Olympiad problems

2015 Mathematical Talent Reward Programme, MCQ: P 7

How many $x$ are there such that $x,[x],\{x\}$ are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here $[x]$ is the largest integer less than equal to $x$ and $\{x\}=x-[ x]$ ) [list=1] [*] 0 [*] 1 [*] 2 [*] 3 [/list]

2016 Mathematical Talent Reward Programme, MCQ: P 14

Tags: function , algebra , greatest integer function , floor function

Let $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Find $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor \rfloor = 88$ [list=1] [*] $\pi$ [*] 3.14 [*] $\frac{22}{7}$ [*] All of these [/list]

2019 Jozsef Wildt International Math Competition, W. 8

Tags: sequence , summation , harmonic numbers , greatest integer function , limit

Let $(a_n)_{n\geq 1}$ be a positive real sequence given by $a_n=\sum \limits_{k=1}^n \frac{1}{k}$. Compute $$\lim \limits_{n \to \infty}e^{-2a_n} \sum \limits_{k=1}^n \left \lfloor \left(\sqrt[2k]{k!}+\sqrt[2(k+1)]{(k+1)!}\right)^2 \right \rfloor$$where we denote by $\lfloor x\rfloor$ the integer part of $x$.

1998 Bundeswettbewerb Mathematik, 4

Tags: greatest integer function , number theory

Prove that $n + \big[ (\sqrt{2} + 1)^n\big] $ is odd for all positive integers $n$. $\big[ x \big]$ denotes the greatest integer function.

2008 CHKMO, 4

Tags: number theory , greatest common divisor , ordered pairs , greatest integer function

Determine if there exist positive integer pairs $(m,n)$, such that (i) the greatest common divisor of m and $n$ is $1$, and $m \le 2007$, (ii) for any $k=1,2,..., 2007$, $\big[\frac{nk}{m}\big]=\big[\sqrt2 k\big]$ . (Here $[x]$ stands for the greatest integer less than or equal to $x$.)