2023 AMC 10A Problems/Problem 1

What is 2*3*239?

% Focused ion beam system % Author: Clemens Helfmeier \documentclass{minimal} \usepackage[utf8]{inputenc} \usepackage{tikz} \usetikzlibrary{arrows,decorations.pathmorphing} \def\english#1{\emph{#1}} \tikzset{

 arrow/.style={-stealth', line width=0.5pt},
 every picture/.append style={line width=1pt},

} \begin{document} \begin{tikzpicture}[

   scale=1.3,
   fib housing horz/.style={top color=black!10, bottom color=black!40, draw=black!50},
   fib housing vert/.style={left color=black!10, right color=black!40, draw=black!50},
   fib housing/.style={fill=black!20, draw=black!50},
   fib background vert/.style={left color=black!40, right color=black!10, draw=black!50},
   fib background/.style={fill=black!30, draw=black!50},
   fib lens thick/.style={draw=black!80, line width=0.04cm},
   fib lens thinn/.style={draw=black!95, line width=0.02cm},
   fib mirror/.style={draw=black!60, line width=0.03cm, fill=black!20},
   fib objective/.style={draw=black!60, line width=0.03cm},
   fib quadrupole/.style={draw=black!60, line width=0.03cm},
   fib blanker/.style={draw=black!90, fill=black!60},
   fib octopole/.style={draw=black!80, line width=0.04cm},
 ]
 % Picture created and copyright by Clemens Helfmeier in 2009 for master thesis.
 % Contact me at: Clemens@Helfmeier.de
 % Aufbau des FIBs
 % "falsch" herum, Probe unten, Ionen oben
 % einzelne Elemente sind rund, insofern auch shading anbringen
 % jedes element ist 1.4cm hoch.
 % Die FIB-Säule
 \begin{scope}[scale=2]
   \begin{scope}[yshift=0.2]
     %% Ionenquelle
     { % Rückseite
       \path[fib background vert]
         (-1,0) arc (180:360:1 and 0.05) -- (1,-1) arc (360:180:1 and 0.05) --cycle;
       \path[fib background vert]
         (-0.8,0.2) arc (180:360:0.8 and 0.04) -- (1,0) arc (360:180:1 and 0.05) --cycle;
       \path[fib background vert]
         (-0.8,-1.2) arc (180:360:0.8 and 0.04) -- (1,-1) arc (360:180:1 and 0.05) --cycle;
     }
     { % Innenleben
       \path[fib housing vert]
         (-0.3,0) -- (0.3,0) -- (0.3,-0.45) -- (0,-0.7) coordinate (source) -- (-0.3,-0.45) --cycle;
       \path[draw=black!80, yslant=-0.1]
         \foreach \x in {0.1,0.2,0.3,0.4} {
           (-0.3,-\x) arc(210:-30:0.35 and 0.06)
         };
       \path[fib lens thinn]
         (-0.8,-0.9) -- (-0.05,-0.9)
         (0,-0.9) coordinate (emitter)
         (0.05,-0.9) -- (0.8,-0.9) coordinate[midway] (extractor);
       \path[fib lens thick]
         (-0.4,-0.4) -- (-0.4,-0.8) -- (-0.1,-0.8) (0.1,-0.8) -- (0.4,-0.8) -- (0.4,-0.4) coordinate[midway] (suppressor);
     }
     \begin{scope} % Gehäuse
       \path[clip,decoration={random steps, segment length=6pt, amplitude=2pt}]
         (-1,0.3) -- (1,0.3) -- (1,-1.3)
         -- (0.6,-1.3) decorate{ ..controls (0.6,0.2) and (-0.6,0.2).. (-0.6,-1.3)}
         -- (-1,-1.3) --cycle;
       \path[fib housing vert]
         (-1,0) arc (180:0:1 and 0.05) -- (1,-1) arc (0:180:1 and 0.05) --cycle;
       \path[fib housing vert]
         (-0.8,0.2) arc (180:0:0.8 and 0.04) -- (1,0) arc (0:180:1 and 0.05) --cycle;
       \path[fib housing vert]
         (-0.8,-1.2) arc (180:0:0.8 and 0.04) -- (1,-1) arc (0:180:1 and 0.05) --cycle;
       \path[fib housing]
         (0,-1.2) ellipse (0.8 and 0.04);
     \end{scope}
     { % Ionenpumpe extern
       \path[fib housing]
         (-1,-0.1) rectangle (-1.2,-0.6);
       \path[fib housing]
         (-1.2,-0.15) rectangle (-1.25,-0.2)
         (-1.2,-0.55) rectangle (-1.25,-0.5);
       \path[fib housing horz]
         (-1.2,-0.25) rectangle (-1.6,-0.45);
     }
   \end{scope}
   %% Beschleunigung und fokussierung
   \begin{scope}[yshift=-1.4cm]
     { % Rückseite
       \path[fib background vert]
         (-0.7,0.2) arc (180:360:0.7 and 0.03) -- (0.7,-1.2) arc (360:180:0.7 and 0.03) --cycle;
     }
     { % Innenleben
       \path[fib lens thinn] % Linse
         (-0.3,-0.25) -- (-0.5,-0.25) -- (-0.5,-0.35) -- (-0.3,-0.35)
         (0.3,-0.25) -- (0.5,-0.25) -- (0.5,-0.35) -- (0.3,-0.35);
       \path[fib lens thick]
         (-0.45,-0.3) -- (-0.35,-0.3) (0.35,-0.3) -- (0.45,-0.3)
         (0,-0.3) coordinate (ionlens1)
         (-0.5,-0.25) coordinate (linse1);
       \path[fib quadrupole] % quadrupole 1
         (0,-0.45) ellipse (0.33cm and 0.01cm)
         (0,-0.65) ellipse (0.33cm and 0.01cm)
         (-0.35,-0.45) -- ++(0,-0.2) (-0.2,-0.45) -- ++(0,-0.2)
         (0.35,-0.45) -- ++(0,-0.2) (0.2,-0.45) -- ++(0,-0.2)
         (-0.35,-0.45) coordinate (quadA);
       \path[fib lens thick]
         (-0.25,-0.45) -- (0.25,-0.45)
         (-0.25,-0.65) -- (0.25,-0.65);
       \path[fib lens thick] % Blende
         (-0.7,-0.75) -- (-0.05,-0.75) (0.05,-0.75) -- (0.7,-0.75) coordinate[midway] (blende)
         (0,-0.75) coordinate (ionblende);
       \path[fib quadrupole] % quadrupole 2
         (0,-0.85) ellipse (0.33cm and 0.01cm)
         (0,-1.05) ellipse (0.33cm and 0.01cm)
         (-0.35,-0.85) -- ++(0,-0.2) (-0.2,-0.85) -- ++(0,-0.2)
         (0.35,-0.85) -- ++(0,-0.2) (0.2,-0.85) -- ++(0,-0.2)
         (-0.35,-0.85) coordinate (quadB);
       \path[fib lens thick]
         (-0.25,-0.85) -- (0.25,-0.85)
         (-0.25,-1.05) -- (0.25,-1.05);
     }
     \begin{scope} % Gehäuse
       \path[clip,decoration={random steps, segment length=6pt, amplitude=2pt}]
         (0.7,0.3) -- (0.7,-1.3)
         -- (0.6,-1.3) decorate{ ..controls (0.5,-0.6).. (0.6,0.3)} --cycle
         (-0.7,0.3) -- (-0.6,0.3) decorate{ ..controls (-0.5,-0.6).. (-0.6,-1.3)}
         -- (-0.7,-1.3) --cycle;
       \path[fib housing vert]
         (-0.7,0.2) arc (180:0:0.7 and 0.03) -- (0.7,-1.2) arc (0:180:0.7 and 0.03) --cycle;
     \end{scope}
     { % Ionenpumpe extern
       \path[fib housing]
         (0.7,-0.1) rectangle (0.9,-0.6);
       \path[fib housing]
         (0.9,-0.15) rectangle (0.95,-0.2) (0.95,-0.15) coordinate (ionenpume)
         (0.9,-0.55) rectangle (0.95,-0.5);
       \path[fib housing horz]
         (0.9,-0.25) rectangle (1.3,-0.45);
     }
   \end{scope}
   \begin{scope}[yshift=-2.8cm]
     % Optik und Ablenkteile
     { %Rückseite
       \path[fib background vert]
         (-0.8,0.2) arc (180:360:0.8 and 0.03) -- (0.8,-1.5) arc (360:180:0.8 and 0.03) --cycle;
       \path[fib background]
         (1.3,0.3) rectangle (2.4,-1.1);
     }
     { % Innenleben
       \path[fib blanker] % blanker
         (-0.3,0.1) rectangle (-0.25,-0.2)
         (0.3,0.1) rectangle (0.25,-0.2)
         (-0.3,0.1) coordinate (blanker);
       \path[fib lens thick]
         (-0.3,-0.05) -- (-0.4,-0.05) (0.3,-0.05) -- (0.4,-0.05);
       \path[fib lens thick] % Faradaykäfig
         (-0.3,-0.25) -- (-0.3,-0.35) -- (-0.15,-0.35) -- (-0.15,-0.25)
         (-0.3,-0.25) coordinate (faraday)
         (0.3,-0.25) -- (0.3,-0.35) -- (0.15,-0.35) -- (0.15,-0.25);
       \path[fib octopole] % octopole 1
         (0,-0.4) ellipse (0.38cm and 0.01cm)
         (0,-0.6) ellipse (0.38cm and 0.01cm)
         (-0.4,-0.4) -- (-0.4,-0.6) (0.4,-0.4) -- (0.4,-0.6)
         (-0.27,-0.4) -- ++(0,-0.2) (-0.1,-0.4) -- ++(0,-0.2)
         (0.27,-0.4) -- ++(0,-0.2) (0.1,-0.4) -- ++(0,-0.2)
         (0,-0.5) coordinate (ionocto1)
         (-0.4,-0.4) coordinate (octoA);
       \path[fib octopole] % octopole 2
         (0,-0.7) ellipse (0.38cm and 0.01cm)
         (0,-1.05) ellipse (0.38cm and 0.01cm)
         (-0.4,-0.7) -- (-0.4,-1.05) (0.4,-0.7) -- (0.4,-1.05)
         (-0.27,-0.7) -- ++(0,-0.35) (-0.1,-0.7) -- ++(0,-0.35)
         (0.27,-0.7) -- ++(0,-0.35) (0.1,-0.7) -- ++(0,-0.35)
         (0,-0.88) coordinate (ionocto2)
         (-0.4,-0.7) coordinate (octoB);
       \path[fib mirror]
         (0.4,-0.4) -- (0.6,-0.2) -- (0.4,-0.2) -- cycle;
       \path[fib mirror]
         (0.6,-1.2) -- (0.4,-1.4) -- (0.6,-1.4) -- cycle;
       \path[fib mirror]
         (-0.1,-1.4) -- (-0.025,-1.325) (0.025,-1.275) -- (0.1,-1.2) -- cycle;
       \path[fib mirror] % kamera innenleben
         (1.6,-0.2) -- (1.8,-0.4);
       \path[fib lens thinn]
         (1.6,-0.6) arc(100:80:0.575) arc(-80:-100:0.575) -- cycle;
       \path[draw=black!60, fill=black!20]
         (1.58,-0.82) -- (1.56,-0.97) -- (1.82,-0.97) -- (1.84,-0.82) --cycle;
       \path[draw=black!60, shading=ball, ball color=black!30]
         (1.7,-0.9) ellipse (0.1cm and 0.05cm);
       \begin{scope}
         \path[clip]
           (2.2,-0.3) circle (0.1cm);
         \path[draw=black!80, line width=0.04cm]
           (2.2,-0.3) circle (0.1cm)
           (2.1,-0.2) -- (2.3,-0.4)
           (2.1,-0.4) -- (2.3,-0.2);
       \end{scope}
       \path[fib lens thinn]
         (1.9,-0.2) arc(170:190:0.575) arc(-10:10:0.575) -- cycle;
     }
     \begin{scope} % externes Gehäuse
       \path[clip,decoration={random steps, segment length=6pt, amplitude=2pt}]
         (0.8,0) -- (1.3,0) -- (1.3,0.3) -- (2.4,0.3) -- (2.4,-1.1) -- (1.3,-1.1) -- (1.3,-0.6) -- (0.8,-0.6) -- (0.8,-0.1) -- (1.4,-0.1)
         decorate{ .. controls (1.4,-0.5) .. (1.6,-1) .. controls (2.3,-1) .. (2.3,-0.1) .. controls (1.8,0.2) .. (1.4,0)}
         -- (1.4,-0.1) -- (0.8,-0.1) --cycle;
       \path[fib housing horz]
         (0.8,0) rectangle (1.3,-0.6);
       \path[fib housing]
         (1.3,0.3) rectangle (2.4,-1.1);
     \end{scope}
     \begin{scope} % Gehäuse
       \path[clip,decoration={random steps, segment length=6pt, amplitude=2pt}]
         (0.8,0.3) -- (0.8,-1.6)
         -- (0.6,-1.6) decorate{ ..controls (0.8,-0.6).. (0.6,0.3)} --cycle
         (-0.8,0.3) -- (-0.6,0.3) decorate{ ..controls (-0.4,-0.6).. (-0.6,-1.6)}
         -- (-0.8,-1.6) --cycle;
       \path[fib housing vert]
         (-0.8,0.2) arc (180:0:0.8 and 0.03) -- (0.8,-1.5) arc (0:180:0.8 and 0.03) --cycle;
     \end{scope}
   \end{scope}
   \begin{scope}[yshift=-4.5cm]
     % Deckel
     { % Rückwand
       \path[fib background vert]
         (-0.8,0.2) arc (180:360:0.8 and 0.03) -- (0.8,-0.6) arc (360:180:0.8 and 0.03) -- cycle;
       \path[fib background vert]
         (0.8,-0.6) arc (360:180:0.8 and 0.03) -- (-0.3,-1) arc (180:360:0.3 and 0.01) -- cycle;
     }
     { % Innenleben
       \path[fib mirror]
         (-0.15,-0.4) arc (-75:-50:0.25) -- (-0.05,-0.4) -- cycle;
       \path[fib mirror]
         (0.15,-0.4) arc (-105:-130:0.25) -- (0.05,-0.4) -- cycle;
       \path[fib mirror]
         (-0.2,-0.1) arc (105:125:0.8) coordinate (tmp) -- (tmp |- -0.2,-0.1) --cycle;
       \path[fib mirror]
         (0.2,-0.1) arc (75:55:0.8) coordinate (tmp) -- (tmp |- 0.2,-0.1) --cycle;
       \path[fib lens thinn] % 2. Linse
         (-0.15,-0.65) -- (-0.3,-0.65) -- (-0.3,-0.75) -- (-0.15,-0.75)
         (0.15,-0.65) -- (0.3,-0.65) -- (0.3,-0.75) -- (0.15,-0.75);
       \path[fib lens thick]
         (-0.27,-0.7) -- (-0.17,-0.7) (0.17,-0.7) -- (0.27,-0.7)
         (0,-0.7) coordinate (ionlens2)
         (-0.27,-0.65) coordinate (linse2);
     }
     \begin{scope} % Gehäuse
       \path[clip,decoration={random steps, segment length=6pt, amplitude=2pt}]
         (-0.8,0.3) -- (-0.6,0.3) decorate{ ..controls (-0.5,-1.3) and (0.5,-1.3).. (0.6,0.3)}
         -- (0.8,0.3) -- (0.8,-1) -- (-0.8,-1) -- cycle;
       \path[fib housing vert]
         (-0.8,0.2) arc (180:0:0.8 and 0.03) -- (0.8,-0.6) arc (0:180:0.8 and 0.03) -- cycle;
       \path[fib housing vert]
         (0.8,-0.6) arc (0:180:0.8 and 0.03) -- (-0.3,-1) arc (180:0:0.3 and 0.01) -- cycle;
     \end{scope}
   \end{scope}
   \begin{scope}[yshift=-6cm, scale=0.5]
     % Probe
     \path[draw, rounded corners=1pt]
       (-1,-0.2) rectangle (1,-0.3);
     \path[draw, fill, top color=black!5, bottom color=black!20, rounded corners=2pt]
       (-0.5,0) -- (-0.5,-0.18) -- (0.5,-0.18) -- (0.5,0) -- (0.2,0) -- (0,-0.07) -- (-0.2,0) -- cycle
       (0,-0.07) coordinate (target);
     \path[draw, fill=black!60]
       (-0.5,-0.07) -- (-0.55,-0.07) to[out=180, in=0] (-0.6,-0.13) {[rounded corners=1pt] -- (-0.68,-0.13) --
         (-0.68,-0.2)} -- (-0.6,-0.2) to[out=0, in=180] (-0.55,-0.14) -- (-0.5,-0.14) -- cycle;
     \path[draw, fill=black!60]
       (0.5,-0.07) -- (0.55,-0.07) to[out=0, in=180] (0.6,-0.13) {[rounded corners=1pt] -- (0.68,-0.13) --
         (0.68,-0.2)} -- (0.6,-0.2) to[out=180, in=0] (0.55,-0.14) -- (0.5,-0.14) -- cycle;
   \end{scope}
   \begin{scope}[yshift=-5.7cm, rotate around={-30:(-0.75,0.2)}]
     % Szintillator
     \path[draw=black!80, line width=0.03cm] % 1. Antenne
       (-0.7,0.2) -- (-0.4,0.15) -- (-0.1,0.2);
     \path[fib housing horz]
       (-0.9,0.3) -- (-0.6,0.25) arc(90:270:0.1 and 0.15) -- (-0.9,0) arc(270:90:0.1 and 0.15) --cycle;
     \path[fib housing]
       (-0.6,0.1) ellipse (0.1 and 0.15);
     \begin{scope}
       \path[fib housing]
         (-0.6,0.1) ellipse (0.07 and 0.11);
       \path[clip]
         (-0.6,0.1) ellipse (0.07 and 0.11);
       \path[shading=ball, ball color=black!30]
         (-0.5,0) circle (0.4);
     \end{scope}
     \path[draw=black!80, line width=0.03cm] % 2. Antenne
       (-0.9,0.05) -- (-0.6,0) -- (-0.4,-0.2);
   \end{scope}
   \begin{scope}[yshift=-6cm]
     % Gasnadel
     \path[fib housing horz]
       (2,0.7) -- (0.35,0.3) -- (0.1,0.1) -- (0.1,0.05) -- (0.4,0.25) -- (2,0.6) -- cycle;
   \end{scope}
   \begin{scope} % Strahlen
     % Ionen
     \path[draw=black!80, dashed]
       (source) -- (emitter)
       (emitter) -- (ionlens1 -| -0.23,0) -- (ionblende -| 0.32,0)
       (ionblende -| 0.02,0) -- (ionocto1 -| 0.028,0) -- (ionocto2 -| 0.03,0) -- (ionlens2 -| 0.07,0) --
       (target) --
       (ionlens2 -| -0.07,0) -- (ionocto2 -| -0.03,0) -- (ionocto1 -| -0.028,0) -- (ionblende -| -0.02,0)
       (ionblende -| -0.32,0) -- (ionlens1 -| 0.23,0) -- (emitter);
     \path[fill=white, opacity=0.3]
       (emitter) -- (ionlens1 -| -0.23,0) -- (ionblende -| 0.32,0) --
       (ionblende -| 0.02,0) -- (ionocto1 -| 0.028,0) -- (ionocto2 -| 0.03,0) -- (ionlens2 -| 0.07,0) --
       (target) --
       (ionlens2 -| -0.07,0) -- (ionocto2 -| -0.03,0) -- (ionocto1 -| -0.028,0) --
       (ionblende -| -0.02,0) -- (ionblende -| -0.32,0) -- (ionlens1 -| 0.23,0) -- cycle;
     % Licht
     \path[draw=red]
       (1.6,-3) -- (0.6,-3) -- (0.6,-4) -- (0.1,-4) -- (0.1,-4.85) -- (0.34,-4.67) -- (target)
       (1.8,-3.2) -- (0.4,-3.2) -- (0.4,-4.2) -- (-0.1,-4.2) -- (-0.1,-4.85) -- (-0.34,-4.67) -- (target);
     \path[draw=red,->]
       (2.1,-3.1) -- (1.9,-3) -- (1.85,-3);
     \path[draw=red,->]
       (2.1,-3.1) -- (1.9,-3.2) -- (1.85,-3.2);
     \path[draw=red]
       (1.85,-3) -- (1.6,-3)
       (1.85,-3.2) -- (1.8,-3.2);
     \path[draw=red,->]
       (1.6,-3) -- (1.6,-3.3);
     \path[draw=red,->]
       (1.8,-3.2) -- (1.8,-3.3);
     \path[draw=red]
       (1.7,-3.7) -- (1.6,-3.4) -- (1.6,-3.3)
       (1.7,-3.7) -- (1.8,-3.4) -- (1.8,-3.3);
   \end{scope}
   % Beschriftung
   \begin{scope}[arrow]
     \path[draw] (0.9,0.3) node[anchor=west] {Galliumquelle mit Heizwendel} to[out=180, in=45] (0.2,-0.3);
     \path[draw] (1.5,-0.6) node[anchor=west] {\english{extractor}} to[out=180, in=45] (extractor);
     \path[draw,shorten >=0.1cm] (1.5,-0.3) node[anchor=west] {\english{suppressor}} to[out=180, in=45] (suppressor);
     \path[draw] (-0.7,0.5) node[anchor=east] (tmp) {Ionenpumpe Quelle} (tmp) to[out=-90, in=180] (-1.3, -0.2);
     \path[draw,shorten >=0.1cm] (-1.1,-1.2) node[anchor=east] {Austritt} to[out=0, in=225] (emitter);
     \path[draw] (1.5,-1.3) node[anchor=west] {Ionenpumpe} to[out=180, in=45] (1,-1.5);
     \path[draw,shorten >=0.1cm] (-0.9,-1.8) node[anchor=east] (quad) {Quadrupole 1, 2} to[out=0, in=160] (quadA);
     \path[draw,shorten >=0.1cm]  (quad) to[out=0, in=160] (quadB);
     \path[draw,shorten >=0.05cm] (-1.1,-1.4) node[anchor=east] {1. Linse} to[out=0, in=135] (linse1);
     \path[draw,shorten >=0.05cm] (1.5,-1.8) node[anchor=west] {Blende} to[out=180, in=60] (blende);
     \path[draw, shorten >=0.1cm] (-1.1,-2.3) node[anchor=east] {\english{blanker}} to[out=0, in=140] (blanker);
     \path[draw, shorten >=0.1cm] (-1.1,-2.7) node[anchor=east] {Faradaykäfig} to[out=0, in=140] (faraday);
     \path[draw, shorten >=0.1cm] (-1.1,-3.1) node[anchor=east] (okto) {Oktopole 1, 2} to[out=0, in=140] (octoA);
     \path[draw, shorten >=0.1cm] (okto) to[out=0, in=140] (octoB);
     \path[draw] (1.5,-2.1) node[anchor=west] {optische Einkopplung} to[out=180, in=70] (0.5, -2.9);
     \path[draw,shorten >=0.2cm] (2,-2.35) node[anchor=east] {Lichtquelle} to[out=0, in=50] (2.2,-3.1);
     \path[draw, shorten >=0.2cm] (2,-4.1) node[anchor=west] {CCD Kamera} to[out=180, in=-70] (1.7,-3.7);
     \path[draw] (1.5,-4.5) node[anchor=west] {Objektiv} to[out=180, in=20] (0.5,-4.7);
     \path[draw] (-1.1,-4.7) node[anchor=east] {2. Linse} to[out=0, in=150] (linse2);
     \path[draw] (1.5,-5.1) node[anchor=west] {Gasinjektionsnadel} to[out=180, in=70] (0.7,-5.6);
     \path[draw] (1.5,-5.8) node[anchor=west] {Probe} to[out=180, in=20] (0.3,-6);
     \path[draw] (-1.1,-5.1) node[anchor=east] {Szintillator} to[out=0, in=110] (-0.8,-5.4);
     \path[draw] (-1.1,-6) node[anchor=east] {Antennen} to[out=0, in=-120] (-0.7,-5.9);
   \end{scope}
 \end{scope}

\end{tikzpicture} \end{document}