I'm going back to the magazine from the previous post.
Starting with this issue, 1 / 1979, a new column appears, PREPARATORY PROBLEMS FOR IMO (see pages 32-33). At that time, I was in my last year of High School and I decided (at the suggestion of a high school classmate, Mariana DIACONESCU) to enroll at the Faculty of Mathematics.
I think that during the summer vacation after my high school diploma, after I passed the college entrance exam, I tried my hand at these problems. Problem O : 1 was a particular case of the article published in the same journal, on pages 5-7, and I imitated the solution there. Problem O : 2 I didn't like at the time, so I moved on to O : 3.
"O : 3. Let $p$ be a prime number, $p>2$ . For each number $k$ from
$1\; to\; p-1$ , we denote by $a_k$ the remainder of the division of $k^p\;by\;p^2$ .
Let it be shown as
$$a_1+a_2+a_3+\dots+a_{p-1}=\frac{p^3-p^2}{2}$$
{Taken from : }( Kvant Magazine, issue 2 / 1978)
ANSWER CiP
$$a_k+a_{p-k}=p^2\;\;\;\;\;,\;\;k=1,\;2,\;\dots,\;p-1 \tag {1}$$
Solution CiP
(in construction)
link-uri la IMO ; GMB-vechi, ; GMB - noi ; KVANT de la mccme.ru ; SIERPINSKY Ce stim si ce nu stim... ;
