test.json

{
    "problem": "Polyhedron $P$ is inscribed in a sphere of radius $36$ (meaning that all vertices of $P$ lie on the sphere surface). What is the least upper bound on the ratio $$\\frac{\\text{volume of }P}{\\text{surface area of }P}~?$$In other words, what is the smallest real number $t$ such that $$\\frac{\\text{volume of }P}{\\text{surface area of }P} \\le t$$must be true for all polyhedra $P$ that can be inscribed in a sphere of radius $36$?",
    "level": "Level 5",
    "type": "Geometry",
    "solution": "Let $O$ be the center of the sphere, and assume for now that $O$ is inside polyhedron $P$. We can carve polyhedron $P$ into pyramids, each of which has a face of $P$ as its base and $O$ as its apex. For example, a cube would be carved into six pyramids, two of which are highlighted in this drawing: [asy]\nsize(4cm);\nimport three;\ntriple A,B,C,D,EE,F,G,H;\nA = (0,0,0);\nB = (1,0,0);\nC = (1,1,0);\nD= (0,1,0);\nEE = (0,0,1);\nF = B+EE;\nG = C + EE;\nH = D + EE;\nO = G/2;\ndraw(surface(B--O--C--cycle),red,nolight);\ndraw(surface(C--O--D--cycle),red+white,nolight);\ndraw(surface(H--O--G--cycle),lightblue,nolight);\ndraw(surface(G--O--F--cycle),blue,nolight);\ndraw(surface(EE--F--G--H--cycle),lightblue+blue,nolight);\ndraw(B--C--D);\ndraw(B--A--D,dashed);\ndraw(EE--F--G--H--EE);\ndraw(A--EE,dashed);\ndraw(B--F);\ndraw(C--G);\ndraw(D--H);\ndraw(A--O--C,dashed);\ndraw(B--O--D,dashed);\ndraw(EE--O--G,dashed);\ndraw(F--O--H,dashed);\ndot(A); dot(B); dot(C); dot(D); dot(EE); dot(F); dot(G); dot(H); dot(O);\nlabel(\"$O$\",O,WSW);\n[/asy] Then if we add up the areas of all the pyramids' bases, we get the surface area of $P$. If we add up the volumes of the pyramids, we get the volume of $P$.\n\nThe volume of each pyramid is equal to $\\frac 13\\cdot\\text{(area of base)}\\cdot\\text{(height)}$. The height of each pyramid must be less than $36$, since the height of each pyramid extends from $O$ to a point inside the sphere. Therefore, the volume of each pyramid is less than $12$ times the area of the base. It follows that the volume of $P$ is less than $12$ times the surface area of $P$. We can, however, make this ratio arbitrarily close to $12$ by selecting polyhedra $P$ with many small faces, so that the height of each pyramid is as close as we wish to $36$.\n\nTherefore, for polyhedra inscribed in a sphere of radius $36$ such that the center of the sphere lies inside the polyhedron, the least upper bound on $$\\frac{\\text{volume of }P}{\\text{surface area of }P}$$is $12$. Finally, we must consider the case of inscribed polyhedra for which the center of the sphere does not lie inside the polyhedron. However, in this case, we can still construct pyramids with apex $O$ whose bases are the faces of $P$; then the surface area of $P$ is still the sum of the areas of the bases, but the volume of $P$ is less than the total volume of the pyramids. This only strengthens the argument for an upper bound of $12$. So, the answer is $\\boxed{12}$."
}