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 2 of 7: The extra hypothesis pins down

f(3)

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In plain words

With the two smallest values of $f$ known, the next one is almost forced -- only the extra condition $f(3)>0$ is needed to choose between two possibilities.

f(3)-f(2)-f(1)\in\{0,1\} \ \Rightarrow\ f(3)\in\{0,1\} \ \Rightarrow\ f(3)=1

Detailed analysis

Apply the functional condition with $m=2,n=1$ : $f(3)-f(2)-f(1)\in\{0,1\}$ . Since $f(2)=f(1)=0$ , this gives $f(3)\in\{0,1\}$ ; the hypothesis $f(3)>0$ then forces $f(3)=1$ .