Difference between revisions of "2025 SSMO Speed Round Problems/Problem 1"
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Thus, the smallest <math>n</math> such that <math>x_n>100</math> is <math>\boxed{6}</math>. | Thus, the smallest <math>n</math> such that <math>x_n>100</math> is <math>\boxed{6}</math>. | ||
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| + | ~Sedro | ||
Latest revision as of 14:39, 9 September 2025
Problem
Define ![\[f(x) = \begin{cases} x + 2 & \text{if } x < 0 \\ x^2 + 1 & \text{if } x \geq 0 \end{cases}\]](http://latex.artofproblemsolving.com/0/c/e/0ce16be4c484b7e04b45e4643012c8b1e9310f95.png)
and let 
. Define 
. Find the least 
such that 
.
Solution
We compute the first few values of 
:
\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 such that 
is 
.
~Sedro
