Problem 1. Prove that for every irrational real number , there are irrational real numbers
and so that and are both rational while and are both irrational.
Problem 2. Let and be positive real numbers such that . Prove that
Problem 3. Prove that there exists a triangle which can be cut into congruent
Problem 4. In a small town, there are houses indexed by for with being the house at the top left corner, where and are the row and column indices, respectively. At time 0, a fire breaks out at the house indexed by , where .
During each subsequent time interval , the fire fighters defend a house which is not
yet on fire while the fire spreads to all undefended neighbors of each house which was on
fire at time . Once a house is defended, it remains so all the time. The process ends when
the fire can no longer spread. At most how many houses can be saved by the fire fighters?
A house indexed by is a neighbor of a house indexed by if .
Problem 5. In a triangle , points and are on sides and , respectively, such that . Let and denote the circumradius and the inradius of the triangle , respectively. Express the ratio in terms of and .