MathDB

1

Part of 2007 IMO Shortlist

Problems(2)

Find all sequences satisfying two conditions

Source: IMO Shortlist 2007, C1, AIMO 2008, TST 1, P1

7/13/2008
Let n>1 n > 1 be an integer. Find all sequences a_1, a_2, \ldots a_{n^2 \plus{} n} satisfying the following conditions: \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 \plus{} n;
\text{ (b) } a_{i \plus{} 1} \plus{} a_{i \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} n} < a_{i \plus{} n \plus{} 1} \plus{} a_{i \plus{} n \plus{} 2} \plus{} \ldots \plus{} a_{i \plus{} 2n} \text{ for all } 0 \leq i \leq n^2 \minus{} n. Author: Dusan Dukic, Serbia
combinatoricsSequenceIMO Shortlistinequality systeminduction
7^a - 3^b divides a^4 + b^2 (from IMO Shortlist 2007)

Source: ISL 2007, N1, VAIMO 2008, P4

4/21/2008
Find all pairs of natural numbers (a,b) (a, b) such that 7^a \minus{} 3^b divides a^4 \plus{} b^2.
Author: Stephan Wagner, Austria
DivisibilityIMO Shortlistnumber theory