Problems(3)
construct the bisector
Source: Ukraine 1997
7/18/2009
Construct the bisector of a given angle using a ruler and a compass, but without marking any auxiliary points inside the angle.
geometryangle bisectorgeometry proposed
find the smallest number
Source: Ukraine 1997 grade 9
7/21/2009
Find the smallest natural number which can be represented in the form 19m\plus{}97n, where are positive integers, in at least two different ways.
number theory unsolvednumber theory
find all solutions
Source: Ukraine 1997 grade 11
7/22/2009
Find all real solutions to the equation:
(2\plus{}\sqrt{3})^x\plus{}1\equal{}\left( 2\sqrt{2\plus{}\sqrt{3}} \right)^x.
algebra unsolvedalgebra