2025 SSMO Speed Round Problems/Problem 1

Problem

Define $f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \geq 0 \end{cases}$ and let $x_0 = -3$ . Define $x_{n+1} = f(x_n)$ . Find the least $n$ such that $x_n > 100$ .

Solution

We compute the first few values of $x_n$ : \begin{align*} x_0 &= -3 \\ x_1 &= f(-3) = -1 \\ x_2 &= f(-1) = 1 \\ x_3 &= f(1) = 2 \\ x_4 &= f(2) = 5 \\ x_5 &= f(5) = 26 \\ x_6 &= f(26) = 677. \end{align*} Thus, the smallest $n$ such that $x_n>100$ is $\boxed{6}$ .

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2025 SSMO Speed Round Problems/Problem 1

Problem

Solution