5
Part of 2007 Estonia National Olympiad
Problems(4)
Estonian Math Competitions 2006/2007
Source: Finalround Problem 2
7/29/2008
Juhan wants to order by weight five balls of pairwise different weight, using only a balance scale. First, he labels the balls with numbers 1 to 5 and creates a list of weighings, such that each element in the list is a pair of two balls. Then, for every pair in the list, he weighs the two balls against each other. Can Juhan sort the balls by weight, using a list with less than 10 pairs?
combinatorics unsolvedcombinatorics
Estonian Math Competitions 2006/2007
Source: Finalround Problem 6
7/29/2008
The identifier of a book is an n-tuple of numbers 0, 1, .... , 9, followed by a checksum. The checksum is computed by a fixed rule that satisfies the following property: whenever one increases a single number in the n-tuple (without modifying the other numbers), the checksum also increases. Find the smallest possible number of required checksums if all possible n-tuples are in use.
combinatorics unsolvedcombinatorics
Estonian Math Competitions 2006/2007
Source: Finalround Problem 11
7/29/2008
Some circles of radius 2 are drawn on the plane. Prove that the numerical value of
the total area covered by these circles is at least as big as the total length of arcs bounding the area.
geometrygeometry unsolved
marked sqaures in a n x n sqaure grid
Source: 2007 Estonia National Olympiad Final Round grade 12 p5
3/12/2020
In a grid of dimensions , a part of the squares is marked with crosses such that in each at least half of the squares are marked. Find the least possible the total number of marked squares in the grid.
combinatoricssquare table