59th Ukrainian National Mathematical Olympiad

Yes, such figure exists.

Consider an L-shaped tromino that consists of three $1\times 1$ cells. $3\times 2$, $2\times 3$ rectangles can be easily split into such figures, as well as $9\times 5$ rectangle (Fig. 16). Now let every $1\times 1$ square be a $2\times 3$ rectangle. Then the tromino became a new figure that will be denoted as $F$ (Fig. 17), then $S=18$.

An $m\times n$ rectangle is called nice, if it can be split into figures $F$. Clearly, if a rectangle can be split into nice rectangles, it is nice itself. Since $3\times 2$, $2\times 3$ and $9\times 5$ rectangles are nice, then so are $18\times 15$, $6\times 6$ and $4\times 9$ rectangles.

We will continue the proof with induction. Consider $k\times 18$ rectangle.

$k=18$, we can split $18\times 18$ into six $6\times 6$.

$k=19$, firstly, from $19\times 18$ we are cutting $18\times 15$ off, then the remainder $4\times 18$ can be split into two $4\times 9$.

$k=20$, can split into ten $9\times 4$.

$k=21$, firstly, from $21\times 18$ we are cutting $18\times 15$ off, then the remainder $6\times 18$ can be split into three $6\times 6$.

Suppose all rectangles of the type $k\times 18$ for $k\ge 21$ are nice. Consider $(k+1)\times 18$ rectangle. Split it into $(k-3)\times 18$ and $4\times 18$.



Fig. 17 Both are nice by the induction hypothesis and by an easy split into other nice rectangles. This completes the proof.