MathDB

2

Part of 2010 Dutch IMO TST

Problems(2)

a_n = A_n + B, M^2 smallest square, M < A +\sqrt{B}

Source: 2010 Dutch IMO TST1 P2

1/10/2020
Let AA and BB be positive integers. De fine the arithmetic sequence a0,a1,a2,...a_0, a_1, a_2, ... by an=An+Ba_n = A_n + B. Suppose that there exists an n≥0n\ge 0 such that ana_n is a square. Let MM be a positive integer such that M2M^2 is the smallest square in the sequence. Prove that M<A+BM < A +\sqrt{B}.
arithmetic sequencealgebra
functional with max, f : R \to R , f(x) = max{y\in R} (2xy - f(y)),

Source: 2010 Dutch IMO TST2 p2

1/10/2020
Find all functions f:R→Rf : R \to R which satisfy f(x)=maxy∈R(2xy−f(y))f(x) = max_{y\in R} (2xy - f(y)) for all x∈Rx \in R.
functional equationmaximumalgebra