UN NUMAR $\pi$ CIUDAT CARE APARE IN REZOLVARE // A STRANGE NUMBER $\pi$ THAT APPEARS IN THE SOLUTION // ΕΝΑΣ ΠΑΡΑΞΕΝΟΣ ΑΡΙΘΜΟΣ $\pi$ ΠΟΥ ΕΜΦΑΝΙΖΕΤΑΙ ΣΤΗ ΛΥΣΗ

This is about the problem exposed in the Post here. The problem has two official solutions (see pages 15-16). I discussed the Second Solution in the Post here. The First Solution inspired me to continue the solution in the Post here(here I also explained why I will use the Greek letter  $\pi$  for the "p" symbol).

               First Solution

               

               Let be  $m=\frac{a^2+b}{a+b-1}=a+1-\frac{ab-1}{a+b-1}$

$n=\frac{b^2+a}{a+b}=b+1-\frac{ab+b}{a+b}$

If  $m\;,n$  are natural number, then the number

$p\;(\pi\; for\;me)=\frac{ab+b}{a+b}-\frac{ab-1}{a+b-1}=\frac{b^2+a}{(a+b)(a+b-1}>0 \tag{1}$

(!! Notice that the number in  (1)  coincides with the number  $k$  in the mentioned Post)

We have  $p<1\;\Leftrightarrow\;b^2+a<a^2+2ab+b^2-a-b\;\Leftrightarrow$

$\Leftrightarrow\;a(a-2)+b(2a-1)>0$

and the last inequality is true for  $a\geqslant 2$,  so in the case  $a\geqslant 2$  there are no

solutions. For  $a=1$  we obtain the solution  $(1,1)$.

$\blacksquare$