Question 1
The sequence $(a_n)$ is defined by $a_1=1$ and, for every integer $n\ge1$,
$$a_{2n}=a_n,\qquad a_{2n+1}=a_n+1.$$
(a) Find all positive integers $n$ such that $a_{kn}=a_n$ for every positive integer $k\le n$.
(b) Prove that there are infinitely many positive integers $m$ such that $a_{km}\ge a_m$ for every positive integer $k$.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days. Question point values reproduce the supplied source labels; no automatic proof grading is implied. Importer scoring multipliers: correct 1, incorrect 0, blank 0.
$$a_{2n}=a_n,\qquad a_{2n+1}=a_n+1.$$
(a) Find all positive integers $n$ such that $a_{kn}=a_n$ for every positive integer $k\le n$.
(b) Prove that there are infinitely many positive integers $m$ such that $a_{km}\ge a_m$ for every positive integer $k$.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days. Question point values reproduce the supplied source labels; no automatic proof grading is implied. Importer scoring multipliers: correct 1, incorrect 0, blank 0.