ITOT 44 Autumn 2022 · THPT · O (Basic)

Find the largest positive integer $m$ such that $m!\cdot2022!$ is the factorial of an integer. Give the value in exact factorial notation and justify maximality.

A large circle is inscribed in a rhombus. Each of two smaller circles is tangent to two sides of the rhombus and to the large circle, as shown. The four dashed lines through the side-contact points shown in the figure form a quadrilateral. Prove that it is a square.

On a line, 2022 equally spaced points are colored, half blue and half red. Can the sum of the lengths of all segments with blue left endpoint and red right endpoint equal the sum for red left endpoint and blue right endpoint? Endpoints need not be adjacent.

Start with an acute scalene triangle. At each step, one existing triangle may be cut along one of its medians into two triangles. Can finitely many such cuts produce only isosceles triangles?

A $2N\times2N$ board is tiled by nonoverlapping $1\times2$ dominoes. A piece visits every cell exactly once, each move going to a cell sharing a side with the preceding one. A move is called a jump if its two cells belong to the same domino. What are (a) the greatest and (b) the least possible numbers of jumps? Original weights: (a) 1 point, (b) 4 points.