MathDB

5

Part of 2024 Brazil National Olympiad

Problems(2)

f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1

Source: Brazilian Mathematical Olympiad 2024, Level 3, Problem 5

10/12/2024
Let R \mathbb{R} be the set of real numbers. Determine all functions f:R→R f: \mathbb{R} \to \mathbb{R} such that, for any real numbers x x and y y , f(x2y−y)=f(x)2f(y)+f(x)2−1. f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1.
functional equationreal numberalgebra
Sequence of quadratic equations

Source: Brazilian Mathematical Olympiad 2024, Level 2, Problem 5

10/12/2024
Esmeralda chooses two distinct positive integers aa and bb, with b>ab > a, and writes the equation
x2−ax+b=0 x^2 - ax + b = 0
on the board. If the equation has distinct positive integer roots cc and dd, with d>cd > c, she writes the equation
x2−cx+d=0 x^2 - cx + d = 0
on the board. She repeats the procedure as long as she obtains distinct positive integer roots. If she writes an equation for which this does not occur, she stops.
a) Show that Esmeralda can choose aa and bb such that she will write exactly 2024 equations on the board. b) What is the maximum number of equations she can write knowing that one of the initially chosen numbers is 2024?
algebraquadraticsboardSequence