UA-UMO 2005 Problem 97

Under the rules of the "Sea battle" game (no two ships may have a common point), can the following sets of rectangular ships be arranged on a $10 \times 10$ square board:

$(a)$ two ships $1 \times 4$ , $4$ ships $1 \times 3$ , $6$ ships $1 \times 2$ , and $8$ ships $1 \times 1$ ;

$(b)$ two ships $1 \times 4$ , $4$ ships $1 \times 3$ , $6$ ships $1 \times 2$ , $6$ ships $1 \times 1$ , and $1$ ship $2 \times 2$ ;

$(c)$ two ships $1 \times 4$ , $4$ ships $1 \times 3$ , $6$ ships $1 \times 2$ , $4$ ships $1 \times 1$ , and $2$ ships $2 \times 2$ ?