57 lines
1.5 KiB
TeX
Executable File
57 lines
1.5 KiB
TeX
Executable File
\documentclass[
|
|
solutions,
|
|
singlenumbering,
|
|
nopagenumber
|
|
]{../../resources/ormc_handout}
|
|
\usepackage{../../resources/macros}
|
|
|
|
|
|
\title{Warm-Up: Partition Products}
|
|
\uptitler{\smallurl{}}
|
|
\subtitle{Prepared by Mark on \today.}
|
|
|
|
\begin{document}
|
|
|
|
\maketitle
|
|
|
|
\problem{}
|
|
Take any positive integer $n$. \par
|
|
Now, write it as sum of smaller positive integers: $n = a_1 + a_2 + ... + a_k$ \par
|
|
Maximize the product $a_1 \times a_2 \times ... \times a_k$
|
|
|
|
|
|
|
|
\begin{solution}
|
|
|
|
\textbf{Interesting Solution:}
|
|
|
|
Of course, all $a_i$ should be greater than $1$. \par
|
|
Also, all $a_i$ should be smaller than four, since $x \leq x(x-2)$ if $x \geq 4$. \par
|
|
Thus, we're left with sequences that only contain 2 and 3. \par
|
|
\note{Note that two twos are the same as one four, but we exclude fours for simplicity.}
|
|
|
|
\vspace{2mm}
|
|
|
|
Finally, we see that $3^2 > 2^3$, so any three twos are better repackaged as two threes. \par
|
|
The best sequence $a_i$ thus consists of a maximal number of threes followed by 0, 1, or 2 twos.
|
|
|
|
\linehack{}
|
|
|
|
|
|
|
|
\textbf{Calculus Solution:}
|
|
|
|
First, solve this problem for equal, non-integer $a_i$:
|
|
|
|
\vspace{2mm}
|
|
|
|
We know $n = \prod{a_i}$, thus $\ln(n) = \sum{\ln(a_i)}$. \par
|
|
If all $a_i$ are equal, we get $\ln(n) = k \times \ln(n / k)$. \par
|
|
Derive wrt $k$ and set to zero to get $\ln(n / k) = 1$ \par
|
|
So $k = n / e$ and $n / k = e \approx 2.7$
|
|
|
|
\vspace{2mm}
|
|
|
|
If we try to approximate this with integers, we get the same solution as above.
|
|
\end{solution}
|
|
\end{document} |