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Found problems: 2

2020 Australian Maths Olympiad, 5
Each term of an infinite sequence $a_1 ,a_2 ,a_3 , \dots$ is equal to 0 or 1. For each positive integer $n$, $$a_n + a_{n+1} \neq a_{n+2} + a_{n+3},\, \text{and}$$ $$a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}.$$ Prove that if $a_1 = 0$, then $a_{2020} = 1$.
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2020 Australian Maths Olympiad, 4
Define the sequence $A_1, A_2, A_3, \dots$ by $A_1 = 1$ and for $n=1,2,3,\dots$ $$A_{n+1}=\frac{A_n+2}{A_n +1}.$$ Define the sequences $B_1, B_2, B_3,\dots$ by $B_1=1$ and for $n=1,2,3,\dots$ $$B_{n+1}=\frac{B_n^2 +2}{2B_n}.$$ Prove that $B_{n+1}=A_{2^n}$ for all non-negative integers $n$.
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