MathDB

4

Part of 2008 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

points in plane

Source: Federation of Bosnia, 2. Grades 2008.

4/23/2008
n n points (no three being collinear) are given in a plane. Some points are connected and they form k k segments. If no three of these segments form triangle ( equiv. there are no three points, such that each two of them are connected) prove that k≤⌊n24⌋ k \leq \left \lfloor \frac {n^{2}}{4}\right\rfloor
floor functionfunctioncombinatorics proposedcombinatorics
A U B = N (combinatorics)

Source: Federation of Bosnia, 1. Grades 2008.

4/23/2008
Given are two disjoint sets A A and B B such that their union is N \mathbb N. Prove that for all positive integers n n there exist different numbers a a and b b, both greater than n n, such that either \{ a,b,a \plus{} b \} is contained in A A or \{ a,b,a \plus{} b \} is contained in B B.
Table 9 x 2008

Source: Federation of Bosnia, 3. and 4. Grades 2008.

4/23/2008
A rectangular table 9 9 rows × \times 2008 2008 columns is fulfilled with numbers 1 1, 2 2, ...,2008 2008 in a such way that each number appears exactly 9 9 times in table and difference between any two numbers from same column is not greater than 3 3. What is maximum value of minimum sum in column (with minimal sum)?
floor functioncombinatorics proposedcombinatorics
Regional Olympiad - FBH 2008 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2008

9/18/2018
Determine is there a function a:N→Na: \mathbb{N} \rightarrow \mathbb{N} such that: i)i) a(0)=0a(0)=0 ii)ii) a(n)=n−a(a(n))a(n)=n-a(a(n)), ∀n∈\forall n \in N \mathbb{N}. If exists prove: a)a) a(k)≥a(k−1)a(k)\geq a(k-1) b)b) Does not exist positive integer kk such that a(k−1)=a(k)=a(k+1)a(k-1)=a(k)=a(k+1).
Sequencefunctionalgebrafunctional equation