This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 3

1983 All Soviet Union Mathematical Olympiad, 356
Tags: sequence , periodical , last digit , digit , floor function , number theory
The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

1985 All Soviet Union Mathematical Olympiad, 405
Tags: periodical , sequence
The sequence $a_1, a_2, ... , a_k, ...$ is constructed according to the rules: $$a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$$Prove that it is non-periodical sequence.

1988 Tournament Of Towns, (183) 6
Tags: periodical , word , combinatorics
Consider a sequence of words , consisting of the letters $A$ and $B$ . The first word in the sequence is "$A$" . The k-th word i s obtained from the $(k-1)$-th by means of the following transformation : each $A$ is substituted by $AAB$ , and each $B$ is substituted by $A$. It is easily seen that every word is an initial part of the next word. The initial parts of these words coincide to give a sequence of letters $AABAABAAA BAABAAB...$ (a) In which place of this sequence is the $1000$-th letter $A$? (b ) Prove that this sequence is not periodic. (V . Galperin , Moscows)