Task 259

The task

In the five-sided star shown, the letters $A$, $B$, $C$, $D$, and $E$ are replaced by the numbers $3$, $5$, $6$, $7$, and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$, $BC$, $CD$, $DE$, and $EA$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence? [asy] unitsize(2 cm); pair A, B, C, D, E; A = dir(90); B = dir(90 + 3*360/5); C = dir(90 + 6*360/5); D = dir(90 + 9*360/5); E = dir(90 + 12*360/5); draw(A--B--C--D--E--cycle); filldraw(Circle(A,0.15),white); filldraw(Circle(B,0.15),white); filldraw(Circle(C,0.15),white); filldraw(Circle(D,0.15),white); filldraw(Circle(E,0.15),white); label("$A$", A); label("$B$", B); label("$C$", C); label("$D$", D); label("$E$", E); [/asy]