1

Part of 2012 Tournament of Towns

Problems(6)

2012 ToT Spring Junior O p1 a treasure is buried under a square 8x8

Source:

A treasure is buried under a square of an

8\times 8

board. Under each other square is a message which indicates the minimum number of steps needed to reach the square with the treasure. Each step takes one from a square to another square sharing a common side. What is the minmum number of squares we must dig up in order to bring up the treasure for sure?

combinatoricsalgebra

2012 ToT Spring Junior A p 1, 2 pears differ by at most 1 gram

Source:

It is possible to place an even number of pears in a row such that the masses of any two neighbouring pears differ by at most

1

gram. Prove that it is then possible to put the pears two in a bag and place the bags in a row such that the masses of any two neighbouring bags differ by at most

1

2012 ToT Spring Senior O p1 polyhedron with 3 edges of same length

Source:

Each vertex of a convex polyhedron lies on exactly three edges, at least two of which have the same length. Prove that the polyhedron has three edges of the same length.

3D geometryconvex polyhedronpolyhedrongeometry

2012 ToT Fall Junior A p1 2 diiferent digits in decimal representation

Source:

The decimal representation of an integer uses only two different digits. The number is at least

10

digits long, and any two neighbouring digits are distinct. What is the greatest power of two that can divide this number?

Digitsnumber theorypower of 2

2012 ToT Fall Junior O p1 difference in ages of 2 out of 5 students

Source:

Five students have the first names Clark, Donald, Jack, Robin and Steve, and have the last names (in a different order) Clarkson, Donaldson, Jackson, Robinson and Stevenson. It is known that Clark is

1

year older than Clarkson, Donald is

2

years older than Donaldson, Jack is

3

years older than Jackson, Robin is

4

years older than Robinson. Who is older, Steve or Stevenson and what is the difference in their ages?

2012 ToT Fall Senior A p1 a_k = a_{k+T}

Source:

Given an infinite sequence of numbers

a_1, a_2, a_3,...

. For each positive integer

k

there exists a positive integer

t = t(k)

a_k = a_{k+t} = a_{k+2t} =...

. Is this sequence necessarily periodic? That is, does a positive integer

T

exist such that

a_k = a_{k+T}

for each positive integer k?

Sequencealgebra