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ceciliachan1979 committed Jul 29, 2024
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4 changes: 2 additions & 2 deletions Exams/IMO/2024/q1.tex
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Expand Up @@ -29,7 +29,7 @@ \subsection*{Solution by Cecilia}

So the first non-negative term in the sum is $ 1 $, but that has to be divisible by $ m > 1 $, so there is a contradiction for case 1.

Case 2: $ 1 < e < 2 $
Case 2: $ 1 \le e < 2 $

Let's consider the sequence $ A_n = \frac{2n - 1}{n} = \{1, \frac{3}{2}, \frac{5}{3}, ...\} $. As we will see, this sequence is going to help us to reason about the floor operators.

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\begin{align*}
A_{q} &\le e \\
(q+1)A_{q} &\le (q+!)e \\
(q+1)A_{q} &\le (q + 1)e \\
\frac{(2q - 1)(q + 1)}{q} &\le (q + 1)e \\
\frac{2q^2 + q - 1}{q} &\le (q + 1)e \\
2q + 1 - \frac{1}{q} &\le (q + 1)e \\
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2 changes: 1 addition & 1 deletion Exams/IMO/IMO-2023.tex
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Expand Up @@ -5,7 +5,7 @@
\usepackage{hyperref}

\title{IMO 2023}
\author{Cecilia Chan}
\author{Cecilia Chan, CY Fung, aoiyamada211}
\date{July 2024}

\begin{document}
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