MathDB
2019 Serbia MO Day 1 P2

Source: 2019 Serbia MO

April 7, 2019
algebrapolynomial

Problem Statement

For the sequence of real numbers a1,a2,…,aka_1,a_2,\dots ,a_k we say it is invested on the interval [b,c][b,c] if there exists numbers x0,x1,…,xkx_0,x_1,\dots ,x_k in the interval [b,c][b,c] such that ∣xi−xi−1∣=ai|x_i-x_{i-1}|=a_i for i=1,2,3,…ki=1,2,3,\dots k . A sequence is normed if all its members are not greater than 11 . For a given natural nn , prove :
a)Every normed sequence of length 2n+12n+1 is invested in the interval [0,2−12n]\left[ 0, 2-\frac{1}{2^n} \right ]. b) there exists normed sequence of length 4n+34n+3 wich is not invested on [0,2−12n]\left[ 0, 2-\frac{1}{2^n} \right ].
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