MathDB

3

Part of 2007 Italy TST

Problems(2)

Nice functional equation

Source: Italian TST , day 1, n°3

6/1/2007
Find all f:R⟶Rf: R \longrightarrow R such that f(xy+f(x))=xf(y)+f(x)f(xy+f(x))=xf(y)+f(x) for every pair of real numbers x,yx,y.
functionalgebrapolynomialalgebra proposedfunctional equation
Strange inequality on primes and factorization...

Source: Italian TST , day 2, n°3

6/2/2007
Let p≥5p \geq 5 be a prime.
(a) Show that exists a prime q≠pq \neq p such that q∣(p−1)p+1q| (p-1)^{p}+1
(b) Factoring in prime numbers (p−1)p+1=∏i=1npiai(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}} show that: ∑i=1npiai≥p22\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2
inequalitiesgeometrygeometric transformationreflectionnumber theoryprime numbersnumber theory proposed