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Part of 2008 Middle European Mathematical Olympiad

Problems(2)

Determine the least possible value of a_(2008)

Source: MEMO 2008, Single, Problem 1

Let (a_n)^{\infty}_{n\equal{}1} be a sequence of integers with a_{n} < a_{n\plus{}1}, \forall n \geq 1. For all quadruple

(i,j,k,l)

of indices such that

1 \leq i < j \leq k < l

and i \plus{} l \equal{} j \plus{} k we have the inequality a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}. Determine the least possible value of

a_{2008}.

inequalitiesalgebra unsolvedalgebra

xf(x + xy) = xf(x) + f(x^2)f(y)

Source: MEMO 2008, Team, Problem 5

Determine all functions

f: \mathbb{R} \mapsto \mathbb{R}

such that x f(x \plus{} xy) \equal{} x f(x) \plus{} f \left( x^2 \right) f(y) \forall x,y \in \mathbb{R}.

functionalgebra unsolvedalgebra