Question 86
Find the number of 6-tuples $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6})$ of distinct positive integers satisfying the following two conditions:
a. $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} = 30$;
b. We can write $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ on sides of a hexagon such that after a finite number of times choosing a vertex of the hexagon and adding 1 to the two numbers written on two sides adjacent to the vertex, we obtain a hexagon with equal numbers on its sides.
a. $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6} = 30$;
b. We can write $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ on sides of a hexagon such that after a finite number of times choosing a vertex of the hexagon and adding 1 to the two numbers written on two sides adjacent to the vertex, we obtain a hexagon with equal numbers on its sides.