2

Part of 2024 Brazil Undergrad MO

Problems(1)

Fixed point in cute polynomial

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

For each pair of integers

j, k \geq 2

, define the function

f_{jk} : \mathbb{R} \to \mathbb{R}

f_{jk}(x) = 1 - (1 - x^j)^k.

(a) Prove that for any integers

j, k \geq 2

, there exists a unique real number

p_{jk} \in (0, 1)

f_{jk}(p_{jk}) = p_{jk}

. Furthermore, defining

\lambda_{jk} := f'_{jk}(p_{jk})

\lambda_{jk} > 1

.
(b) Prove that

p^j_{jk} = 1 - p_{kj}

for any integers

j, k \geq 2

.
(c) Prove that

\lambda_{jk} = \lambda_{kj}

for any integers

j, k \geq 2

real analysisderivativeFixed pointalgebrapolynomial