Answer: No since 10 bills with value 1,3 or 5 will produce an even number.
Problem 2. Can a 
square checkers board be covered by
dominoes?
Answer: No since we will cover even number of squares.
Problem 3. Can we draw a closed path of 9 segments, each of which intersects exactly one of the other segments?
Answer: No we need even number of squares for that.
Problem 4. Can a knight start in the (a1) square of a chess board and go to (h8) (the opposite square on the diagonal) and visit each of the remaining squares only ones?
Answer: No since we need an even number of moves to stay in the same color. We need to make 63 moves for that and the knight changes the color in each of the moves.
Homework:
Problem 1. Show that any axis of symmetry of a convex 101-gon passes through one of it vertexes. How about an 10-gon.
Problem 2. The product of 22 integers is one. Show that their sum cannot be zero.
Problem 3. The numbers between 0 and 10 are written in a row. Can we put plus and minus signs between them so that at the end we get zero.
Problem 4. Create two problems similar to the above.