Solve 3 problems (Combinatorics)

Need help with my Mathematics question – I’m studying for my class.

3. Prove that among 1002 positive integers, there are always two integers whose sum or difference is a multiple of 2000.

4. Suppose every point in N^2 is colored using one of 8 colors.

a) Prove that there exists a rectangle whose vertices are monochromatic

b) Suppose N^2 is colored using one of r colors, where r > 0. For which values of r does part (a) still hold?

Challenge problems. Challenge problems are not required for submission, but bonus points will be awarded for submitting a partial attempt or a complete solution.

5) Suppose the function g : Z≥1 → Z satisfies g(1) = 1 and

X d|n g(d) = 0

for all n ≥ 2. Find a closed form for g(n) (your answer may use cases, but not sums).


I attached the screenshot for problems, the mathematical symbols did not appear above there.