1

Part of 2022 239 Open Mathematical Olympiad

Problems(2)

Source: 239-School Open Olympiad (Senior Level)

A piece is placed in the lower left-corner cell of the

15 \times 15

board. It can move to the cells that are adjacent to the sides or the corners of its current cell. It must also alternate between horizontal and diagonal moves

(

the first move must be diagonal

).

What is the maximum number of moves it can make without stepping on the same cell twice

?

combinatoricsboarddiagonalcells

Polynomial with many different integer roots

Source: 239-School Open Olympiad 2022, Junior League P1

Egor and Igor take turns (Igor starts) replacing the coefficients of the polynomial

a_{99}x^{99} + \cdots + a_1x + a_0

with non-zero integers. Egor wants the polynomial to have as many different integer roots as possible. What is the largest number of roots he can always achieve?

algebragamepolynomial