1

Part of 2005 Spain Mathematical Olympiad

Problems(2)

Spanish Mathematical Olympiad 2005, Problem 1

Source: 2005 Spanish Mathematical Olympiad

a

b

be integers. Demonstrate that the equation

(x-a)(x-b)(x-3) +1 = 0

has an integer solution.

algebranumber theory

decimal expression of 1/n+1/(n+1)+1/(n+2) is periodic

Source: 2005 Spanish Mathematical Olympiad P4

Prove that for every positive integer

n

, the decimal expression of

\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}

number theoryperiodic decimal number