MathDB

11.8

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(2)

(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},... (I Soros Olympiad 1994-95 R1 11.8)

Source:

8/1/2021
A polynomial with rational coefficients is called integer, if it takes integer values ​​for all integer values ​​of the variable. For an integer polynomial PP, consider the sequence (−1)P(1),(−1)P(2),(−1)P(3),...(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...
a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer kk and for all natural ii, the ii-th and (i+2ki+2^k)th members of the sequence are equal).
b) Prove that for any periodic sequence consisting of (−1)(- 1) and 1 1 and with a period of some power of two, there exists a integer, polynomial P for which this sequence is (−1)P(1),(−1)P(2),(−1)P(3),...(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...
polynomialalgebraperiodic
min length of arithm. progression (I Soros Olympiad 1994-99 Round 2 11.8)

Source:

5/26/2024
Let's write down a segment of a series of integers from 00 to 19951995. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining 19941994 numbers. Let KK be the length of the progression. Which two numbers must be crossed out so that the value of KK is the smallest?
algebraArithmetic Progression