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Difference between revisions of "2025 SSMO Speed Round Problems/Problem 1"

(Created page with "==Problem== Define <cmath>f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \geq 0 \end{cases}</cmath> and let <math>x_0 = -3</math>. Define <math>x_{n+...")
 
m (→‎Solution)
 
(One intermediate revision by the same user not shown)
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==Solution==
 
==Solution==
 +
 +
We compute the first few values of <math>x_n</math>:
 +
\begin{align*}
 +
x_0 &= -3 \\
 +
x_1 &= f(-3) = -1 \\
 +
x_2 &= f(-1) = 1 \\
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x_3 &= f(1) = 2 \\
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x_4 &= f(2) = 5 \\
 +
x_5 &= f(5) = 26 \\
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x_6 &= f(26) = 677.
 +
\end{align*}
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Thus, the smallest <math>n</math> such that <math>x_n>100</math> is <math>\boxed{6}</math>.
 +
 +
~Sedro

Latest revision as of 14:39, 9 September 2025

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 $\boxed{6}$.

~Sedro

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