MathDB

2

Part of 2011 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2011 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
At the round table there are 1010 students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were 11, 22, 33, 44, 55, 66, 77, 88, 99 and 1010, respectively, which number thought student who told publicly number 66
combinatoricsRound Table
Regional Olympiad - FBH 2011 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
If p>2p>2 is prime number and mm and nn are positive integers such that mn=1+12+13+...+1p−1\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1} Prove that pp divides mm
number theoryprime numbers
Regional Olympiad - FBH 2011 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
For positive integers aa and bb holds a3+4a=b2a^3+4a=b^2. Prove that a=2t2a=2t^2 for some positive integer tt
number theoryequation
Regional Olympiad - FBH 2011 Grade 12 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/27/2018
If for real numbers xx and yy holds (x+1+y2)(y+1+x2)=1\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1 prove that (x+1+x2)(y+1+y2)=1\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1
algebraInequalityinequalities proposed