Question

starting from the point $(0,0)$, a grasshopper makes a series of moves on the coordinate plane. the grasshopper's first move takes her to $(3,4)$. she then proceeds via this rule: move rule: after a move which adds $(m,n)$ to the grasshopper's coordinates, her next move adds either $(m,n-1)$ or $(m-1,n)$ to her coordinates. thus, her second move adds either $(3,3)$ or $(2,4)$ to her position, landing her at $(6,7)$ or $(5,8)$ accordingly. if the grasshopper's $x$ and $y$ coordinates never decrease, and if she ends at the point $(a,a)$ where $a$ is as large as possible, then what is the value of $a$?

Answer 1

the grasshopper ends at the point (5, 8) where a = 8.

The grasshopper starts at the point (0,0) and makes a move to (3,4) in the first step. According to the move rule, her next move can add either (3,3) or (2,4) to her position. Let's consider both cases:

1. If the grasshopper adds (3,3) to her position, she will be at (3+3, 4+3) = (6, 7).

2. If the grasshopper adds (2,4) to her position, she will be at (3+2, 4+4) = (5, 8).

To maximize the value of a, the grasshopper needs to choose the move that increases both the x and y coordinates as much as possible. In this case, the grasshopper should choose the second move, adding (2,4) to her position. This will give her the highest possible value for a.