By Bouyukliev I., Fack V., Winne J.

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**Additional info for 2-(31,15,7), 2-(35,17,8) and 2-(36,15,6) designs with automorphisms of odd prime order, and their related Hadamard matrices and codes**

**Example text**

What remains to do is to check that so obtained function satisfies the given equation. 13. It is obvious that the identity function f (n) = n satisfies the given equation. We may suspect that there are no other functions than that. First we observe that f (n) is injective. For suppose f (n) = f (m). Then obviously f f (n) = f f (m) and consequently f f f (n) = f f f (m) . e. 3n = 3m and n = m. For n = 1 we get f f f (1) + f f (1) + f (1) = 3, which can only mean that f (1) = 1. Hence f (2) ≥ 2, f (3) ≥ 3, and so on.

Instead, after the problems there are some hints and answers. Problems. 27. (Poland, 1992) Find all functions f : R → R such that f (x + y) − f (x − y) = f (x)f (y) for all x, y ∈ R. 1 28. Find all functions that satisfy the equation f (1 − x) + xf (x − 1) = for all x real x = 0, x = 1 and x = −1. 29. Find all continous functions f : R → R which satisfy the equation f (x+y) = f (x) + f (y) + xy for all x, y ∈ R. 30. Find all functions f : R → R satsifying xf (y) + yf (x) = (x + y)f (x)f (y) for all x, y ∈ R.

46. Hint: Start by taking y = x. Find out that x + f (x) + xf (x) is a fixed point of f (x) for each x ∈ S. How many fixed points can f (x) has at most? −x . Answer: f (x) = x+1 47. Hint: Prove that f (0) = 0. Show that f (f (y)) = y for all real y. Show thereafter that f (x + y) = f (x) + f (y). Answer: f (x) = x. 48. Hint: Start with y = x and then change x to f (x). Assume that f f (x) > x and don’t forget to use the fact that f (x) is strictly decreasing. 49. e. let n2 be the binary representation of n.