diff --git a/practice/prep3.py b/practice/prep3.py
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+"""CSC111 Winter 2022 Prep 3: Programming Exercises
+
+Instructions (READ THIS FIRST!)
+===============================
+
+This Python module contains a few functions that you should implement.
+Each function represents a recursively-defined mathematical function,
+corresponding to one in the starter file prep3_functions.pdf.
+(You do not need to modify or submit prep3_functions.pdf, but you'll
+need to read it to complete this programming exercise.)
+
+We have marked each place you need to write code with the word "TODO".
+As you complete your work in this file, delete each TODO comment.
+
+You do not need to add additional doctests. However, you should test your work carefully
+before submitting it!
+
+Copyright and Usage Information
+===============================
+
+This file is provided solely for the personal and private use of students
+taking CSC111 at the University of Toronto St. George campus. All forms of
+distribution of this code, whether as given or with any changes, are
+expressly prohibited. For more information on copyright for CSC111 materials,
+please consult our Course Syllabus.
+
+This file is Copyright (c) 2022 Mario Badr and David Liu.
+"""
+
+
+def formula(n: int) -> float:
+    """Return the value given by Definition 1 in prep3_functions.py.
+
+    Preconditions:
+        - n >= 0
+
+    >>> formula(0)
+    4.0
+    >>> formula(1) == 4 - 4/3
+    True
+    >>> formula(2) == 4 - 4/3 + 4/5
+    True
+    """
+    if n == 0:
+        return 4.0
+    elif n > 0:
+        return formula(n - 1) + ((4 * (-1) ** n) / (2 * n + 1))
+    else:
+        raise ValueError(f'Precondition n >= 0 is violated. (n = {n})')
+
+
+def formula_multiple_args(n: int, a: float, b: float) -> float:
+    """Return the value given by Definition 2 in prep3_functions.py.
+
+    Preconditions:
+        - n >= 0
+
+    >>> formula_multiple_args(0, -1, -1)
+    0.0
+    >>> formula_multiple_args(1, 3, 2)
+    2.0
+    """
+    if n == 0:
+        return 0.0
+    elif n > 0:
+        return a * formula_multiple_args(n - 1, a, b) + b
+    else:
+        raise ValueError(f'Precondition n >= 0 is violated. (n = {n})')
+
+
+def formula_double_recursion(n: int, m: int) -> int:
+    """Return the value given by Definition 3 in prep3_functions.py.
+
+    Preconditions:
+        - n >= 0
+        - m >= 0
+
+    >>> formula_double_recursion(0, 155)
+    155
+    >>> formula_double_recursion(1, 1) == 3 + (2 * 1)
+    True
+    """
+    if n == 0:
+        return m
+    elif n > 0:
+        if m == 0:
+            return 2 * formula_double_recursion(n - 1, n)
+        elif m > 0:
+            return 3 + formula_double_recursion(n, m - 1)
+        else:
+            raise ValueError(f'Precondition m >= 0 is violated. (m = {m})')
+    else:
+        raise ValueError(f'Precondition n >= 0 is violated. (n = {n})')
+
+
+# Note: the type annotation list[int] requires that you have Python 3.9 installed!
+# If you haven't upgraded already, now is an excellent time to do so (check the
+# "Software Installation/Upgrading Instructions" page on Quercus).
+def create_list1(n: int) -> list[int]:
+    """Return the value given by Definition 4 in prep3_functions.py.
+
+    Preconditions:
+        - n >= 0
+
+    >>> create_list1(0)
+    [0]
+    >>> create_list1(1)
+    [0, 0, 1]
+    >>> create_list1(2)
+    [0, 0, 1, 0, 0, 1, 2]
+    """
+    if n == 0:
+        return [0]
+    elif n > 0:
+        return 2 * create_list1(n - 1) + [n]
+    else:
+        raise ValueError(f'Precondition n >= 0 is violated. (n = {n})')
+
+
+def create_list2(n: int, m: int) -> list:
+    """Return the value given by Definition 5 in prep3_functions.py.
+
+    Preconditions:
+        - n >= 0
+        - m >= 0
+
+    TESTING NOTE: because this function makes two recursive calls instead of
+    just one, it will be a lot slower than your other functions!
+    We recommend testing your work with very small numbers, e.g. m, n <= 5.
+
+    >>> create_list2(0, 155)
+    [155]
+    >>> create_list2(155, 0)
+    [155]
+    >>> create_list2(1, 1)
+    [[1], [1]]
+    >>> create_list2(2, 2)
+    [[[2], [[1], [1]]], [[[1], [1]], [2]]]
+    """
+    if n == 0:
+        return [m]
+    elif m == 0:
+        return [n]
+    elif n > 0 and m > 0:
+        return [create_list2(n - 1, m), create_list2(n, m - 1)]
+    else:
+        raise ValueError(f'Precondition (n >= 0 and m >= 0) is violated. (n = {n}, m = {m})')
+
+
+if __name__ == '__main__':
+    import python_ta.contracts
+    python_ta.contracts.check_all_contracts()
+
+    import doctest
+    doctest.testmod()
+
+    import python_ta
+    python_ta.check_all(config={
+        'max-line-length': 100
+    })
diff --git a/practice/prep3_functions.pdf b/practice/prep3_functions.pdf
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