POTD 2007-04

Table of contents

April 2007

Thursday, 26th of April

from yannick

Find a polynomial f(n)\, which is equivalent to \sum_{k=1}^{n^2}{\left \lfloor \sqrt{k} \right \rfloor+  \left \lceil \sqrt{k} \right \rceil} for all n.\,

Help from landen based on number crunching. Not sure if these help:

f(n)={{4\,n^3}\over{3}}+{{2\,n}\over{3}}

f(n)=4\,f\left(n-1\right)-6\,f\left(n-2\right)+4\,f\left(n-3\right)-f  \left(n-4\right)

HiLander found a solution without these hints. Very clever.

Tuesday, 17th of April

posted by Crito

Source: Magnuz's Problem solving class.

Compare tan(sin x) and sin(tan x) for x \in (0, \pi/2).

Thursday, 12th of April

posted by Crito

Find the number of positive integers x < 102006 such that x2x is divisible by 102006.

Solution by int-e.

Tuesday, 10th of April

posted by Crito


Let a sequence be defined as follows: a1 = 3, a2 = 3, and for n \ge 2, a_{n+1}a_{n-1}= a_{n}^{2}+2007. Find the largest integer less than or equal to \frac{a_{2007}^{2}+a_{2006}^{2}}{a_{2007}a_{2006}}.

Solution by int-e