MathDB

1

Part of 1994 Polish MO Finals

Problems(2)

triples (x,y,z) in Q+

Source: Problem 1, Polish NO 1994

10/7/2005
Find all triples (x,y,z)(x,y,z) of positive rationals such that x+y+zx + y + z, 1x+1y+1z\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} and xyzxyz are all integers.
algebrapolynomialRational Root Theoremalgebra unsolved
jugs with water

Source: Problem 4, Polish NO 1994

10/7/2005
m,nm, n are relatively prime. We have three jugs which contain mm, nn and m+nm+n liters. Initially the largest jug is full of water. Show that for any kk in {1,2,...,m+n}\{1, 2, ... , m+n\} we can get exactly kk liters into one of the jugs.
algorithmnumber theorygreatest common divisornumber theory unsolved