1. We are given a trapezoid with bases of lengths 1 and 4, respectively. We divide it into two trapezoids by a cut parallel to the bases, of length 3. We now want to further divide these two new trapezoids, always by cuts parallel to the bases, into $m$ and $n$ trapezoids, respectively, so that all $m+n$ trapezoids obtained have the same area. Determine the minimum possible value for $m+n$ and the lengths of the cuts to be made to achieve this minimum value.
Solution: Let $ABCD$ be the trapezoid, with $AB$ as the larger base, and let $P$ and $Q$ be the endpoints of the cut already made, located on $AD$ and $BC$ respectively. Extend the oblique sides until they meet at a point we call $E$; the triangles $DCE$, $PQE$, and $ABE$ are similar (they all have congruent angles due to the parallelism of the lines $DC$, $PQ$, and $AB$). We know that $PQ / DC = 3$ and $AB / DC = 4$, from which, denoting $S$ as the measure of the area of triangle $DCE$, we obtain the area ratios: $\operatorname{Area}(PQE) / S = 3^2 = 9$, Area $(ABE) / S = 4^2 = 16$.

It follows that, by difference, $\operatorname{Area}(PQCD) = 8S$ and $\operatorname{Area}(ABQP) = 7S$. Suppose we divide the trapezoid $PQCD$ into $m$ parts, and the trapezoid $ABPQ$ into $n$ parts; for all parts to have equal area, it must be that $8S / m = 7S / n$, or $8n = 7m$, and thus $m$ must be a multiple of 8 and $n$ a multiple of 7. At a minimum, $m + n$ must equal $7 + 8 = 15$.

To divide the initial trapezoid into 15 parts, we need to make 14 cuts (including the initial cut of length 3); let $P_i$ and $Q_i$ (on $AD$ and $BC$ respectively) be the endpoints of the $i$-th cut (in order of length: in this way $P$ will coincide with $P_8$ and $Q$ with $Q_8$). The area of the trapezoid $P_iQ_iCD$ is $i$ times the area of a single part, which is equal to $S$, so $\operatorname{Area}(P_iQ_iE) = (i+1)S$. As before, the triangles $P_iQ_iE$ and $DCE$ are similar; the ratio of corresponding sides is equal to the square root of the ratio of the areas (which is $i+1$); thus $P_iQ_i = \sqrt{i+1}$. In conclusion, the cuts are of lengths $\sqrt{2}, \sqrt{3}, \cdots, \sqrt{15}$, and allow the initial trapezoid to be divided into exactly 15 parts.