MathDB

4

Part of 2011 Bosnia And Herzegovina - Regional Olympiad

Problems(3)

Regional Olympiad - FBH 2011 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
For positive integer nn, prove that at least one of the numbers A=2n−1,B=5n−1,C=13n−1A=2n-1 , B=5n-1, C=13n-1 is not perfect square
number theoryPerfect Squares
Regional Olympiad - FBH 2011 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
Let nn be a positive integer and set S={n,n+1,n+2,...,5n}S=\{n,n+1,n+2,...,5n\} a)a) If set SS is divided into two disjoint sets , prove that there exist three numbers xx, yy and zz(possibly equal) which belong to same subset of SS and x+y=zx+y=z b)b) Does a)a) hold for set S={n,n+1,n+2,...,5n−1}S=\{n,n+1,n+2,...,5n-1\}
combinatoricsSets
Regional Olympiad - FBH 2011 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/27/2018
Prove that among any 66 irrational numbers you can pick three numbers aa, bb and cc such that numbers a+ba+b, b+cb+c and c+ac+a are irrational
combinatoricsirrationalirrational number