6

Part of 2002 Tournament Of Towns

Problems(4)

Sequence and divisibility

Source: Tournament of Towns,Spring 2002, Junior A Level, P6

In an infinite increasing sequence of positive integers, every term from the

2002^{\text{th}}

term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.

inductioninequalitiesnumber theory proposednumber theory

Source: Tournament of Towns,Spring 2002, Senior A Level, P6

52

cards of a standard deck are placed in a

13\times 4

array. If every two adjacent cards, vertically or horizontally, have the same suit or have the same value, prove that all

13

cards of the same suit are in the same row.

combinatorics proposedcombinatorics

Source: Tournament of Towns, Fall 2002, Junior A Level, P6

There's a large pile of cards. On each card a number from

1,2,\ldots n

is written. It is known that sum of all numbers on all of the cards is equal to

k\cdot n!

k

. Prove that it is possible to arrange cards into

k

stacks so that sum of numbers written on the cards in each stack is equal to

n!

combinatorics proposedcombinatorics

A bijection on N

Source: Tournament of Towns, Fall 2002, Senior A Level, P6

Define a sequence

\{a_n\}_{n\ge 1}

a_1=1,a_2=2

a_{n+1}

is the smallest positive integer

m

m

hasn't yet occurred in the sequence and also

\text{gcd}(m,a_n)\neq 1

. Show all positive integers occur in the sequence.

number theorygreatest common divisornumber theory proposed