MathDB

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

P

, consider the sequence

(-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

k

and for all natural

i

, the

i

-th and (

i+2^k

)th members of the sequence are equal).
b) Prove that for any periodic sequence consisting of

(- 1)

and

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)},...

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

0

1995

. Among the numbers written out, two have been crossed out. Let's consider the longest arithmetic progression contained among the remaining

1994

numbers. Let

K

be the length of the progression. Which two numbers must be crossed out so that the value of

K

is the smallest?

algebraArithmetic Progression