Question 37
Kiara wrote 20 nonzero integers on a blackboard. For each pair of numbers written by Kiara, Yndira then wrote their product on the same board, repeating it even if that result was already written.
For example, if $2,3,4,6$ were among Kiara's numbers, Yndira would write $6,8,12,12,18,24$, since $6=2\times3$, $8=2\times4$, $12=2\times6=3\times4$, $18=3\times6$ and $24=4\times6$. The number 6 is written again even though Kiara already wrote it, and Yndira writes 12 twice.
a) How many numbers did Kiara and Yndira write in total?
b) Suppose exactly 120 of all the numbers on the board are positive. If Kiara wrote more positive than negative numbers, how many of Kiara's numbers were positive?
For example, if $2,3,4,6$ were among Kiara's numbers, Yndira would write $6,8,12,12,18,24$, since $6=2\times3$, $8=2\times4$, $12=2\times6=3\times4$, $18=3\times6$ and $24=4\times6$. The number 6 is written again even though Kiara already wrote it, and Yndira writes 12 twice.
a) How many numbers did Kiara and Yndira write in total?
b) Suppose exactly 120 of all the numbers on the board are positive. If Kiara wrote more positive than negative numbers, how many of Kiara's numbers were positive?