MathDB

3

Part of 1996 Irish Math Olympiad

Problems(2)

function

Source: Ireland 1996

7/1/2009
A function f f from [0,1] [0,1] to R \mathbb{R} has the following properties: (i) (i) f(1)\equal{}1; (ii) (ii) f(x)≥0 f(x) \ge 0 for all x∈[0,1] x \in [0,1]; (iii) (iii) If x,y,x\plus{}y \in [0,1], then f(x\plus{}y) \ge f(x)\plus{}f(y). Prove that f(x)≤2x f(x) \le 2x for all x∈[0,1] x \in [0,1].
functionalgebra proposedalgebra
prime number

Source: Ireland 1996

7/1/2009
Suppose that p p is a prime number and a a and n n positive integers such that: 2^p\plus{}3^p\equal{}a^n. Prove that n\equal{}1.
number theory unsolvednumber theory