Olympiad problems

2008 Miklós Schweitzer, 5

Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which $$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$ and show that there is no set for which the error term is $o(\log\log x)$ in the above formula. (translated by Miklós Maróti)

2021 EGMO, 6

Tags: equation , asymptotics , number theory , counting , algebra

Does there exist a nonnegative integer $a$ for which the equation \[\left\lfloor\frac{m}{1}\right\rfloor + \left\lfloor\frac{m}{2}\right\rfloor + \left\lfloor\frac{m}{3}\right\rfloor + \cdots + \left\lfloor\frac{m}{m}\right\rfloor = n^2 + a\] has more than one million different solutions $(m, n)$ where $m$ and $n$ are positive integers? [i]The expression $\lfloor x\rfloor$ denotes the integer part (or floor) of the real number $x$. Thus $\lfloor\sqrt{2}\rfloor = 1, \lfloor\pi\rfloor =\lfloor 22/7 \rfloor = 3, \lfloor 42\rfloor = 42,$ and $\lfloor 0 \rfloor = 0$.[/i]

2020 Jozsef Wildt International Math Competition, W20

Tags: real analysis , sequence , asymptotics

Let $p\in(0,1)$ and $a>0$ be real numbers. Determine the asymptotic behavior of the sequence $\{a_n\}_{n=1}^\infty$ defined recursively by $$a_1=a,a_{n+1}=\frac{a_n}{1+a_n^p},n\in\mathbb N$$ [i]Proposed by Arkady Alt[/i]