BMO 2010 Shortlist

Since the only representations of $98$ and $149$ as sums of two squares are $7^2+7^2$ and $7^2+10^2$, we conclude that every odd move of the grasshopper is of the form $(x,y)\to (x\pm 7,y\pm 7)$ and every even one - of the form $(x,y)\to (x\pm 7,y\pm 10)$ or $(x\pm 10,y\pm 7)$. Let the starting point be $(0,0)$. We need the grasshopper to get in an even number of moves to some of the points $(\pm 7,\pm 7)$, e.g. to $(7,7)$.

After any pair of two consecutive moves the grasshopper gets from the point $(a,b)$ to the point $(c,d)$, where $c\in \{a,a\pm 14\}$, $d\in \{b\pm 17,b\pm 3\}$ or $c\in \{a\pm 17,a\pm 3\}$, $d\in \{b,b\pm 14\}$. It means that after any pair of consecutive moves one of the coordinates remains the same modulo $14$, and another changes by $3$ modulo $14$. This implies that each coordinate may obtain a value equivalent to $7$ modulo $14$, in particular precisely the value $7$, only after at least $7$ pairs of moves, e.g. $0+3+3+3+3+3+3+3\equiv 7(\mathrm{mod}14)$.

To obtain a similar result for another coordinate one needs at least another $7$ pairs of moves. Therefore, one needs at least $2\times 14=28$ moves to get the point $(7,7)$, and totally at least $28+1=29$ moves to get the initial point $(0,0)$.

An example of such $29$ moves is the following: all $15$ odd moves are as $(x-7,y-7)$, and $14$ even moves consist of six moves as $(x+10,y+7)$, six moves as $(x+7,y+10)$, one move $(x+10,y-7)$ and one move $(x-7,y+10)$.

$\boxed{29}$