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6

Part of 1994 IMC

Problems(2)

IMC 1994 D1 P6

Source:

3/6/2017
Let f∈C2[0,N]f\in C^2[0,N] and ∣f′(x)∣<1|f'(x)|<1, f′′(x)>0f''(x)>0 for every x∈[0,N]x\in [0, N]. Let 0≤m0 <m1<⋯<mk≤N0\leq m_0\ <m_1 < \cdots < m_k\leq N be integers such that ni=f(mi)n_i=f(m_i) are also integers for i=0,1,…,ki=0,1,\ldots, k. Denote bi=ni−ni−1b_i=n_i-n_{i-1} and ai=mi−mi−1a_i=m_i-m_{i-1} for i=1,2,…,ki=1,2,\ldots, k.
a) Prove that −1<b1a1<b2a2<⋯<bkak<1-1<\frac{b_1}{a_1}<\frac{b_2}{a_2}<\cdots < \frac{b_k}{a_k}<1
b) Prove that for every choice of A>1A>1 there are no more than N/AN / A indices jj such that aj>Aa_j>A.
c) Prove that k≤3N2/3k\leq 3N^{2/3} (i.e. there are no more than 3N2/33N^{2/3} integer points on the curve y=f(x)y=f(x), x∈[0,N]x\in [0,N]).
IMCreal analysis
IMC 1994 D2 P6

Source:

3/6/2017
Find lim⁡N→∞ln⁡2NN∑k=2N−21ln⁡k⋅ln⁡(N−k)\lim_{N\to\infty}\frac{\ln^2 N}{N} \sum_{k=2}^{N-2} \frac{1}{\ln k \cdot \ln (N-k)}
IMCreal analysislimitcollege contests