MathDB

3

Part of 2004 Regional Olympiad - Republic of Srpska

Problems(4)

2 quadratics

Source: RS2004

3/20/2005
Determine all pairs of positive integers (a,b)(a,b), such that the roots of the equations x2−ax+a+b−3=0,x^2-ax+a+b-3=0, x2−bx+a+b−3=0,x^2-bx+a+b-3=0, are also positive integers.
quadraticsalgebra proposedalgebra
Regional Olympiad - Republic of Srpska 2004 Grade 9 Problem 3

Source: Regional Olympiad - Republic of Srpska 2004

9/19/2018
Let ABCABC be an isosceles triangle with ∠A=∠B=80∘\angle A=\angle B=80^\circ. A straight line passes through BB and through the circumcenter of the triangle and intersects the side ACAC at DD. Prove that AB=CDAB=CD.
geometryisoscelesangles
tiling by dominoes and coprime numbers

Source: RS2004

3/20/2005
An 8×88\times8 chessboard is completely tiled by 2×12\times1 dominoes. Prove that we can place positive integers in all cells of the table in such a way that the sums of numbers in every domino are equal and the numbers placed in two adjacent cells are coprime if and only if they belong to the same domino. (Two cells are called adjacent if they have a common side.) Well this can belong to number theory as well...
number theorycombinatorics proposedcombinatorics
periodic seq

Source: RS2004

3/20/2005
Given a sequence (an)(a_n) of real numbers such that the set {an}\{a_n\} is finite. If for every k>1k>1 subsequence (akn)(a_{kn}) is periodic, is it true that the sequence (an)(a_n) must be periodic?
modular arithmeticfunctionarithmetic sequencenumber theory proposednumber theory