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299

Part of 1980 All Soviet Union Mathematical Olympiad

Problems(1)

ASU 299 All Soviet Union MO 1980 x<1/3(p/4-sqrt(d^2-s/2)) in parallelepiped

Source:

7/19/2019
Let the edges of rectangular parallelepiped be x,yx,y and zz (x<y<zx<y<z). Let p=4(x+y+z),s=2(xy+yz+zx)   and   d=x2+y2+z2p=4(x+y+z), s=2(xy+yz+zx) \,\,\, and \,\,\, d=\sqrt{x^2+y^2+z^2} be its perimeter, surface area and diagonal length, respectively. Prove that x<13(p4−d2−s2)   and   z>13(p4−d2−s2)x < \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )\,\,\, and \,\,\, z > \frac{1}{3}\left( \frac{p}{4}- \sqrt{d^2 - \frac{s}{2}}\right )
geometric inequalityinequalitiesgeometry3D geometryparallelepiped