Subcontests
(3)Sum of B-entries equals the sum of corresponding A-entries
Given real numbers xi,yi(i=1,2,…,n), let A be the n×n matrix given by aij=1 if xi≥yj and aij=0 otherwise. Suppose B is a n×n matrix whose entries are 0 and 1 such that the sum of entries in any row or column of B equals the sum of entries in the corresponding row or column of A. Prove that B=A. Infintiely many polygons coinciding with P_0
Let P0=A0A1…An−1 be a convex polygon such that AiAi+1=2[i/2] for i=0,1,…,n−1 (where An=A0). Define the sequence of polygons Pk=A0kA1k…An−1k as follows: Ai1 is symmetric to Ai with respect to A0, Ai2 is symmetric to Ai1 with respect to A11, Ai3 is symmetric to Ai2 with respect to A22 and so on. Find the values of n for which infinitely many polygons Pk coincide with P0. Italian TST 2004 - Problem 1
At the vertices A,B,C,D,E,F,G,H of a cube, 2001,2002,2003,2004,2005,2008,2007 and 2006 stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at A,B,…,H can be obtained?
(\text{a}) 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005;
(\text{b}) 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007;
(\text{c}) 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.