Problem 1

The function

f(n)

is defined on the positive integers and takes non-negative integer values.

f(2)=0

f(3)>0

f(9999)=3333

, and for all

m,n

f(m+n)-f(m)-f(n)=0 \text{ or } 1.

Determine

f(1982)

Step 5 of 7: Narrowing

f(1982)

to two candidates

In plain words

The answer is now trapped between two consecutive integers; the last step must rule one of them out.

f(1982)-f(1980)-f(2)\in\{0,1\} \ \Rightarrow\ f(1982)\in\{660,661\}

Detailed analysis

Apply the functional condition with $m=1980,n=2$ , using $f(1980)=660$ (Step 4) and $f(2)=0$ .