MathDB

1

Part of 2004 Italy TST

Problems(2)

Italian TST 2004 - Problem 1

Source: Italian TST 2004 - Problem 1

6/19/2004
At the vertices A,B,C,D,E,F,G,HA, B, C, D, E, F, G, H of a cube, 2001,2002,2003,2004,2005,2008,20072001, 2002, 2003, 2004, 2005, 2008, 2007 and 20062006 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,…,HA, B, \ldots , 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.
geometry3D geometrylinear algebramatrixtetrahedronrectangleinvariant
The lines r,s,t and the two γ circles

Source:

11/21/2010
Two circles γ1\gamma_1 and γ2\gamma_2 intersect at AA and BB. A line rr through BB meets γ1\gamma_1 at CC and γ2\gamma_2 at DD so that BB is between CC and DD. Let ss be the line parallel to ADAD which is tangent to γ1\gamma_1 at EE, at the smaller distance from ADAD. Line EAEA meets γ2\gamma_2 in FF. Let tt be the tangent to γ2\gamma_2 at FF. (a)(a) Prove that tt is parallel to ACAC. (b)(b) Prove that the lines r,s,tr,s,t are concurrent.
geometry proposedgeometry