The InfinityDots Project

Every week, we will select three notable or interesting problems, marked with E,M,H (“easy”, “medium”, “hard”) for the relative difficulty. Easy problems will be around TMO or easier than TMO; medium problems will be around Oct Camp / easy IMO, and hard problems will be around medium / hard IMO.

E1 [Classical]
Starting from the origin $(0,0)$ . In each turn, you can move either up, down, or right (by one unit) but you cannot visit the same point twice and the points you visit must have coordinate $(x,y)$ such that $0 \leqslant x \leqslant m , 0 \leqslant y \leqslant n$ . How many paths are there to reach the point $(m,n)$ ?

M1 [Kvant Magazine]
Prove that for all positive integers $n>1,$

$\displaystyle \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt[3]{n}\right\rfloor + \dotsc + \left\lfloor\sqrt[n]{n} \right\rfloor = \left\lfloor \log_{2}{n} \right\rfloor + \left\lfloor \log_{3}{n}\right\rfloor + \dotsc + \left\lfloor\log_{n}{n}\right\rfloor.$

H1 [China MO 1993]
Let $f$ be a function from the positive real numbers to the positive real numbers such that $f(xy)\leqslant f(x)f(y)$ for any positive real numbers $x$ and $y$ . Prove that for any positive real number $x$ and all positive integers $n$ ,