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% Document Info
\author{peterwongau}
\pdfinfo{
  /Title (mathematics-olympiad.pdf)
  /Creator (Cheatography)
  /Author (peterwongau)
  /Subject (Mathematics Olympiad Cheat Sheet)
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Mathematics Olympiad Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{peterwongau} via \textcolor{DarkBackground}{\uline{cheatography.com/133991/cs/32226/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}peterwongau \\
  \uline{cheatography.com/peterwongau} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 28th May, 2022.\\
   Updated 30th May, 2022.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.1 Shortcuts in Computation}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{1. Quicker Counting Methods}} & Grouping numbers that add up to 5 or 10 & 73+74+27+26\{\{nl\}\}=(73+27)+(74+26) \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
 & Round off numbers that are close to 5 or 10 & 73+74+27+26\{\{nl\}\}=75{\bf{-2}}+75{\bf{-1}}+25{\bf{+2}}+25{\bf{+1}} \tn 
% Row Count 8 (+ 5)
% Row 2
\SetRowColor{LightBackground}
{\bf{2. Sum of numbers that form a pattern}} & {\bf{For patterns where}}: numbers \seqsplit{increase/decrease} by same value & 1. rewrite sum in reverse order underneath\{\{nl\}\}2. pair up and sum\{\{nl\}\}3. sums of pairs are the same\{\{nl\}\}4. Since sums are the same, multiple sum by number of pairs\{\{nl\}\}5. Divide by 2 \tn 
% Row Count 23 (+ 15)
% Row 3
\SetRowColor{white}
 & {\bf{Example:}} & Find 2+4+6+...+78+80\{\{nl\}\}\{\{nl\}\}2+4+6+...+78+80\{\{nl\}\}80+78+...+6+4+2\{\{nl\}\}82*40=3280\{\{nl\}\}3280/2=1640\{\{nl\}\}\{\{nl\}\}2+4+6+...+78+80={\bf{1640}} \tn 
% Row Count 34 (+ 11)
\end{tabularx}
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\columnbreak
\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.1 Shortcuts in Computation (cont)}}  \tn
% Row 4
\SetRowColor{LightBackground}
{\bf{3. Quicker \seqsplit{Muliplication} Methods}} & Remember numbers in their expanded form & \seqsplit{3526=3000+500+20+6} \tn 
% Row Count 3 (+ 3)
% Row 5
\SetRowColor{white}
 & {\bf{3.1 Multiples of 10}} & 30x25\{\{nl\}\}=3x10x25\{\{nl\}\}=3x250\{\{nl\}\}=750 \tn 
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% Row 6
\SetRowColor{LightBackground}
 & {\bf{ 3.2 Multiples of 5}} & 25x6\{\{nl\}\}=5x5x6\{\{nl\}\}=5x30\{\{nl\}\}=5x3x10\{\{nl\}\}=15x10\{\{nl\}\}=150 \tn 
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\begin{tabularx}{8.4cm}{x{2.28 cm} x{2.052 cm} x{3.268 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.2 Number Logic}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Properties of Numbers}} & Primes & factor of 1 and itself only \tn 
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% Row 1
\SetRowColor{white}
 & \seqsplit{Composites} & factors other than itself \tn 
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% Row 2
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{\bf{Divisbility Rules}} & \seqsplit{Divisibility} rule of 2 & EVEN\{\{nl\}\}ends with 0,2,4,6,8 \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
 & \seqsplit{Divisibility} rule of 3 & {\bf{sum}} of its digits divisble by 3 \tn 
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% Row 4
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 & \seqsplit{Divisibility} rule of 4 & {\bf{last 2}} digits divisible by 4 \tn 
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% Row 5
\SetRowColor{white}
 & \seqsplit{Divisibility} rule of 5 & ends with 0 or 5 \tn 
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% Row 6
\SetRowColor{LightBackground}
 & \seqsplit{Divisibility} rule of 6 & EVEN AND divisible by 3 \tn 
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\SetRowColor{white}
 & Divisible by 8 & {\bf{last 3}} digits divisible by 8 \tn 
% Row Count 22 (+ 2)
% Row 8
\SetRowColor{LightBackground}
 & Divisible by 9 & {\bf{sum}} of its digits divisble by 9 \tn 
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% Row 9
\SetRowColor{white}
 & Divisible by 10 & ends with 0 \tn 
% Row Count 27 (+ 2)
% Row 10
\SetRowColor{LightBackground}
{\bf{Squared Numbers}} & NxN=N\textasciicircum{}2\textasciicircum{} & eg 2x2=2\textasciicircum{}2\textasciicircum{}=4 \tn 
% Row Count 29 (+ 2)
% Row 11
\SetRowColor{white}
{\bf{Cubed Numbers}} & NxNxN=N\textasciicircum{}3\textasciicircum{} & eg 2x2x2=2\textasciicircum{}3\textasciicircum{}=8 \tn 
% Row Count 31 (+ 2)
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\begin{tabularx}{8.4cm}{x{1.824 cm} x{2.888 cm} x{2.888 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.3 Developing Patterns and Shortcuts}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Factorising Numbers}} & A number is factorised when expressed as a product of {\bf{prime}} numbers & 250\{\{nl\}\}=2x125\{\{nl\}\}=2x5x25\{\{nl\}\}=2x5x5x5 \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
 & Find prime factors &  \tn 
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{\bf{HCF}} & {\bf{largest}} counting number that divides into both exactly &  \tn 
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% Row 3
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Highest Common Factor & {\bf{method}} & 1. factorise\{\{nl\}\}2. multiply the factors that are {\bf{common}}\{\{nl\}\}only those factors that have a pair \tn 
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% Row 4
\SetRowColor{LightBackground}
 & {\bf{example}} & HCF of 240 and 924\{\{nl\}\}240={\bf{2}}x5x{\bf{2}}x2x{\bf{3}}x2\{\{nl\}\}924={\bf{3}}x{\bf{2}}x7x11x{\bf{2}}\{\{nl\}\}HCF=2x3x2=12 \tn 
% Row Count 25 (+ 7)
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{\bf{LCM}} & Of all the multiples of the 2 numbers, its the {\bf{smallest}} multiple they have in common &  \tn 
% Row Count 31 (+ 6)
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\vfill
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\begin{tabularx}{8.4cm}{x{1.824 cm} x{2.888 cm} x{2.888 cm} }
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\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.3 Developing Patterns and Shortcuts (cont)}}  \tn
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Lowest Common Multiple & {\bf{method}} & 1. factorise\{\{nl\}\}2. multiply the factors that are common {\bf{and}} factors they dont have in common \tn 
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\SetRowColor{white}
 & {\bf{example}} & LCM of 120 and 140\{\{nl\}\}120={\bf{2}}x{\bf{2}}x2x{\bf{3}}x5\{\{nl\}\}140={\bf{2}}x{\bf{2}}x{\bf{3}}x7\{\{nl\}\}LCM={\bf{2}}x{\bf{2}}x{\bf{3}}x2x5x7 \tn 
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Question (find \seqsplit{multiples)} & Jack, Art, Fran and Megan work as volunteers at the local kennel. Jack gives the dogs baths every 4 days. Art cleans out cages every 6 days. Frand feeds the animals in section b every 2 days. Megan helps the receptionist every 3 days. {\bf{How many times in 12 weeks will all 4 helpers be at the clinic on the same day?}} & {\bf{how to solve}}\{\{nl\}\}Find all the common multiples from 6 days to 84 days (12 weeks) of 4, 6, 2, 3 \tn 
% Row Count 37 (+ 22)
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\begin{tabularx}{8.4cm}{x{1.824 cm} x{2.888 cm} x{2.888 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.3 Developing Patterns and Shortcuts (cont)}}  \tn
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Question (LCM) & Two buses leave the terminal at {\bf{8am}}. Bus A takes {\bf{60mins}} to complete its route and Bus B takes {\bf{75mins}}. When is the next time the two buses will arrive together at the terminal (if they are on time)? & {\bf{how to solve}}\{\{nl\}\}1. Find LCM of 60 and 75. 2. Add LCM to 8am \tn 
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Question (HCM and LCM) & Dennis has a choice between two house numbers on Small Street. The two house numbers have their highest common factor of 6. Their least common multiple is 36. One of the house numbers is 12. What is the other number? & {\bf{how to solve}}\{\{nl\}\}work backwards \tn 
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\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.4 Logic Deduction}}  \tn
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Logic \seqsplit{Deduction} Problems & If need to add groups of things, use biggest numbers first &  \tn 
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% Row 1
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{\bf{Question}} & What is the minimum number of coins needed to make \$4.85 from only 5c, 20c, 50c coins & Start with biggest coins first, working through to smaller coins\{\{nl\}\}9x50c=\$4.50\{\{nl\}\}1x20c=20c\{\{nl\}\}3x5c=15c \tn 
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% Row 2
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 & assume worse case scenario &  \tn 
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 & investigate standard case &  \tn 
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% Row 4
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 & write relations between numbers down &  \tn 
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\hhline{>{\arrayrulecolor{DarkBackground}}---}
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\begin{tabularx}{8.4cm}{x{1.976 cm} x{2.736 cm} x{2.888 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.5 Space, Area and Volume}}  \tn
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Area of Rectangle & length x width &  \tn 
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Area of Triangle & A = base x height / 2 &  \tn 
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Volume of Cube & V = a\textasciicircum{}3\textasciicircum{} where a is length of a side &  \tn 
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\SetRowColor{white}
Volume of \seqsplit{Rectangular} Prism & V = length x height x depth &  \tn 
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% Row 4
\SetRowColor{LightBackground}
1m & = 100cm &  \tn 
% Row Count 11 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{3}{x{8.4cm}}{{\bf{Finding Area of Rectangular Shapes}}} \tn 
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% Row 6
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{\bf{Method 1}} & Divide shape into rectangles & Find area of each and find total \tn 
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% Row 7
\SetRowColor{white}
{\bf{Method 2}} & Extend shape into one larger rectangle & 1. Find area of larger rectangle (X)\{\{nl\}\}2. Find area of missing rectangle (Y)\{\{nl\}\}3. Larger rectangle (X) - Missing rectangle (Y) \tn 
% Row Count 24 (+ 9)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
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\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.6 Equations}}  \tn
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{\bf{Pronumerals}} & Boxes to store missing numbers &  \tn 
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 & Letters to represent unknown numbers &  \tn 
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 & Use {\bf{x}}, {\bf{y}} and {\bf{z}} &  \tn 
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{\bf{Rearranging Equations}} & = is like a balancing scale &  \tn 
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 & solving an equation & {\bf{aim}} of finding the unknown number \tn 
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 & rearranging equations & {\bf{how}} to solve an equation \tn 
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 &  & {\bf{how}} if we do something to one side, we need to do the same thing to the other side \tn 
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 &  & eg. if we {\bf{add 3}} to one side, we need to {\bf{add 3}} to the other side \tn 
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\end{tabularx}
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\vfill
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\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.6 Equations (cont)}}  \tn
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 &  & eg. if we {\bf{times by}} 3 to one side, we need to {\bf{times by}} 3 to the other side \tn 
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 & + & - \tn 
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 & x & / \tn 
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{\bf{Simultaneous Equations}} & if there are 2 unknowns, need 2 equations &  \tn 
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% Row 12
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{\bf{1. Solving by Adding and Subtracting Equations}} & {\bf{example}} & 5x - y = 4 (1)\{\{nl\}\}2x + y = 10\{\{nl\}\}(1)+(2)\{\{nl\}\}7x = 14\{\{nl\}\}x = 2\{\{nl\}\}y = 6 \tn 
% Row Count 20 (+ 7)
% Row 13
\SetRowColor{white}
 & {\bf{example}} & 7x + y = 18 (1)\{\{nl\}\}2x + 2y = 12 (2)\{\{nl\}\} (1) x 2\{\{nl\}\}14x + 2y = 36 (1a)\{\{nl\}\}(1a) - (2)\{\{nl\}\}12x = 24\{\{nl\}\}x = 2\{\{nl\}\}y = 4 \tn 
% Row Count 30 (+ 10)
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\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.6 Equations (cont)}}  \tn
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{\bf{2. Solving by Substitution}} & {\bf{method}} & 1. rearrange one equation for y\{\{nl\}\}2. substitute y into other equation \tn 
% Row Count 6 (+ 6)
% Row 15
\SetRowColor{white}
 & {\bf{example}} & 5x - y = 4 (1)\{\{nl\}\}2x + y = 10 (2)\{\{nl\}\}{\bf{rearrange (1)}}\{\{nl\}\}y = 5x - 4 (1a)\{\{nl\}\}{\bf{substitute (1a) into (2)}}\{\{nl\}\}2x + (5x - 4) = 10\{\{nl\}\}x = 2\{\{nl\}\}y = 6 \tn 
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% Row 16
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\mymulticolumn{3}{x{8.4cm}}{{\bf{Turning word problems into an equation}}} \tn 
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% Row 17
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{\bf{Step 1}} & What are the unknowns? & Give each a letter, {\bf{x}}, {\bf{y}} \tn 
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% Row 18
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{\bf{Step 2}} & Find the equations to solve &  \tn 
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% Row 19
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{\bf{Step 3}} & Solve the simultaneous equations &  \tn 
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% Row 20
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\mymulticolumn{3}{x{8.4cm}}{Example Questions} \tn 
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\begin{tabularx}{8.4cm}{x{2.584 cm} x{2.508 cm} x{2.508 cm} }
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\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.6 Equations (cont)}}  \tn
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\mymulticolumn{3}{x{8.4cm}}{The quotient of two numbers is 4 and their difference is 39. What is the smaller number of the two} \tn 
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\mymulticolumn{3}{x{8.4cm}}{The sum of the ages of Alan and Bill is 25; the sum of the ages of Alan and Carl is 20; the sum of the ages of Bill and Carl is 31. Who is the oldest of the three boys and how old is he?} \tn 
% Row Count 6 (+ 4)
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\begin{tabularx}{8.4cm}{x{1.444 cm} x{3.116 cm} x{3.04 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.7 Probability, Venn Diagrams and Whodunits}}  \tn
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\mymulticolumn{3}{x{8.4cm}}{{\bf{1. Certainty Problems}}} \tn 
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% Row 1
\SetRowColor{white}
\seqsplit{Typical} \seqsplit{Question} & Suppose that there are ten black and ten navy socks in your drawer. Your room is dark and you cannot turn on the light. What is the smallest number of socks that you must take out of your drawer to be certain that you have a pair of the same colour? & Basically, to be certain of "an outcome", what is the smallest number of "actions" required to take \tn 
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% Row 2
\SetRowColor{LightBackground}
\seqsplit{Strategy} & {\bf{Start from smallest and go up}} &  \tn 
% Row Count 20 (+ 3)
% Row 3
\SetRowColor{white}
 & 1 sock & can't be certain \tn 
% Row Count 21 (+ 1)
% Row 4
\SetRowColor{LightBackground}
 & 2 socks & can't be certain \tn 
% Row Count 22 (+ 1)
% Row 5
\SetRowColor{white}
 & 3 socks & can be certain \tn 
% Row Count 23 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{2. Certainty Problems with Restrictions}}} \tn 
% Row Count 24 (+ 1)
% Row 7
\SetRowColor{white}
\seqsplit{Typical} \seqsplit{Question} & As above question, but what is the smallest number of socks needed to ensure we get a pair of black socks & Restriction is it must be black socks \tn 
% Row Count 31 (+ 7)
\end{tabularx}
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\begin{tabularx}{8.4cm}{x{1.444 cm} x{3.116 cm} x{3.04 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.7 Probability, Venn Diagrams and Whodunits (cont)}}  \tn
% Row 8
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\seqsplit{Strategy} & {\bf{Think Worst Case Scenario}} &  \tn 
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% Row 9
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 & Worst case is you could in 10 picks, pick only Navy socks. 2 more picks you'll be certain to get a pair of black socks &  \tn 
% Row Count 10 (+ 8)
% Row 10
\SetRowColor{LightBackground}
 & 12 socks & can be certain \tn 
% Row Count 11 (+ 1)
% Row 11
\SetRowColor{white}
{\bf{ Venn \seqsplit{Diagrams} }} & circle represents {\bf{sets}} or groups of {\emph{things}} that are same &  \tn 
% Row Count 15 (+ 4)
% Row 12
\SetRowColor{LightBackground}
\seqsplit{Example} \seqsplit{Question} & There are 160 students in Year 5. Of these students, 69 walked to school and 57 caught a train to school. If 148 students either walked to school or caught the train, how many students walked {\bf{and}} caught a train to school? &  \tn 
% Row Count 30 (+ 15)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{1.444 cm} x{3.116 cm} x{3.04 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.7 Probability, Venn Diagrams and Whodunits (cont)}}  \tn
% Row 13
\SetRowColor{LightBackground}
 & Draw a Venn diagram with a circle for students that walked and students that caught the train &  \tn 
% Row Count 6 (+ 6)
% Row 14
\SetRowColor{white}
 & Where they overlap, are the number of students that walked and caught the train &  \tn 
% Row Count 11 (+ 5)
% Row 15
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{Whodunits}}} \tn 
% Row Count 12 (+ 1)
% Row 16
\SetRowColor{white}
\seqsplit{Strategy} & Use a table, with different {\bf{characteristics}} in columns and {\bf{members}} of a group in rows & Usually the answer needed are the {\bf{characteristics}} \tn 
% Row Count 18 (+ 6)
% Row 17
\SetRowColor{LightBackground}
\seqsplit{Example} \seqsplit{Question} & Martin, Bill and Dave (members of a group) play first base, second base, and third base \seqsplit{(characteristics)} on their school softball team, but not necessarily in that order. Martin and the third baseman took Dave to the movies yesterday. Martin does not play first base. Who's on first base? &  \tn 
% Row Count 37 (+ 19)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.672 cm} x{2.964 cm} x{2.964 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.8 Motions, Books, Clocks and Work Problems}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{1. Motion Problems}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\seqsplit{distance} & = rate x time &  \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{Example \seqsplit{Question} 1}} & Two trains leave the same station at the same time, but in {\bf{opposite}} directions. One train averages 56 km/h and the other averages 64 km/h. How far apart will the trains be when three hours have passed? &  \tn 
% Row Count 16 (+ 14)
% Row 3
\SetRowColor{white}
\seqsplit{Strategy} & Step 1 & Whats the distance after 1hr? (Draw a diagram) \tn 
% Row Count 20 (+ 4)
% Row 4
\SetRowColor{LightBackground}
 &  & 56km + 64km = 120km \tn 
% Row Count 22 (+ 2)
% Row 5
\SetRowColor{white}
 &  & 56km/hr + 64km/hr = 120km/hr \tn 
% Row Count 24 (+ 2)
% Row 6
\SetRowColor{LightBackground}
 & Step 2 & Whats the distance after 3hrs? \tn 
% Row Count 26 (+ 2)
% Row 7
\SetRowColor{white}
 &  & 120km x 3 = 360km \tn 
% Row Count 28 (+ 2)
% Row 8
\SetRowColor{LightBackground}
 & if {\bf{opposite}} direction, & {\bf{add}} \tn 
% Row Count 30 (+ 2)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{1.672 cm} x{2.964 cm} x{2.964 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.8 Motions, Books, Clocks and Work Problems (cont)}}  \tn
% Row 9
\SetRowColor{LightBackground}
{\bf{Example \seqsplit{Question} 2}} & Suppose that these two trains start from the same station at the same time, this time in the same direction. How far apart will the fronts of the trains be at the end of the three hours? &  \tn 
% Row Count 13 (+ 13)
% Row 10
\SetRowColor{white}
 & Step 1 & Whats the distance after 1hr? (Draw a diagram) \tn 
% Row Count 17 (+ 4)
% Row 11
\SetRowColor{LightBackground}
 &  & 64km/hr - 56km/hr = 8km/hr \tn 
% Row Count 19 (+ 2)
% Row 12
\SetRowColor{white}
 &  & 64km - 56km = 8km \tn 
% Row Count 21 (+ 2)
% Row 13
\SetRowColor{LightBackground}
 & Step 2 & Whats the distance after 3hrs? \tn 
% Row Count 23 (+ 2)
% Row 14
\SetRowColor{white}
 &  & 8km x 3 = 24km \tn 
% Row Count 24 (+ 1)
% Row 15
\SetRowColor{LightBackground}
 & if {\bf{opposite}} direction, & {\bf{subtract}} \tn 
% Row Count 26 (+ 2)
% Row 16
\SetRowColor{white}
\mymulticolumn{3}{x{8.4cm}}{{\bf{ 2. Book Problems }}} \tn 
% Row Count 27 (+ 1)
% Row 17
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{look at the structure of counting numbers used for book pages} \tn 
% Row Count 29 (+ 2)
% Row 18
\SetRowColor{white}
{\bf{Example Question}} & A printer uses an old-style printing press and needs one piece of type for each digit in the page numbers of a book. How many 2s will the printer need to print page numbers from 1 to 250 &  \tn 
% Row Count 42 (+ 13)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{1.672 cm} x{2.964 cm} x{2.964 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.8 Motions, Books, Clocks and Work Problems (cont)}}  \tn
% Row 19
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{consider the numbers place by place} \tn 
% Row Count 1 (+ 1)
% Row 20
\SetRowColor{white}
 & number of times 2s appear in the 1s place & 25 \tn 
% Row Count 4 (+ 3)
% Row 21
\SetRowColor{LightBackground}
 & number of times 2s appear in the 10s place & 30 \tn 
% Row Count 7 (+ 3)
% Row 22
\SetRowColor{white}
 & number of times 2s appear in the 100s place & 51 \tn 
% Row Count 10 (+ 3)
% Row 23
\SetRowColor{LightBackground}
 & answer & =25+30+51\{\{nl\}\}=106 \tn 
% Row Count 12 (+ 2)
% Row 24
\SetRowColor{white}
\mymulticolumn{3}{x{8.4cm}}{{\bf{3. Clock Problems}}} \tn 
% Row Count 13 (+ 1)
% Row 25
\SetRowColor{LightBackground}
elapse time & amount of time that has passed &  \tn 
% Row Count 15 (+ 2)
% Row 26
\SetRowColor{white}
solve using & facts about time &  \tn 
% Row Count 17 (+ 2)
% Row 27
\SetRowColor{LightBackground}
{\bf{Example Question}} & A certain clock gains one minute of time every hour. If the clock shows the correct time now, in how many hours will it next show the correct time again without regard to am or pm? &  \tn 
% Row Count 29 (+ 12)
% Row 28
\SetRowColor{white}
Fact 1 & A clock that has stopped & Will show the correct time every 12 hrs. As it stopped at 6.03am on Monday. It was correct at the time it stopped. It will be correct again, when the time is at 6.03pm \tn 
% Row Count 41 (+ 12)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{1.672 cm} x{2.964 cm} x{2.964 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.8 Motions, Books, Clocks and Work Problems (cont)}}  \tn
% Row 29
\SetRowColor{LightBackground}
Fact 2 & The clock in the problem must gain 12 hours to show correct time again &  \tn 
% Row Count 5 (+ 5)
% Row 30
\SetRowColor{white}
thus & 12 hrs & =60mins x 12\{\{nl\}\}= 720mins\{\{nl\}\} \tn 
% Row Count 8 (+ 3)
% Row 31
\SetRowColor{LightBackground}
thus & as clock gains 1 min in 1hr & the clock will gain 720min in 720hrs \tn 
% Row Count 11 (+ 3)
% Row 32
\SetRowColor{white}
 &  & 720/24=30days \tn 
% Row Count 12 (+ 1)
% Row 33
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{4. Work Problems}}} \tn 
% Row Count 13 (+ 1)
% Row 34
\SetRowColor{white}
solving using & fractional parts of whole numbers and draw diagrams &  \tn 
% Row Count 17 (+ 4)
% Row 35
\SetRowColor{LightBackground}
{\bf{Example Question}} & Paul can do a certain job in 3hrs and John can do the same job in 2hrs. At these rates, how long would it take Paul and John to do this job if they work together &  \tn 
% Row Count 28 (+ 11)
% Row 36
\SetRowColor{white}
\seqsplit{Strategy} & Step 1 & Draw a diagram for Paul and John. Fractional parts done in each hour \tn 
% Row Count 33 (+ 5)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{1.672 cm} x{2.964 cm} x{2.964 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.8 Motions, Books, Clocks and Work Problems (cont)}}  \tn
% Row 37
\SetRowColor{LightBackground}
 & Step 2 & Using the diagram, in one hour they can complete 1/3 + 1/2 = 5/6 of the job \tn 
% Row Count 5 (+ 5)
% Row 38
\SetRowColor{white}
 & Step 3 & Work out how long to complete job \tn 
% Row Count 8 (+ 3)
% Row 39
\SetRowColor{LightBackground}
 &  & 1/5 of job left \tn 
% Row Count 9 (+ 1)
% Row 40
\SetRowColor{white}
 &  & 60min / 5 = 12mins to complete 1/5 of job \tn 
% Row Count 12 (+ 3)
% Row 41
\SetRowColor{LightBackground}
 & answer & =1hr 12mins \tn 
% Row Count 13 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.72 cm} x{5.28 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{1.9 Problem Solving Strategies}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{{\bf{1. Drawing a picture or diagram}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{ Example Question}} & The lengths of three rods are 5cm, 7cm, and 15cm. How can you use these rods to measure a length of 13cm? \tn 
% Row Count 6 (+ 5)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{{\bf{2. Making an organised list}}} \tn 
% Row Count 7 (+ 1)
% Row 3
\SetRowColor{white}
{\bf{ Example Question}} & Five students hold a chess tournament. Each of the students plays each of the other students just once. How many different games are played? \tn 
% Row Count 13 (+ 6)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{{\bf{3. Making a table}}} \tn 
% Row Count 14 (+ 1)
% Row 5
\SetRowColor{white}
{\bf{ Example Question}} & Two dice both have faces numbered from 1 through to 6. Suppose that you role the two dice. What is the probability of rolling a sum of 8 in the uppermost faces? \tn 
% Row Count 21 (+ 7)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{{\bf{4. Solving a simpler related problem}}} \tn 
% Row Count 22 (+ 1)
% Row 7
\SetRowColor{white}
{\bf{ Example Question}} & The houses on Thomas Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7? \tn 
% Row Count 27 (+ 5)
% Row 8
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{{\bf{5. Finding a pattern}}} \tn 
% Row Count 28 (+ 1)
% Row 9
\SetRowColor{white}
{\bf{ Example Question}} & What is the sum of the following series of numbers? \tn 
% Row Count 30 (+ 2)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{2.72 cm} x{5.28 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{1.9 Problem Solving Strategies (cont)}}  \tn
% Row 10
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{{\bf{6. Guessing and Checking}}} \tn 
% Row Count 1 (+ 1)
% Row 11
\SetRowColor{white}
{\bf{ Example Question}} & Arrange the counting numbers from 1 to 6 in the circles so that the sum of the numbers along each side of the triangle is 10. \tn 
% Row Count 6 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.496 cm} x{3.344 cm} p{0.76 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.10 Problem Solving Strategies}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{1. Acting out the problem}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{Example Question}} & Suppose that you buy a rare stamp for \$16, sell it for \$22, buy it back for \$30, and finally sell it for \$35. How much money did you make or lose? &  \tn 
% Row Count 10 (+ 9)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{2. Working backwards}}} \tn 
% Row Count 11 (+ 1)
% Row 3
\SetRowColor{white}
{\bf{Example Question}} & At the end of a school day, a teacher had 15 crayons left. The teacher remembered giving out 13 of all her crayons in the morning, getting 8 back at recess, and giving out 9 crayons after lunch. How many crayons did the teacher have at the start of the day? &  \tn 
% Row Count 27 (+ 16)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{3. Writing an Equation}}} \tn 
% Row Count 28 (+ 1)
% Row 5
\SetRowColor{white}
{\bf{Example Question}} & The triple of what number is sixteen greater than the number? &  \tn 
% Row Count 32 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{3.496 cm} x{3.344 cm} p{0.76 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{1.10 Problem Solving Strategies (cont)}}  \tn
% Row 6
\SetRowColor{LightBackground}
{\bf{4. Changing your point of view}} & Change your approach &  \tn 
% Row Count 2 (+ 2)
% Row 7
\SetRowColor{white}
 & Are you assuming something thats not in the question &  \tn 
% Row Count 6 (+ 4)
% Row 8
\SetRowColor{LightBackground}
{\bf{Example Question}} & Draw four continuous line segments through the nine dots &  \tn 
% Row Count 10 (+ 4)
% Row 9
\SetRowColor{white}
\mymulticolumn{3}{x{8.4cm}}{{\bf{5. Using Reasoning}}} \tn 
% Row Count 11 (+ 1)
% Row 10
\SetRowColor{LightBackground}
{\bf{Example Question}} & A school has 731 students. Prove that there must be at least 3 students who have the same birthday. &  \tn 
% Row Count 17 (+ 6)
% Row 11
\SetRowColor{white}
\mymulticolumn{3}{x{8.4cm}}{{\bf{6. Miscellaneous}}} \tn 
% Row Count 18 (+ 1)
% Row 12
\SetRowColor{LightBackground}
{\bf{Example Question}} & Three apples and two pears cost 78 cents. But two apples and three pears cost 82 cents. What is the total cost of one apple and one pear? &  \tn 
% Row Count 27 (+ 9)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.204 cm} x{2.812 cm} x{2.584 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{2.1 Logical Approach to Problem Solving}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{8.4cm}}{{\bf{4 Steps to Problem Solving}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
{\bf{Step 1}} & Understand the problem &  \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
{\bf{Step 2}} & Develop a plan & {\bf{choose a problem solving strategy}} \tn 
% Row Count 6 (+ 3)
% Row 3
\SetRowColor{white}
{\bf{Step 3}} & Carry out the plan &  \tn 
% Row Count 8 (+ 2)
% Row 4
\SetRowColor{LightBackground}
{\bf{Step 4}} & Reflect &  \tn 
% Row Count 9 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{3}{x{8.4cm}}{{\bf{Mathematical Terms used in the Olympiad}}} \tn 
% Row Count 10 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Standard Form & 1358 &  \tn 
% Row Count 12 (+ 2)
% Row 7
\SetRowColor{white}
Expanded Form & \seqsplit{1x1000+3x100+5x10+8x1} &  \tn 
% Row Count 14 (+ 2)
% Row 8
\SetRowColor{LightBackground}
\seqsplit{Exponential} Form & 1x10\textasciicircum{}3\textasciicircum{}+3x10\textasciicircum{}2+5x10+8x1 &  \tn 
% Row Count 16 (+ 2)
% Row 9
\SetRowColor{white}
Whole numbers & 0,1,2,3,... &  \tn 
% Row Count 18 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Counting numbers & 1,2,3,... &  \tn 
% Row Count 20 (+ 2)
% Row 11
\SetRowColor{white}
\seqsplit{Divisibility} & A is divisible by B, if B divides into A with zero remainder & If so, B is a factor of A \tn 
% Row Count 25 (+ 5)
% Row 12
\SetRowColor{LightBackground}
Prime number & counting number greater than 1, which is divisible only by itself and &  \tn 
% Row Count 30 (+ 5)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{2.204 cm} x{2.812 cm} x{2.584 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{2.1 Logical Approach to Problem Solving (cont)}}  \tn
% Row 13
\SetRowColor{LightBackground}
Composite number & counting number greater than 1 which is divisible by a counting number other than 1 and itself &  \tn 
% Row Count 7 (+ 7)
% Row 14
\SetRowColor{white}
A number is factored completely & when it is a product of prime numbers &  \tn 
% Row Count 10 (+ 3)
% Row 15
\SetRowColor{LightBackground}
Order of Operation & BODMAS &  \tn 
% Row Count 12 (+ 2)
% Row 16
\SetRowColor{white}
common or simple fraction & a/b where a and b are whole numbers and b is no zero &  \tn 
% Row Count 16 (+ 4)
% Row 17
\SetRowColor{LightBackground}
unit fraction & common fraction with a numerator of 1 &  \tn 
% Row Count 19 (+ 3)
% Row 18
\SetRowColor{white}
proper fraction & a/b where a \textless{} b &  \tn 
% Row Count 21 (+ 2)
% Row 19
\SetRowColor{LightBackground}
improper fraction & a/b where a \textgreater{} b &  \tn 
% Row Count 23 (+ 2)
% Row 20
\SetRowColor{white}
complex fraction & numerator or denominator contains a fraction &  \tn 
% Row Count 27 (+ 4)
% Row 21
\SetRowColor{LightBackground}
20th century & 100 year period 1901-2000 inclusive &  \tn 
% Row Count 30 (+ 3)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{2.204 cm} x{2.812 cm} x{2.584 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{8.4cm}}{\bf\textcolor{white}{2.1 Logical Approach to Problem Solving (cont)}}  \tn
% Row 22
\SetRowColor{LightBackground}
average of a set of N numbers & sum of the N numbers divided by N &  \tn 
% Row Count 3 (+ 3)
% Row 23
\SetRowColor{white}
acute angle & less than 90 degrees &  \tn 
% Row Count 5 (+ 2)
% Row 24
\SetRowColor{LightBackground}
right angle & 90 degrees &  \tn 
% Row Count 6 (+ 1)
% Row 25
\SetRowColor{white}
obtuse angle & greater than 90 degrees &  \tn 
% Row Count 8 (+ 2)
% Row 26
\SetRowColor{LightBackground}
straight angle & 180 degrees &  \tn 
% Row Count 10 (+ 2)
% Row 27
\SetRowColor{white}
reflex angle & more than 180 degrees and less than 360 degrees &  \tn 
% Row Count 14 (+ 4)
% Row 28
\SetRowColor{LightBackground}
scalene triangle & no equal angles &  \tn 
% Row Count 16 (+ 2)
% Row 29
\SetRowColor{white}
isosceles triangle & 2 equal angles &  \tn 
% Row Count 18 (+ 2)
% Row 30
\SetRowColor{LightBackground}
\seqsplit{equilateral} & 3 equal angles &  \tn 
% Row Count 19 (+ 1)
% Row 31
\SetRowColor{white}
\seqsplit{right-angled} & 90 angle &  \tn 
% Row Count 21 (+ 2)
% Row 32
\SetRowColor{LightBackground}
congruent shapes & shapes on the same plane whose sides and angles are the same &  \tn 
% Row Count 26 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.2 cm} x{4.8 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{2.2 Types of Problems}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{1. Translation Problems}} & translate word sentences to mathematical sentences \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
Example Question & Farmer Joe bought 2 bags of feed for \$4 each and 1 bag of feed for \$3. How much did the feed bags cost altogether? \tn 
% Row Count 8 (+ 5)
% Row 2
\SetRowColor{LightBackground}
{\bf{2. Application Problems}} & 'real-world' problems, usually involve calculations with money, to find {\bf{discounts}}, {\bf{profits}} or {\bf{cost}} of items \tn 
% Row Count 13 (+ 5)
% Row 3
\SetRowColor{white}
Example Question & Shop A is offering a 10\% discount on 34cm colour TV sets priced normally at \$379. Meanwhile Shop B is offering 15\% discount on the same sets priced normally at \$409. Which shop should you purchase the TV from? \tn 
% Row Count 22 (+ 9)
% Row 4
\SetRowColor{LightBackground}
{\bf{3. Process Problems}} & Usually require using general problem solving steps and specific strategies. May use short-cuts when aware of patterns \tn 
% Row Count 27 (+ 5)
% Row 5
\SetRowColor{white}
Example Question & The first 4 triangular numbers are 1, 3, 6, 10. What will the 10th triangular number be? \tn 
% Row Count 31 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{8.4cm}{x{3.2 cm} x{4.8 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{2.2 Types of Problems (cont)}}  \tn
% Row 6
\SetRowColor{LightBackground}
{\bf{4. Puzzle Problems}} & like riddles \tn 
% Row Count 2 (+ 2)
% Row 7
\SetRowColor{white}
Example Question & Three Australian students who were born in different countries have last names Brown, Black and Bright. Their first names are Jim, John and Jane but not necessarily in that order. Using the information below can you determine the full name of each student?\{\{nl\}\}Brown was born in Australia\{\{nl\}\}Bright has never been to Malaysia\{\{nl\}\}Jane was born in England\{\{nl\}\}Jim was born in Malaysia \tn 
% Row Count 19 (+ 17)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}