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a game played on the integers in the closed interval [1, n]

Source: 2018 SMT - Stanford Math Tournament , Team Round, Proof Question 2

1/26/2022

Consider a game played on the integers in the closed interval

[1, n]

. The game begins with some tokens placed in

[1, n]

. At each turn, tokens are added or removed from

[1, n]

using the following rule: For each integer

k \in [1, n]

, if exactly one of

k - 1

and

k + 1

has a token, place a token at

k

for the next turn, otherwise leave k blank for the next turn. We call a position static if no changes to the interval occur after one turn. For instance, the trivial position with no tokens is static because no tokens are added or removed after a turn (because there are no tokens). Find all non-trivial static positions.

combinatoricsgamegame strategy

SMT 2018 Geometry #12 trapezoid

Source:

8/8/2023

Let

ABCD

be a trapezoid with

AB

parallel to

CD

and perpendicular to

BC

. Let

M

be a point on

BC

such that

\angle AMB = \angle DMC

. If

AB = 3

,

BC = 24

, and

CD = 4

, what is the value of

AM + MD

?

geometrytrapezoid

SMT 2018 Geometry Tiebreaker #2 3D

Source:

8/8/2023

What is the largest possible height of a right cylinder with radius

3

that can fit in a cube with side length

12

?

geometry3D geometry

2

Problems(3)