The Exercise Supplement of the Mathematical Gazette // Príloha k cvičeniam v Matematickom vestníku

 Click on the image and use the password : ogeometrie .The QR code on page 2 contains a link to the Books in my Electronic Library. (Click on the year of publication. Same password if needed.) A collection of important magazines is at the letter G : GAZETA MATEMATICĂ seria B.

       

                 To see how simple the exercises and problems are, we will take the following :

    

               "S.E26.100.  The degree measures of the angles of an acute-angled triangle

 are expressed as prime numbers. Prove that the triangle is isosceles.

* * * {no author}"

ANSWER CiP

The triangle has angle measures  $2^{\circ}\;,\;\;89^{\circ}\;,\;\;89^{\circ}$

    Solution CiP

               If  $x^{\circ},\;y^{\circ},\;z^{\circ}$  are the angles of this triangle, then :

$x^{\circ}\;,\;y^{\circ},\;z^{\circ}\;<90^{\circ}\;\;and\;\;x^{\circ}+y^{\circ}+z^{\circ}=180^{\circ} \tag{1}$

          Being prime numbers, if they were all odd then their sum would be odd. We contradict (1). So one of the numbers, let's say  $z^{\circ}$  is  $2^{\circ}$.  Then the second condition in (1) becomes :

$x^{\circ}+y^{\circ}=178^{\circ} \tag{2}$

     An obvious solution for  (2) , which meets the requirements of the problem, is  $x^{\circ}=y^{\circ}=89^{\circ}$.  This triangle is isosceles.

     $x^{\circ}\underset{(1)}{<}89^{\circ}\Rightarrow y\underset{(2)}{=}178^{\circ}-x^{\circ}>89^{\circ}$.  And we can still have  $y^{\circ}=90^{\circ}$ ,  which violates all the requirements.

$\blacksquare$