IMO LL 1992

[SANZEED]

A sequence of positive integers $\left \{ a_{n} \right \}$ is defined as follows:
$ a_{n}=\left [ n+\sqrt{n}+\frac{1}{2} \right ] $
Find the positive integers that occur in this series.

[SANZEED]

My ideas an solution:

I used the idea of inverse function & complementary function. If we check & observe then we will see that the sequence defined here has all the non-square positive integers as its terms. So we can define another series $\left \{ b_{n} \right \}$ such that these two contains all the positive integers together. Clearly $\left \{ b_{n} \right \}$ contains all the squares. Now these two sequences are complementary, so there exists $f(n),f^{*}(n)$ such that $a_{n}=f(n)+n,b_{n}=f^{*}(n)+n$ and $f^{*}(n)=k\Rightarrow f(k)< n\leq f(k+1),f^{*}(n)=n^{2}-n$. Now calculation yields the result.

[*Mahi*]

For others, I can provide two hints

(probably everyone knows)
1. $\left [ a+x \right ] = a+\left [x \right ] $ with $a$ integer.
2. $\left [ x+ \frac 12 \right ]$ is the closest integer to $x$

[SANZEED]

Yes,...,In my solution I used them.Thanks to Mahi vaia.