2007 Turkey Junior National Olympiad

1

Let $ABCD$ be a trapezoid such that $AD\parallel BC$ and $|AB|=|BC|$. Let $E$ and $F$ be the midpoints of $[BC]$ and $[AD]$, respectively. If the internal angle bisector of $\triangle ABC$ passes through $F$, find $|BD|/|EF|$.

2

In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding $N$. If there are at least $7$ teams qualified, find the possible least value of $N$.

3

Find all odd postive integers less than $2007$ such that the sum of all of its positive divisors is odd.