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IMC 1994 D1 P6

Source:

March 6, 2017

IMCreal analysis

Problem Statement

Let

f\in C^2[0,N]

and

x\in [0, N]

,

f''(x)>0

for every

x\in [0, N]

. Let

0\leq m_0\ <m_1 < \cdots < m_k\leq N

be integers such that

n_i=f(m_i)

are also integers for

i=0,1,\ldots, k

. Denote

b_i=n_i-n_{i-1}

and

a_i=m_i-m_{i-1}

for

i=1,2,\ldots, k

.
a) Prove that

-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>1

there are no more than

N / A

indices

j

such that

a_j>A

.
c) Prove that

k\leq 3N^{2/3}

(i.e. there are no more than

3N^{2/3}

integer points on the curve

y=f(x)

,

x\in [0,N]

).

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