MathDB

3

Part of 2016 Azerbaijan BMO TST

Problems(3)

combinatorics

Source:

2/20/2016
kk is a positive integer. AA company has a special method to sell clocks. Every customer can reason with two customers after he has bought a clock himself ;; it's not allowed to reason with an agreed person. These new customers can reason with other two persons and it goes like this.. If both of the customers agreed by a person could play a role (it can be directly or not) in buying clocks by at least kk customers, this person gets a present. Prove that, if nn persons have bought clocks, then at most nk+2\frac{n}{k+2} presents have been accepted.
combinatorics
Balkan TSTp4.3

Source: Azerbaijan Balkan TST 2016 no 4

10/20/2016
a,ba,b are positive integers and (a!+b!)∣a!b!(a!+b!)|a!b!.Prove that 3a≥2b+23a\ge 2b+2.
number theory
Balkan TSTp3.3

Source: Azerbaijan Balkan TST 2016 no 3

10/20/2016
There are some checkers in n⋅nn\cdot n size chess board.Known that for all numbers 1≤i,j≤n1\le i,j\le n if checkwork in the intersection of ii th row and jj th column is empty,so the number of checkers that are in this row and column is at least nn.Prove that there are at least n22\frac{n^2}{2} checkers in chess board.
combinatorics