MathDB

4

Part of 2016 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2016 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
Let AA be a set of 6565 integers with pairwise different remainders modulo 20162016. Prove that exists a subset B={a,b,c,d}B=\{a,b,c,d\} of set AA such that a+b−c−da+b-c-d is divisible with 20162016
Combinatorial Number Theoryremaindersetcombinatorics
Regional Olympiad - FBH 2016 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
Let aa and bb be distinct positive integers, bigger that 10610^6, such that (a+b)3(a+b)^3 is divisible with abab. Prove that ∣a−b∣>104 \mid a-b \mid > 10^4
number theorydivisible
Regional Olympiad - FBH 2016 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
Find all functions f:Q→Rf : \mathbb{Q} \rightarrow \mathbb{R} such that: a)a) f(1)+2>0f(1)+2>0 b)b) f(x+y)−xf(y)−yf(x)=f(x)f(y)+f(x)+f(y)+xyf(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy, ∀x,y∈Q\forall x,y \in \mathbb{Q} c)c) f(x)=3f(x+1)+2x+5f(x)=3f(x+1)+2x+5, ∀x∈Q\forall x \in \mathbb{Q}
functionfunctional equationalgebra
Regional Olympiad - FBH 2016 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

9/22/2018
It is given circle with center in center of coordinate center with radius of 20162016. On circle and inside it are 540540 points with integer coordinates such that no three of them are collinear. Prove that there exist two triangles with vertices in given points such that they have same area
geometryincenteranalytic geometry