MathDB

5

Part of 2005 Ukraine National Mathematical Olympiad

Problems(4)

integers

Source: Ukraine 2005 grade 8

7/24/2009
Are there integers a,b,c,d,x,y,z,t a,b,c,d,x,y,z,t such that each of the numbers: |ay\minus{}bx|,|az\minus{}cx|,|at\minus{}dx|,|bz\minus{}cy|,|bt\minus{}dy|,|ct\minus{}dz| equals either 1 1 or 2005 2005?
number theory unsolvednumber theory
sum of terms

Source: Ukraine 2005 grade 9

7/26/2009
Can 1 1 be written as a sum of 2005 2005 different terms of the form \frac{1}{3n\minus{}1}, where n n is a positive integer?
number theory unsolvednumber theory
find all functions

Source: Ukraine 2005 grade 10

7/28/2009
Find all functions f: \mathbb{R}^{\plus{}}\rightarrow \mathbb{R} that satisfy: f(x)f(y)\equal{}f(xy)\plus{}2005 \left( \frac{1}{x}\plus{}\frac{1}{y}\plus{}2004 \right) for all x,y>0. x,y>0.
functionalgebra proposedalgebra
find all positive integers

Source:

7/28/2009
Find all positive integers n n for which \cos (\pi \sqrt{n^2\plus{}n}) \ge 0.
trigonometryalgebra proposedalgebra