2. The generalized Schur number S(4,5) is defined as the smallest positive integer n such that any blue/red colouring of the set {1, 2, ..., n} contains a blue solution to the equation L(4): X1 + x2 + x3 = 24 or a red solution to the equation L(5): x1 + x2 + x3 + 24 = X5. (a) Consider the following 2-colouring of the interval [1, 13] = {1, 2, ...,13}: B = {12, 12, 13} and R= [3, 11]. Check that this colouring does not not contain a blue solution to L(4) or a red solution to L(5) (b) To show that any blue/red colouring of [1, 14] contains a blue solution to L(4) or a red solution to L(5) do the following. i. Suppose that 1 E B and build a blue/red colouring trying to avoid a blue solution to L(4) AND a red solution to L(5). ii. Suppose that 1 R and build a blue/red colouring trying to avoid a blue solution to L(4) AND a red solution to L(5). (c) Carefully justify your conclusion that S(4,5) = 14. 2. The generalized Schur number S(4,5) is defined as the smallest positive integer n such that any blue/red colouring of the set {1, 2, ..., n} contains a blue solution to the equation L(4): X1 + x2 + x3 = 24 or a red solution to the equation L(5): x1 + x2 + x3 + 24 = X5. (a) Consider the following 2-colouring of the interval [1, 13] = {1, 2, ...,13}: B = {12, 12, 13} and R= [3, 11]. Check that this colouring does not not contain a blue solution to L(4) or a red solution to L(5) (b) To show that any blue/red colouring of [1, 14] contains a blue solution to L(4) or a red solution to L(5) do the following. i. Suppose that 1 E B and build a blue/red colouring trying to avoid a blue solution to L(4) AND a red solution to L(5). ii. Suppose that 1 R and build a blue/red colouring trying to avoid a blue solution to L(4) AND a red solution to L(5). (c) Carefully justify your conclusion that S(4,5) = 14