December 2005 1

Functional Equations (S)

Problems

1. (Russia 2000) Find all functions f : ℝ ⟶ ℝ such that the following equation

holds for all real x, y and z:

f(x + y) + f(y + z) + f(z + x) ≥ 3f(x + 2y + 3z).

2. Suppose f is a real-valued function defined on the real numbers satisfying the following properties:

(a) f(x + y) + f(x − y) = 2f(x)f(y) for all x, y,

(b) There exists x_{0} ∈ ℝ with f(x_{0}) = −1.

Prove that f is periodic.

3. (Balkan 1987) Consider a function f : ℝ⟶ℝ such that f(0) = 1/2 and there

exists a such that for all x and y,

f(x + y) = f(x)f(a − y) + f(y)f(a − x).

Prove f is constant.

4. Find all surjective functions from the set of positive integers to itself with the property that m|n if and only if f(m)|f(n).

5. (IMO 1992) Find all functions f : ℝ⟶ℝ such that f(x^{2} + f(y)) = y + f(x)^{2}.

6. Find all increasing functions f : ℕ⟶ℕ with the property that f(f(n)) = 3n for all n.

7. f : ℕ⟶ℕ (0∈ ℕ)

xf(y) + yf(x) = (x + y)f(x^{2} + y^{2})

8. f : ℚ⟶ℚ,

f(x + f(y)) = f(x) + y

9. f : ℝ⟶ℝ,

(f(x) + f(z))(f(y) + f(t)) = f(xy − zt) + f(xt + yz)

10. (IMO 1999) Find all functions f : ℝ ⟶ ℝ such that for all real x and y we

have

f(x − f(y)) = f(f(y)) + xf(y) + f(x) − 1.