Problem A. 766. (December 2019)

A. 766. Let \(\displaystyle T\) be any triangle such that its side-lengths \(\displaystyle a\), \(\displaystyle b\), and \(\displaystyle c\) and its circumradius \(\displaystyle R\) are positive integers. Show that

\(\displaystyle a)\) the inradius \(\displaystyle r\) of \(\displaystyle T\) is a positive integer;

\(\displaystyle b)\) the perimeter \(\displaystyle P\) of \(\displaystyle T\) is a multiple of four; and

\(\displaystyle c)\) all three of \(\displaystyle a\), \(\displaystyle b\), and \(\displaystyle c\) are even.

Proposed by Nikolai Beluhov, Bulgaria

(7 pont)

Deadline expired on January 10, 2020.

Statistics:

8 students sent a solution.
7 points:	Bán-Szabó Áron, Beke Csongor, Tóth 827 Balázs, Várkonyi Zsombor, Weisz Máté.
4 points:	2 students.
0 point:	1 student.

Problems in Mathematics of KöMaL, December 2019