Subcontests
(9)Centroids of tetrahedra, volume
Let P be a point inside a tetrahedron ABCD and let SA,SB,SC,SD be the centroids (i.e. centers of gravity) of the tetrahedra PBCD,PCDA,PDAB,PABC. Show that the volume of the tetrahedron SASBSCSD equals 1/64 the volume of ABCD. Graph construction given degrees
Suppose that n≥8 persons P1,P2,…,Pn meet at a party. Assume that Pk knows k+3 persons for k=1,2,…,n−6. Further assume that each of Pn−5,Pn−4,Pn−3 knows n−2 persons, and each of Pn−2,Pn−1,Pn knows n−1 persons. Find all integers n≥8 for which this is possible.(It is understood that "to know" is a symmetric nonreflexive relation: if Pi knows Pj then Pj knows Pi; to say that Pi knows p persons means: knows p persons other than herself/himself.) Maximize 4-variable ratio
Find an upper bound for the ratio
x12+x22+x32+x42x1x2+2x2x3+x3x4
over all quadruples of real numbers (x1,x2,x3,x4)=(0,0,0,0).Note. The smaller the bound, the better the solution. Labelling of triangulation
A convex n-gon A0A1…An−1 has been partitioned into n−2 triangles by certain diagonals not intersecting inside the n-gon. Prove that these triangles can be labeled △1,△2,…,△n−2 in such a way that Ai is a vertex of △i, for i=1,2,…,n−2. Find the number of all such labellings. Pseudo-incenter of a convex polygon
We are given a convex polygon. Show that one can find a point Q inside the polygon and three vertices A1,A2,A3 (not necessarily consecutive) such that each ray AiQ (i=1,2,3) makes acute angles with the two sides emanating from Ai.