## Task A-2.5.

How many elements can the largest subset of the set $\{1,2,3, \ldots, 2017\}$ have such that for any two elements $a$ and $b$ of this subset, the number $a+b$ is not divisible by $a-b$?
## Solution.

We claim that the answer is 673.
In the subset, we cannot have consecutive elements $a$ and $b=a+1$ because 1 divides $a+b$. ..... 1 point
In the subset, we cannot have $a$ and $b=a+2$ because 2 divides $a+b=2a+2$. ..... 1 point
Therefore, among three consecutive numbers in the subset, there can be at most one. Hence,
the subset with the desired property can have at most 673 elements.

The subset $A=\{1,4,7, \ldots, 2014,2017\}$, which consists of numbers that give a remainder of 1 when divided by 3, has 673 elements. Thus, $a-b$ does not divide $a+b$.

## COUNTY COMPETITION IN MATHEMATICS

## 3rd grade - high school - A variant

February 28, 2017.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH APPROPRIATELY.