2

Part of ICMC 7

Problems(2)

Unusual movement on the lattice grid

Source: ICMC 7 Round 1 Problem 2

Fredy starts at the origin of the Euclidean plane. Each minute, Fredy may jump a positive integer distance to another lattice point, provided the jump is not parallel to either axis. Can Fredy reach any given lattice point in 2023 jumps or less?
Proposed by Tony Wang

combinatoricslattice pointsICMC

Operations on circular necklaces

Source: ICMC 7 Round 2 Problem 2

n\geqslant 3

be a positive integer. A circular necklace is called fun if it has

n{}

black beads and

n{}

white beads. A move consists of cutting out a segment of consecutive beads and reattaching it in reverse. Prove that it is possible to change any fun necklace into any other fun necklace using at most

(n-1)

moves.
Note: Necklaces related by rotations or reflections are considered to be the same.
Proposed by Dylan Toh