Problem 1: 1. Show that the following mathematical program max x1 + 72 subject to min{221, 3x2} 21 -22 +1 and 0 1,2, $ 1 (1) can be cast as a mixed-integer linear program (where R denotes the set of real numbers). 2. Show that the following mathematical program 2x1 + 3x2 2 1 max |z1 + 12 - 1 subject to 4x1 + 312 $ 5 (2) can be cast as a mixed-integer linear program. 3. Formulate the following question as an integer program: how many non-mutually attacking queens can you fit into a chessboard? See example in Figure 1. A queen can attack horizontally, vertically, and diagonally, regardless of how far the other piece is on the board. Solve the integer program using gurobi. Use gurobi to generate at least two distinct ways to position the queens on the board Figure 1: Non-mutually attacking queens (they must be distinct up to rotation)