MathDB

3

Part of 1998 IberoAmerican

Problems(2)

13rd ibmo - rep. dominicana 1998/q3.

Source: Spanish Communities

4/16/2006
Find the minimum natural number nn with the following property: between any collection of nn distinct natural numbers in the set {1,2,…,999}\{1,2, \dots,999\} it is possible to choose four different a, b, c, da,\ b,\ c,\ d such that: a+2b+3c=da + 2b + 3c = d.
number theory unsolvednumber theory
13rd ibmo - rep. dominicana 1998/q6.

Source: Spanish Communities

4/16/2006
Let λ\lambda the positive root of the equation t2−1998t−1=0t^2-1998t-1=0. It is defined the sequence x0,x1,x2,…,xn,…x_0,x_1,x_2,\ldots,x_n,\ldots by x_0=1,\ x_{n+1}=\lfloor\lambda{x_n}\rfloor\mbox{ for }n=1,2\ldots Find the remainder of the division of x1998x_{1998} by 19981998. Note: ⌊x⌋\lfloor{x}\rfloor is the greatest integer less than or equal to xx.
floor functionmodular arithmeticalgebra solvedalgebra