NL-NWO 1997 Problem 3

a. View the second-degree quadratic equation $x^2+? x +? = 0$ Two players successively put an integer each at the location of a question mark. Show that the second player can always ensure that the quadratic gets two integer solutions. Note: we say that the quadratic also has two integer solutions, even when they are equal (for example if they are both equal to $3$ ).

b.View the third-degree equation $x^3 +? x^2 +? x +? = 0$ Three players successively put an integer each at the location of a question mark. The equation appears to have three integer (possibly again the same) solutions. It is given that two players each put a $3$ in the place of a question mark. What number did the third player put? Determine that number and the place where it is placed and prove that only one number is possible.