MathDB

1

Part of 2015 Czech-Polish-Slovak Match

Problems(2)

Show that an angle is obtuse

Source: Czech-Polish-Slovak Match 2015, Problem 1

6/19/2015
On a circle of radius rr, the distinct points AA, BB, CC, DD, and EE lie in this order, satisfying AB=CD=DE>rAB=CD=DE>r. Show that the triangle with vertices lying in the centroids of the triangles ABDABD, BCDBCD, and ADEADE is obtuse.
Proposed by Tomáš Jurík, Slovakia
geometry
A strange calculator

Source: Czech-Polish-Slovak Match 2015, Problem 4

6/19/2015
A strange calculator has only two buttons with positive itegers, each of them consisting of two digits. It displays the number 1 at the beginning. Whenever a button with number NN is pressed, the calculator replaces the displayed number XX with the number X⋅NX\cdot N or X+NX+N. Multiplication and addition alternate, multiplication is the first. (For example,if the number 10 is on the 1st button, the number 20 is on the 2nd button, and we consecutively press the 1st, 2nd, 1st and 1st button, we get the results 1⋅10=101\cdot 10=10, 10+20=3010+20=30, 30⋅10=30030\cdot 10=300, and 300+10=310300+10=310.) Decide whether there exist particular values of the two-digit nubers on the buttons such that one can display infinitely many numbers (without cleaning the display, i.e. you must keep going and get infinitel many numbers) ending with (a) 20152015, (b) 58135813.
Proposed by Michal Rolínek and Peter Novotný
number theory