MathDB

9

Part of 1989 IMO Longlists

Problems(2)

Infinitely many pos. integers m such that m - f(m) = 1989

Source: IMO Longlist 1989, Problem 9

9/18/2008
Let m m be a positive integer and define f(m) f(m) to be the number of factors of 2 2 in m! m! (that is, the greatest positive integer k k such that 2k∣m! 2^k|m!). Prove that there are infinitely many positive integers m m such that m \minus{} f(m) \equal{} 1989.
floor functionnumber theory unsolvednumber theory
Do there exist two sequences of real numbers

Source: IMO Longlist 1989, Problem 108

9/18/2008
Do there exist two sequences of real numbers {ai},{bi}, \{a_i\}, \{b_i\}, i∈N, i \in \mathbb{N}, satisfying the following conditions: 3⋅π2≤ai≤bi \frac{3 \cdot \pi}{2} \leq a_i \leq b_i and \cos(a_i x) \minus{} \cos(b_i x) \geq \minus{} \frac{1}{i} ∀i∈N \forall i \in \mathbb{N} and all x, x, with 0<x<1? 0 < x < 1?
algebra unsolvedalgebra