1

Part of 2006 Spain Mathematical Olympiad

Problems(2)

integer polynomial , k does not divide P(i), i=1,..., k

Source: 2006 Spanish Mathematical Olympiad P1

P(x)

be a polynomial with integer coefficients. Prove that if there is an integer

k

such that none of the integers

P(1),P(2), ..., P(k)

is divisible by

k

P(x)

does not have integer roots.

polynomialinteger rootInteger Polynomialalgebra

f(x)f(y)+f\(\lambda /x})f(\lambda/y)=2f(xy) w f(\lambda)=1

Source: 2006 Spanish Mathematical Olympiad P4

Find all the functions

f:(0,+\infty) \to R

that satisfy the equation

f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)

for all pairs of

x,y

real and positive numbers, where

\lambda

is a positive real number such that

f(\lambda )=1

functional equationreal numberalgebra