MathDB

6

Part of 2009 Iran MO (3rd Round)

Problems(1)

Numbers in base (1+i)

Source: Iran 3rd round 2009 - final exam problem 6

1/2/2015
Let zz be a complex non-zero number such that Re(z),Im(z)∈ZRe(z),Im(z)\in \mathbb{Z}. Prove that zz is uniquely representable as a0+a1(1+i)+a2(1+i)2+⋯+an(1+i)na_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n where n≥0n\geq 0 and aj∈{0,1}a_j \in \{0,1\} and an=1a_n=1.
Time allowed for this problem was 1 hour.
inductionalgorithmfunctiontrigonometrynumber theory unsolvednumber theory