Putnam TNG - Functional Equations & Iteration

1: Let f: R² → R be a function such that f(x, y) + f(y, z) + f(z, x) = 0 for all real numbers x, y, and z. Prove that there exists a function g: R → R such that f(x, y) = g(x) - g(y) for all real numbers x and y.

2: Find all functions f: R⁺ → R such that for all x, y ∈ R⁺,

f(x + y) = f(x² + y²).

Here, R⁺ is the set of positive real numbers.

3: Determine all real functions f that satisfy

f(x² - y²) = xf(x) - yf(y)

for all pairs of real numbers x and y.

4: Let f(x) be a continuous function such that f(2x² - 1) = 2xf(x) for all x. Show that f(x) = 0 for -1 ≤ x ≤ 1.

5: Let S denote the set of rational numbers different from {-1, 0, 1}. Define f: S → S by f(x) = x - 1/x. Prove or disprove that

_n=1∩^∞f⁽ⁿ⁾(S) = ∅,

where f⁽ⁿ⁾ denotes f composed with itself n times.

6: Let f(x) be a function defined on x > 0 such that f(x) > 0. Find all functions that satisfy: f(x)f(yf(x)) = f(x + y), for all x, y > 0.

7: An (ordered) triple (x₁, x₂, x₃) of positive irrational numbers with x₁ + x₂ + x₃ = 1 is called balanced if each x_i < 1/2. If a triple is not balanced, say x_j > 1/2, one performs the balancing act

B(x₁, x₂, x₃) = (x'₁, x'₂, x'₃)

where x'_i = 2x_i if i ≠ j and x'_j = 2x_j - 1. If the new triple is not balanced, one performs the balancing act on it again. Does continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?

8: Let F: [0, 1] → [0, 1] be a continuous function such that f(f(f(x))) = x for all x ∈ [0, 1]. Prove that f(x) = x for all x ∈ [0, 1].