\section*{Problem 1 - 111021}

Five students \(A, B, C, D, E\) are playing the following game, whose rules are known to all of them: One of them, for example, student \(A\), leaves the room. Now, exactly 10 quadrilaterals are drawn on a piece of paper. The drawing is hidden, and \(A\) is called back in. Each of the students \(B, C, D,\) and \(E\) makes exactly one statement about the drawn quadrilaterals. Of these statements, exactly one is false. They are:

(1) There is not only one square on the drawing.

(2) There are exactly twice as many rectangles as squares on the drawing.

(3) Among the quadrilaterals on the drawing, there is exactly one parallelogram.

(4) There are exactly twice as many trapezoids as rectangles on the drawing.

A is now supposed to determine which statement is false. Additionally, he should specify the exact number of squares, rectangles, and trapezoids.

How can this be done?
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It holds: Every square is simultaneously a rectangle, a parallelogram, and a trapezoid. Every rectangle is also a parallelogram and a trapezoid.

A can therefore conclude as follows:

Assume that statement (1) is false, i.e., there is either no square or exactly 1 square on the drawing.

According to the rules of the game, statements (2) to (4) would then be true. If there were exactly 1 square, then according to (2) there would be another rectangle, thus at least 2 parallelograms, in contradiction to (3). If there were no square on the drawing, then according to (2) and (4) there would also be no rectangle and no trapezoid, in contradiction to (3).

Therefore, (1) is true. Hence, there are at least 2 squares on the drawing.

Therefore, (3) is false, and (2), (4) are true.

If there were 3 or more squares on the drawing, then due to (2) and (4) the number of trapezoids would have to be at least 12, which is not possible. Therefore, it follows:

There are exactly 2 squares, exactly 4 rectangles, and exactly 8 trapezoids on the drawing.