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Part of 2004 Rioplatense Mathematical Olympiad, Level 3

Problems(2)

Functional equation: x p(y/x) + y p(x/y) = x + y

Source: XIII Rioplatense Mathematical Olympiad (2004), Level 3

Find all polynomials

P(x)

with real coefficients such that

xP\bigg(\frac{y}{x}\bigg)+yP\bigg(\frac{x}{y}\bigg)=x+y

for all nonzero real numbers

x

y

algebrapolynomialalgebra unsolved

Integers n such that n divides x^13-x for all x in N

Source: XIII Rioplatense Mathematical Olympiad (2004), Level 3

How many integers

n>1

are there such that

n

x^{13}-x

for every positive integer

x

number theorynumber theory unsolved