AMC10 2018 B

Answer (C): The set of all possible ordered pairs $(x,y)$ occupies the unit square $0\le x\le 1,\ 0\le y\le 1$ in the Cartesian plane. The numbers $x$, $y$, and $1$ are the side lengths of a triangle if and only if $x+y>1$, which means that $(x,y)$ lies above the line $y=1-x$. By a generalization of the Pythagorean Theorem, the triangle is obtuse if and only if, in addition, $x^2+y^2<1^2$, which means that $(x,y)$ lies inside the circle of radius $1$ centered at the origin. Within the unit square, the region inside the circle of radius $1$ centered at the origin has area $\frac{\pi}{4}$, and the region below the line $y=1-x$ has area $\frac{1}{2}$. Therefore the ordered pairs that meet the required conditions occupy a region with area $\frac{\pi}{4}-\frac{1}{2}=\frac{\pi-2}{4}$. The area of the unit square is $1$, so the required probability is also $\frac{\pi-2}{4}\approx \frac{1.14}{4}=0.285$, which is closest to $0.29$.