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39th Austrian Mathematical Olympiad

Source: round1, problem2 - next smaller integer and

February 10, 2009
algebrasystem of equationsalgebra unsolved

Problem Statement

For a real number x x is [x] [x] the next smaller integer to x x, that is the integer g g with g≦<g+1 g\leqq<g+1, and {x}=x−[x] \{x\}=x-[x] is the “decimal part” of x x. Determine all triples (a,b,c) (a,b,c) of real numbers, which fulfil the following system of equations: {a}++{c}=2,9 \{a\}++\{c\}=2,9{b}+[c]+{a}=5,3 \{b\}+[c]+\{a\}=5,3{c}+[a]+{b}=4,0\{c\}+[a]+\{b\}=4,0
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