2025 USA-NA-AIO Round 1, Problem 1, Part 1

AI Olympiads 2025 USA-NA-AIO Round 1
1

Problem 1 (100 points)

Let us consider the following sequence:

F_n = F_{n-1} + F_{n-2} , \ \forall \ n \geq 2 .

Before starting this problem, make sure to run the following code first without any change:

# DO NOT CHANGE

import numpy as np

import matplotlib.pyplot as plt

""" END OF THIS PART """

\color{red}{\text{WARNING !!!}}

  • Beyond importing libraries/modules/classes/functions in the preceeding cell, you are NOT allowed to import anything else for the following purposes:

    • As a part of your final solution. For instance, if a problem asks you to build a model without using sklearn but you use it, then you will not earn points.

    • Temporarily import something to assist you to get a solution. For instance, if a problem asks you to manually compute eigenvalues but you temporarily use np.linalg.eig to get an answer and then delete your code, then you violate the rule.

    Rule of thumb: Each part has its particular purpose to intentionally test you something. Do not attempt to find a shortcut to circumvent the rule.

  • All coding tasks shall run on CPUs, not GPUs.

Part 1 (10 points, non-coding task)

Let F_0 = 3, F_1 = 1.

Manually write down F_n for n = 2, 3, 4 , 5.

  • Reasoning is not required.
2

\color{green}{\text{### WRITE YOUR SOLUTION HERE ###}}

\begin{align*} \boxed{F_2 = 4, F_3 = 5, F_4 = 9, F_5 = 14 .} \end{align*}

\color{red}{\text{""" END OF THIS PART """}}