MathDB

1

Part of 2006 IMO Shortlist

Problems(2)

Italian WinterCamps test07 Problem5

Source: ISL 2006, A1, AIMO 2007, TST 1, P1

1/29/2007
A sequence of real numbers a0, a1, a2,… a_{0},\ a_{1},\ a_{2},\dots is defined by the formula a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}  i\geq 0; here a0a_0 is an arbitrary real number, ⌊ai⌋\lfloor a_i\rfloor denotes the greatest integer not exceeding aia_i, and ⟨ai⟩=ai−⌊ai⌋\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor. Prove that ai=ai+2a_i=a_{i+2} for ii sufficiently large.
Proposed by Harmel Nestra, Estionia
floor functionalgebraSequencerecurrence relationPeriodic sequenceIMO Shortlist
n lamps

Source: IMO Shortlist 2006, Combinatorics 1, AIMO 2007, TST 2, P1

6/28/2007
We have n≥2 n \geq 2 lamps L1,...,Ln L_{1}, . . . ,L_{n} in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp Li L_{i} and its neighbours (only one neighbour for i \equal{} 1 or i \equal{} n, two neighbours for other i i) are in the same state, then Li L_{i} is switched off; – otherwise, Li L_{i} is switched on. Initially all the lamps are off except the leftmost one which is on. (a) (a) Prove that there are infinitely many integers n n for which all the lamps will eventually be off. (b) (b) Prove that there are infinitely many integers n n for which the lamps will never be all off.
combinatoricsIMO Shortlist