b , c a,b,c .

Then we can set up the equations as:

b a + b = 2 5 \dfrac{b}{a+b}=\dfrac{2}{5}

c c + b = 2 5 \dfrac{c}{c+b}=\dfrac{2}{5}

Solving in terms of b b , we will get:

a = 3 a 2 a=\dfrac{3a}{2}

c = 2 b 3 c=\dfrac{2b}{3}

Thus, the ratio a : b : c a:b:c = 9 : 6 : 4 9:6:4 .

Since all distances are in integers and the total distance is prime, the constant among the ratio is one and total distance is 9 + 6 + 4 = 19 9+6+4 =\boxed{19} .

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