COMC Functions and Sequences

Question 1

A function f satisfies f(x + y) = f(x) + f(y) for all real numbers x and y, and f(1) = 4. What is f(7)?

Accepted Answer

28

Answer

By the Cauchy functional equation, f(n) = 4n for all integers, so f(7) = 4 × 7 = 28.

Question 2

If f(2x) = f(x) + x for all x > 0 and f(1) = 0, what is f(8)?

Accepted Answer

7

Answer

f(2) = f(1)+1 = 1, f(4) = f(2)+2 = 3, f(8) = f(4)+4 = 7.

Question 3

Let f(x) = ax + b with a > 0. If f(f(x)) = 9x + 16, what is a + b?

Accepted Answer

7

Answer

f(f(x)) = a²x + b(a+1) = 9x + 16, so a² = 9 → a = 3, then 4b = 16 → b = 4, giving a + b = 7.

Question 4

How many functions f: {1, 2, 3} → {1, 2, 3} satisfy f(f(x)) = x for all x in the domain?

Accepted Answer

4

Answer

Such involutions are: the identity, swap(1,2) fix 3, swap(1,3) fix 2, swap(2,3) fix 1 — exactly 4 functions.

Question 5

The function f satisfies f(1) = 1 and f(n+1) = f(n) + 2n for all positive integers n. What is f(6)?

Accepted Answer

31

Answer

f(2)=3, f(3)=7, f(4)=13, f(5)=21, f(6)=31; equivalently f(n) = n² − n + 1.

COMC - Canadian Open Mathematics Challenge Practice Test