HMMT February

We claim that a point in the first quadrant satisfies the desired property if the point is below the line $x+y=3$ and does not satisfy the desired property if it is above the line. To see this, for a point inside the region, $x+y<3$ and $\lfloor x\rfloor+\lfloor y\rfloor \leq x+y<3$ However, $\lfloor x\rfloor+\lfloor y\rfloor$ must equal to an integer. Thus, $\lfloor x\rfloor+\lfloor y\rfloor \leq 2$. Adding these two equations, $x+y+\lfloor x\rfloor+\lfloor y\rfloor<5$, which satisfies the desired property. Conversely, for a point outside the region, $\lfloor x\rfloor+\lfloor y\rfloor+\{x\}+\{y\}=x+y>3$ However, $\{x\}+\{y\}<2$. Thus, $\lfloor x\rfloor+\lfloor y\rfloor>1$, so $\lfloor x\rfloor+\lfloor y\rfloor \geq 2$, implying that $x+y+\lfloor x\rfloor+\lfloor y\rfloor>5$. To finish, $R$ is the region bounded by the x -axis, the y -axis, and the line $x+y=3$ is a right triangle whose legs have length 3. Consequently, $R$ has area $\frac{9}{2}$.