MathDB

2

Part of 2016 Danube Mathematical Olympiad

Problems(2)

Divisors Yield Primes

Source: Mathematical Danube Competition 2016, Juniors P2

4/21/2022
Determine all positive integers n>1n>1 such that for any divisor dd of n,n, the numbers d2āˆ’d+1d^2-d+1 and d2+d+1d^2+d+1 are prime.
Lucian Petrescu
number theoryprime numbersDivisorsromania
Combinatorics

Source: Danube Mathematical Competition,Seniors #2

10/29/2016
A bank has a set S of codes formed only with 0 and 1,each one with length n.Two codes are 'friends' if they are different on only one position.We know that each code has exactly k 'friends'.Prove that: 1)S has an even number of elements 2)S contains at least 2k2^k codes
combinatorics