MathDB

1

Part of 2001 Federal Competition For Advanced Students, Part 2

Problems(2)

Show that the sum is an integer divisible by 25

Source: Austrian Mathematical Olympiad 2001, Part 2, D1, P1

6/25/2011
Prove that 125∑k=02001[2k25]\frac{1}{25} \sum_{k=0}^{2001} \left[ \frac{2^k}{25}\right] is a positive integer.
modular arithmeticfloor functionalgebra proposedalgebra
Find all R to R functions with f(f(x)^2 + f(y)) = xf(x) + y

Source: Austrian Mathematical Olympiad 2001, Part 2, D2, P1

6/25/2011
Find all functions f:R→Rf :\mathbb R \to \mathbb R such that for all real x,yx, y f(f(x)2+f(y))=xf(x)+y.f(f(x)^2 + f(y)) = xf(x) + y.
functionalgebra unsolvedalgebra