There are cups labeled , where the -th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most steps.
Prove that in at most steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
Circumcircle of XYZ is tangent to circumcircle of ABC
mathuz39
N2 hours ago
by zuat.e
Source: ARMO 2013 Grade 11 Day 2 P4
Let be the incircle of the triangle and with centre . Let be the circumcircle of the triangle . Circles and intersect at the point and . Let be the intersection of the common tangents of the circles and . Show that the circumcircle of the triangle is tangent to the circumcircle of the triangle .
In the acute-angled triangle , the point is the foot of the altitude from , and is a point on the segment . The lines through parallel to and meet at and , respectively. Points and lie on the circles and , respectively, such that and .
Prove that and are concyclic.
Let be an acute triangle and let be the midpoint of . A circle passing through and meets the sides and at points and respectively. Let be the point such that is a parallelogram. Suppose that lies on the circumcircle of . Determine all possible values of .
Let be a triangle with circumcircle . Let and respectively denote the midpoints of the arcs and that do not contain the third vertex. Let denote the midpoint of arc (the arc including ). Let be the incenter of . Let be the circle that is tangent to and internally tangent to at , and let be the circle that is tangent to and internally tangent to at . Show that the line , and the lines through the intersections of and , meet on .
Let be a circle with centre , and a convex quadrilateral such that each of the segments and is tangent to . Let be the circumcircle of the triangle . The extension of beyond meets at , and the extension of beyond meets at . The extensions of and beyond meet at and , respectively. Prove that Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland