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
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.