MathDB

6

Part of 2015 Tuymaada Olympiad

Problems(2)

Form of integer

Source: Tuymaada 2015, Day 2, Problem 6, Senior League

7/10/2017
Let 0≤b≤c≤d≤a0 \leq b \leq c \leq d \leq a and a>14a>14 are integers. Prove, that there is such natural nn that can not be represented as n=x(ax+b)+y(ay+c)+z(az+d)n=x(ax+b)+y(ay+c)+z(az+d) where x,y,zx,y,z are some integers.
K. Kohas
number theoryalgebra
Interesting sequence

Source: Tuymaada 2015, Day 1, Problem 6, Junior League

7/10/2017
Is there sequence (an)(a_n) of natural numbers, such that differences {an+1−an}\{a_{n+1}-a_n\} take every natural value and only one time and differences {an+2−an}\{a_{n+2}-a_n\} take every natural value greater 20152015 and only one time ?
A. Golovanov
number theory