7
Part of 2015 Tournament of Towns
Problems(3)
Circumcircle-Incircle Analogy for 3-D Cuboid
Source: Tournament of Towns Spring 2015 Senior A-level
2/24/2017
It is well-known that if a quadrilateral has the circumcircle and the incircle with the same centre then it is a square. Is the similar statement true in 3 dimensions: namely, if a cuboid is inscribed into a sphere and circumscribed around a sphere and the centres of the spheres coincide, does it imply that the cuboid is a cube? (A cuboid is a polyhedron with 6 quadrilateral faces such that each vertex belongs to edges.)
( points)
geometry3-Dimensional Geometrycircumcircle
Santa Clause Problem
Source: Tournament of Towns 2015 Junior A-Level Question 7
3/27/2017
Santa Clause had sorts of candies, candies of each sort. He distributed them at random between gift bags, candies per a bag and gave a bag to everyone of children at Christmas party. The children learned what they had in their bags and decided to trade. Two children trade one candy for one candy in case if each of them gets the candy of the sort which was absent in his/her bag. Prove that they can organize a sequence of trades so that finally every child would have candies of each sort.
Tournament of Towns
Heights in Descending order
Source: Tournament of Towns Fall 2015 Senior A-level
2/23/2017
children no two of the same height stand in a line. The following two-step procedure is applied: first, the line is split into the least possible number of groups so that in each group all children are arranged from the left to the right in ascending order of their heights (a group may consist of a single child). Second, the order of children in each group is reversed, so now in each group the children stand in descending order of their heights. Prove that in result of applying this procedure times the children in the line would stand from the left to the right in descending order of their heights.
(12 points)
combinatorics