Problem 2.1.19. For each positive integer n, define by s(n) the greatest integer such that for all positive integer k ≤ s(n), n² can be expressed as a sum of squares of k positive integers. (a) Prove that s(n) ≤ n² - 14 for all n ≥ 4. (b) Find a number n such that s(n) = n² - 14. (c) Prove that there exist infinitely many positive integers n such that s(n) = n² - 14. (33rd IMO) Problem 2.1.20. Let A be the set of positive integers representable in the form a² + 2b² for integers a, b with b ≠ 0. Show that if p² ∈ A for a prime p, then p ∈ A. (1997 Romanian IMO Team Selection Test)

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Problem 2.1.19. For each positive integer n, define by s(n) the greatest integer such that for all positive integer k ≤ s(n), n² can be expressed as a sum of squares of k positive integers.

(a) Prove that s(n) ≤ n² - 14 for all n ≥ 4.
(b) Find a number n such that s(n) = n² - 14.
(c) Prove that there exist infinitely many positive integers n such that
s(n) = n² - 14.

(33rd IMO)

Problem 2.1.20. Let A be the set of positive integers representable in the form a² + 2b² for integers a, b with b ≠ 0. Show that if p² ∈ A for a prime p, then p ∈ A.

(1997 Romanian IMO Team Selection Test)