MathDB

1

Part of 2020 JBMO Shortlist

Problems(3)

JBMO Shortlist 2020 C1

Source: JBMO Shortlist 2020

7/4/2021
Alice and Bob play the following game: starting with the number 22 written on a blackboard, each player in turn changes the current number nn to a number n+pn + p, where pp is a prime divisor of nn. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greater than 22...2⏟2020\underbrace{22...2}_{2020}. Assuming perfect play, who will win the game.
JuniorBalkanshortlist2020combinatoricsgame
JBMO Shortlist 2020 G1

Source: JBMO Shortlist 2020

7/4/2021
Let △ABC\triangle ABC be an acute triangle. The line through AA perpendicular to BCBC intersects BCBC at DD. Let EE be the midpoint of ADAD and ω\omega the the circle with center EE and radius equal to AEAE. The line BEBE intersects ω\omega at a point XX such that XX and BB are not on the same side of ADAD and the line CECE intersects ω\omega at a point YY such that CC and YY are not on the same side of ADAD. If both of the intersection points of the circumcircles of △BDX\triangle BDX and △CDY\triangle CDY lie on the line ADAD, prove that AB=ACAB = AC.
JuniorBalkanshortlist2020geometry
JBMO Shortlist 2020 N1

Source: JBMO Shortlist 2020

7/4/2021
Determine whether there is a natural number nn for which 8n+478^n + 47 is prime.
JuniorBalkanshortlist2020number theory