MathDB

15

Part of 2022 Durer Math Competition Finals

Problems(2)

Doofy duck buy tangerines in the store

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E15

12/12/2022
Doofy duck buy tangerines in the store. All tangerines have equal weight and are divided into 99, 1010, 1111, 1212, or 1313 equal wedges, although this cannot be seen without peeling them. How many tangerines does Doofy duck need to buy if he wishes to eat exactly one tangerine’s worth while eating at most one wedge from every tangerine? Doofy duck only peels the tangerines at home.
combinatorics
ant crawls along the grid lines of an infinite quadrille notebook

Source: (2021-) 2022 XV 15th Dürer Math Competition Finals Day 2 E+15

12/12/2022
An ant crawls along the grid lines of an infinite quadrille notebook. One grid point is marked red, this is its starting point. Every time the ant reaches a grid point, it continues forward with probability 13\frac13 , left with probability 13\frac13 , and right with probability 13\frac13. What is the chance that it is after its third turn, but not after its fourth turn that it returns to the red point? If the answer is pq\frac{p}{q} , where pp and qq are coprime positive integers, then your answer should be p+qp + q. The steps of the ant are independent.
combinatorics