1

Part of 2003 IMO Shortlist

Problems(2)

IMO ShortList 2003, number theory problem 1

Source: IMO ShortList 2003, number theory problem 1

m

be a fixed integer greater than

1

x_0

x_1

x_2

\ldots

is defined as follows:

x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}

Find the greatest

k

for which the sequence contains

k

consecutive terms divisible by

m

.
Proposed by Marcin Kuczma, Poland

modular arithmeticnumber theorySequenceDivisibilityIMO Shortlist

Imo shortlist 2003, algebra problem 1

Source: German TST 2004, exam I, problem 1

a_{ij}

i=1,2,3

j=1,2,3

be real numbers such that

a_{ij}

is positive for

i=j

and negative for

i\neq j

.
Prove the existence of positive real numbers

c_{1}

c_{2}

c_{3}

such that the numbers

a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}

are either all negative, all positive, or all zero.
Proposed by Kiran Kedlaya, USA

geometry3D geometrytetrahedronlinear algebraalgebraIMO ShortlistVectors