1

Part of 2014 Spain Mathematical Olympiad

Problems(2)

Consecutive 3 Numbers on a Circle

Source: OME 2014 1

Is it possible to place the numbers

0,1,2,\dots,9

on a circle so that the sum of any three consecutive numbers is a) 13, b) 14, c) 15?

inductionsymmetrygraph theorycombinatorics unsolvedcombinatorics

Sequence with cubes

Source: OME 2014 4

(x_n)

be a sequence of positive integers defined by

x_1=2

x_{n+1}=2x_n^3+x_n

for all integers

n\ge1

. Determine the largest power of

5

x_{2014}^2+1

geometrynumber theory unsolvednumber theory