MathDB

2

Part of 2001 Vietnam Team Selection Test

Problems(2)

H be a point symmetric to B under PT

Source: Vietnam TST 2001 for the 42th IMO, problem 2

6/26/2005
In the plane let two circles be given which intersect at two points A,BA, B; Let PTPT be one of the two common tangent line of these circles (P,TP, T are points of tangency). Tangents at PP and TT of the circumcircle of triangle APTAPT meet each other at SS. Let HH be a point symmetric to BB under PTPT. Show that A,S,HA, S, H are collinear.
geometrycircumcircletrigonometrygeometric transformationreflectionpower of a pointradical axis
painted orthogonal coordinate system

Source: Vietnam TST 2001 for the 42th IMO, problem 5

6/26/2005
Let an integer n>1n > 1 be given. In the space with orthogonal coordinate system OxyzOxyz we denote by TT the set of all points (x,y,z)(x, y, z) with x,y,zx, y, z are integers, satisfying the condition: 1≤x,y,z≤n1 \leq x, y, z \leq n. We paint all the points of TT in such a way that: if the point A(x0,y0,z0)A(x_0, y_0, z_0) is painted then points B(x1,y1,z1)B(x_1, y_1, z_1) for which x1≤x0,y1≤y0x_1 \leq x_0, y_1 \leq y_0 and z1≤z0z_1 \leq z_0 could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.
analytic geometryfloor functionceiling functionIMO Shortlistcombinatorics unsolvedcombinatorics