NL-NWO 1980

$f(x) = x^3-ax+1$ , $a \in R$ has three different zeros in $R$ . Prove that for the zero $x_o$ with the smallest absolute value holds: $\frac{1}{a}< x_0 < \frac{2}{a}$

Find the product of all divisors of $1980^n$ , $n \ge 1$ .

Given is the non-right triangle $ABC$ . $D,E$ and $F$ are the feet of the respective altitudes from $A,B$ and $C$ . $P,Q$ and $R$ are the respective midpoints of the line segments $EF$ , $FD$ and $DE$ . $p \perp BC$ passes through $P$ , $q \perp CA$ passes through $Q$ and $r \perp AB$ passes through $R$ . Prove that the lines $p, q$ and $r$ pass through one point.

In Venetiania, the smallest currency is the ducat. The finance minister instructs his officials as follows: "I wish six kinds of banknotes, each worth a whole number of ducats. Those six values must be such that there exists a number N with the following property: Any amount of money of $n$ ducats ( $n$ positive and integer) where $n \le N$ may be paid in such a way that no more than two copies of each kind are required either to pay or to return. I also wish those six values to be as large as possible for $N$ . Determine those six values and provide proof that all conditions have been met." Solve the problem of those officials