MathDB

4

Part of 2013 Saudi Arabia IMO TST

Problems(3)

numbers 1, 2,...,2012 in a circle

Source: 2013 Saudi Arabia IMO TST I p4

7/23/2020
Determine whether it is possible to place the integers 1,2,...,20121, 2,...,2012 in a circle in such a way that the 20122012 products of adjacent pairs of numbers leave pairwise distinct remainders when divided by 20132013.
number theorycombinatoricsremainder
each pos. integer exactly once in |a_1- a_2|,|a_2 - a_3|, ...,|a_k- a_{k+1}|,...

Source: 2013 Saudi Arabia IMO TST II p4

7/23/2020
Determine if there exists an infinite sequence of positive integers a1,a2,a3,...a_1,a_2, a_3, ... such that (i) each positive integer occurs exactly once in the sequence, and (ii) each positive integer occurs exactly once in the sequence ∣a1−a2∣,∣a2−a3∣,...,∣a+k−ak+1∣,... |a_1 - a_2|, |a_2 - a_3|, ..., |a+k - a_{k+1}|, ...
Sequencealgebranumber theory
2^n - 1 is divisible by p(n)

Source: 2013 Saudi Arabia IMO TST III p4

7/23/2020
Find all polynomials p(x)p(x) with integer coefficients such that for each positive integer nn, the number 2n−12^n - 1 is divisible by p(n)p(n).
algebrapolynomialIntegerdivisible