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pretraining/mathematica/geometry/solids/10221.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0.819 & 0.478 \\
+ 0.732 & 0.084 & 0.601 \\
+ 0.903 & 0.188 & 0.201 \\
+ 0.006 & 0.413 & 0.913 \\
+ 0.414 & 0.078 & 0.368 \\
+ 0.691 & 0.881 & 0.124 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.65$ 
+Volume: $0.1$ 
+Surface Area: $1.72$

pretraining/mathematica/geometry/solids/12353.txt

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+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.517 & 0.549 & 0.95 \\
+ 0.202 & 0.296 & 0.582 \\
+ 0.893 & 0.414 & 0.164 \\
+ 0.916 & 0.174 & 0.774 \\
+ 0.198 & 0.085 & 0.945 \\
+ 0.113 & 0.634 & 0.241 \\
+ 0.848 & 0.244 & 0.006 \\
+ 0.49 & 0.986 & 0.435 \\
+ 0.623 & 0.705 & 0.685 \\
+ 0.924 & 0.573 & 0.435 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $2.38$ 
+Surface Area: $2.11$ 
+Volume: $0.21$

pretraining/mathematica/geometry/solids/12990.txt

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+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(25-11 \sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(25-11 \sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{11}{40 \sqrt{5}}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{11}{40 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{5-2 \sqrt{5}} & -\frac{1}{2}-\frac{2}{\sqrt{5}} & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{5-2 \sqrt{5}} & \frac{1}{2}+\frac{2}{\sqrt{5}} & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{5-2 \sqrt{5}} & -\frac{1}{2}-\frac{2}{\sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ \frac{1}{10} \sqrt{5-2 \sqrt{5}} & \frac{1}{2}+\frac{2}{\sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ -\frac{1}{5} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} & \frac{1}{10} \left(-5-3 \sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ -\frac{1}{5} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} & \frac{1}{10} \left(5+3 \sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ \sqrt{\frac{1}{10}-\frac{1}{10 \sqrt{5}}} & \frac{1}{10} \left(-5-3 \sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{10}-\frac{1}{10 \sqrt{5}}} & \frac{1}{10} \left(5+3 \sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} & \frac{1}{20} \left(-5-\sqrt{5}\right) & -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} & -\frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} & \frac{1}{20} \left(5+\sqrt{5}\right) & -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} & \frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} & -\frac{3}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} & \frac{3}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{20} \left(-5-3 \sqrt{5}\right) & \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{20} \left(5+3 \sqrt{5}\right) & \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{20} \left(-5-3 \sqrt{5}\right) & -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{20} \left(5+3 \sqrt{5}\right) & -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2 \sqrt{5}} & \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{5+2 \sqrt{5}} & \frac{1}{2 \sqrt{5}} & \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2 \sqrt{5}} & -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{5+2 \sqrt{5}} & \frac{1}{2 \sqrt{5}} & -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ -\frac{1}{5} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & 0 & -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ \frac{1}{5} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & 0 & \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{25+2 \sqrt{5}} & -\frac{1}{2}-\frac{1}{\sqrt{5}} & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{25+2 \sqrt{5}} & \frac{1}{2}+\frac{1}{\sqrt{5}} & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{25+2 \sqrt{5}} & -\frac{1}{2}-\frac{1}{\sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ \frac{1}{10} \sqrt{25+2 \sqrt{5}} & \frac{1}{2}+\frac{1}{\sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(-3-\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{1}{8 \sqrt{5}}} & \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & -\frac{3}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & \frac{3}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & -\frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} & \frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ -\frac{1}{5} \sqrt{5+2 \sqrt{5}} & 0 & \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ \frac{1}{5} \sqrt{5+2 \sqrt{5}} & 0 & -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} \\
+ -\frac{1}{5} \sqrt{2 \left(5+\sqrt{5}\right)} & \frac{1}{10} \left(-5-3 \sqrt{5}\right) & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ -\frac{1}{5} \sqrt{2 \left(5+\sqrt{5}\right)} & \frac{1}{10} \left(5+3 \sqrt{5}\right) & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ \frac{1}{5} \sqrt{2 \left(5+\sqrt{5}\right)} & \frac{1}{10} \left(-5-3 \sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ \frac{1}{5} \sqrt{2 \left(5+\sqrt{5}\right)} & \frac{1}{10} \left(5+3 \sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} & -\frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} & \frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} & -\frac{3}{20} \left(5+\sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} & \frac{3}{20} \left(5+\sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \frac{1}{2} & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ \sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & 0 & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{65}{2}+\frac{29 \sqrt{5}}{2}} & \frac{1}{20} \left(-5-7 \sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{65}{2}+\frac{29 \sqrt{5}}{2}} & \frac{1}{20} \left(5+7 \sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{13}{40}+\frac{29}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-7 \sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{13}{40}+\frac{29}{40 \sqrt{5}}} & \frac{1}{20} \left(5+7 \sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} & -\frac{3}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} & \frac{3}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} & -\frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} & \frac{3}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{5}{8}+\frac{41}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-3 \sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{5}{8}+\frac{41}{40 \sqrt{5}}} & \frac{1}{20} \left(5+3 \sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ \sqrt{\frac{5}{8}+\frac{41}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-3 \sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{41}{40 \sqrt{5}}} & \frac{1}{20} \left(5+3 \sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ -\frac{1}{5} \sqrt{\frac{25}{2}+\frac{11 \sqrt{5}}{2}} & -\frac{1}{\sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ -\frac{1}{5} \sqrt{\frac{25}{2}+\frac{11 \sqrt{5}}{2}} & \frac{1}{\sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ \sqrt{\frac{1}{2}+\frac{11}{10 \sqrt{5}}} & -\frac{1}{\sqrt{5}} & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ \sqrt{\frac{1}{2}+\frac{11}{10 \sqrt{5}}} & \frac{1}{\sqrt{5}} & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{65+22 \sqrt{5}} & -\frac{1}{2 \sqrt{5}} & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ -\frac{1}{10} \sqrt{65+22 \sqrt{5}} & \frac{1}{2 \sqrt{5}} & \sqrt{\frac{17}{40}+\frac{31}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{65+22 \sqrt{5}} & -\frac{1}{2 \sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ \frac{1}{10} \sqrt{65+22 \sqrt{5}} & \frac{1}{2 \sqrt{5}} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(85+31 \sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \sqrt{\frac{37}{40}+\frac{59}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{29}{40}+\frac{61}{40 \sqrt{5}}} & \frac{1}{20} \left(5-\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ -\sqrt{\frac{29}{40}+\frac{61}{40 \sqrt{5}}} & \frac{1}{20} \left(\sqrt{5}-5\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(65+19 \sqrt{5}\right)} \\
+ \sqrt{\frac{29}{40}+\frac{61}{40 \sqrt{5}}} & \frac{1}{20} \left(5-\sqrt{5}\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ \sqrt{\frac{29}{40}+\frac{61}{40 \sqrt{5}}} & \frac{1}{20} \left(\sqrt{5}-5\right) & \sqrt{\frac{13}{40}+\frac{19}{40 \sqrt{5}}} \\
+ -\frac{2}{5} \sqrt{5+2 \sqrt{5}} & -\frac{1}{\sqrt{5}} & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\frac{2}{5} \sqrt{5+2 \sqrt{5}} & \frac{1}{\sqrt{5}} & -\sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \frac{2}{5} \sqrt{5+2 \sqrt{5}} & -\frac{1}{\sqrt{5}} & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ \frac{2}{5} \sqrt{5+2 \sqrt{5}} & \frac{1}{\sqrt{5}} & \sqrt{\frac{1}{8}+\frac{11}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ -\sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} & \frac{1}{20} \left(-5-\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ \sqrt{\frac{41}{40}+\frac{71}{40 \sqrt{5}}} & \frac{1}{20} \left(5+\sqrt{5}\right) & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{85+38 \sqrt{5}} & -\frac{1}{2} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ -\frac{1}{10} \sqrt{85+38 \sqrt{5}} & \frac{1}{2} & -\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \\
+ \frac{1}{10} \sqrt{85+38 \sqrt{5}} & -\frac{1}{2} & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ \frac{1}{10} \sqrt{85+38 \sqrt{5}} & \frac{1}{2} & \sqrt{\frac{1}{40}-\frac{1}{40 \sqrt{5}}} \\
+ -\sqrt{1+\frac{2}{\sqrt{5}}} & 0 & \sqrt{\frac{1}{8}-\frac{1}{8 \sqrt{5}}} \\
+ \sqrt{1+\frac{2}{\sqrt{5}}} & 0 & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} \\
+\end{array}
+\right)$. Determine the SurfaceArea.
+Answer:
+$18 \sqrt{5 \left(5+2 \sqrt{5}\right)}$

pretraining/mathematica/geometry/solids/15739.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.735 & 0.866 & 0.968 \\
+ 0.461 & 0.81 & 0.438 \\
+ 0.274 & 0.939 & 0.649 \\
+ 0.945 & 0.128 & 0.702 \\
+ 0.225 & 0.888 & 0.708 \\
+ 0.708 & 0.213 & 0.419 \\
+ 0.763 & 0.802 & 0.584 \\
+ 0.162 & 0.36 & 0.403 \\
+ 0.849 & 0.589 & 0.329 \\
+ 0.589 & 0.459 & 0.817 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.5$ 
+Volume: $0.12$ 
+Solid Angle: $1.12$

pretraining/mathematica/geometry/solids/17878.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.386 & 0.237 & 0.937 \\
+ 0.76 & 0.656 & 0.182 \\
+ 0.266 & 0.347 & 0.549 \\
+ 0.086 & 0.525 & 0.28 \\
+ 0.985 & 0.471 & 0.669 \\
+ 0.762 & 0.062 & 0.663 \\
+ 0.916 & 0.92 & 0.243 \\
+ 0.124 & 0.644 & 0.315 \\
+ 0.386 & 0.4 & 0.332 \\
+ 0.613 & 0.698 & 0.506 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.52$ 
+Volume: $0.11$ 
+Solid Angle: $0.97$

pretraining/mathematica/geometry/solids/19290.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & 0.588 \\
+ 0.425 & -0.309 & 0.263 \\
+ -0.162 & 0.5 & 0.263 \\
+ 0.162 & -0.5 & -0.263 \\
+ -0.425 & 0.309 & -0.263 \\
+ 0. & 0. & -0.588 \\
+ 0.425 & 0.309 & 0.263 \\
+ -0.425 & -0.309 & -0.263 \\
+ -0.162 & -0.5 & 0.263 \\
+ -0.526 & 0. & 0.263 \\
+ 0.526 & 0. & -0.263 \\
+ 0.162 & 0.5 & -0.263 \\
+\end{array}
+\right)$. Determine the Incenter.
+Answer:
+$\{0.,0.,0.\}$

pretraining/mathematica/geometry/solids/25680.txt

@@ -0,0 +1,39 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -2.227 & 0. & 1.114 \\
+ -1.802 & -1.309 & -1.114 \\
+ -1.802 & 1.309 & -1.114 \\
+ -1.376 & 0. & -0.263 \\
+ -1.114 & -0.809 & 0.263 \\
+ -1.114 & 0.809 & 0.263 \\
+ -0.851 & 0. & -1.114 \\
+ -0.688 & -2.118 & 1.114 \\
+ -0.688 & -0.5 & 1.114 \\
+ -0.688 & 0.5 & 1.114 \\
+ -0.688 & 2.118 & 1.114 \\
+ -0.425 & -1.309 & -0.263 \\
+ -0.425 & 1.309 & -0.263 \\
+ -0.263 & -0.809 & -1.114 \\
+ -0.263 & 0.809 & -1.114 \\
+ 0. & 0. & -2.49 \\
+ 0. & 0. & 2.49 \\
+ 0.263 & -0.809 & 1.114 \\
+ 0.263 & 0.809 & 1.114 \\
+ 0.425 & -1.309 & 0.263 \\
+ 0.425 & 1.309 & 0.263 \\
+ 0.688 & -2.118 & -1.114 \\
+ 0.688 & -0.5 & -1.114 \\
+ 0.688 & 0.5 & -1.114 \\
+ 0.688 & 2.118 & -1.114 \\
+ 0.851 & 0. & 1.114 \\
+ 1.114 & -0.809 & -0.263 \\
+ 1.114 & 0.809 & -0.263 \\
+ 1.376 & 0. & 0.263 \\
+ 1.802 & -1.309 & 1.114 \\
+ 1.802 & 1.309 & 1.114 \\
+ 2.227 & 0. & -1.114 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$90.$

pretraining/mathematica/geometry/solids/32216.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.926 & 0.271 & 0.605 \\
+ 0.392 & 0.722 & 0.148 \\
+ 0.605 & 0.759 & 0.257 \\
+ 0.396 & 0.104 & 0.607 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.$ 
+Surface Area: $0.55$ 
+Solid Angle: $0.07$

pretraining/mathematica/geometry/solids/37709.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.741 & 0.726 & 0.829 \\
+ 0.662 & 0.682 & 0.382 \\
+ 0.606 & 0.467 & 0.182 \\
+ 0.849 & 0.007 & 0.738 \\
+ 0.034 & 0.999 & 0.718 \\
+ 0.245 & 0.012 & 0.102 \\
+ 0.509 & 0.872 & 0.826 \\
+ 0.284 & 0.839 & 0.039 \\
+ 0.747 & 0.353 & 0.891 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $2.24$ 
+Surface Area: $2.24$ 
+Volume: $0.2$

pretraining/mathematica/geometry/solids/38525.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.049 & 0.68 & 0.071 \\
+ 0.22 & 0.392 & 0.064 \\
+ 0.242 & 0.774 & 0.624 \\
+ 0.759 & 0.879 & 0.831 \\
+ 0.828 & 0.265 & 0.416 \\
+ 0.971 & 0.884 & 0.12 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.12$ 
+Surface Area: $1.62$ 
+Solid Angle: $0.93$

pretraining/mathematica/geometry/solids/40646.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.669 & 0.019 & 0.311 \\
+ 0.004 & 0.155 & 0.551 \\
+ 0.179 & 0.239 & 0.13 \\
+ 0.728 & 0.916 & 0.167 \\
+ 0.845 & 0.745 & 0.979 \\
+ 0.277 & 0.974 & 0.557 \\
+ 0.488 & 0.863 & 0.058 \\
+ 0.005 & 0.608 & 0.862 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.3$ 
+Surface Area: $2.41$ 
+Volume: $0.26$

pretraining/mathematica/geometry/solids/44719.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.26 & 0.001 & 0.915 \\
+ 0.143 & 0.155 & 0.535 \\
+ 0.307 & 0.857 & 0.437 \\
+ 0.041 & 0.217 & 0.55 \\
+ 0.873 & 0.626 & 0.907 \\
+ 0.481 & 0.711 & 0.784 \\
+ 0.723 & 0.432 & 0.339 \\
+ 0.938 & 0.134 & 0.895 \\
+ 0.704 & 0.357 & 0.922 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.64$ 
+Solid Angle: $1.01$ 
+Volume: $0.13$

pretraining/mathematica/geometry/solids/49716.txt

@@ -0,0 +1,5 @@
+Problem:
+A sphere centered at $\{3.507,-3.927,4.165\}$ has radius $4.48$. Estimate the sphere's surface area and volume.
+Answer:
+Surface Area: $252.25$ 
+Volume: $376.71$

pretraining/mathematica/geometry/solids/49939.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.732 & 0.606 & 0.652 \\
+ 0.013 & 0.886 & 0.547 \\
+ 0.619 & 0.852 & 0.982 \\
+ 0.822 & 0.987 & 0.067 \\
+ 0.151 & 0.895 & 0.858 \\
+ 0.767 & 0.539 & 0.393 \\
+ 0.338 & 0.923 & 0.838 \\
+ 0.046 & 0.212 & 0.077 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.16$ 
+Solid Angle: $2.85$ 
+Surface Area: $1.87$

pretraining/mathematica/geometry/solids/52051.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.701 & 0.384 & 0.264 \\
+ 0.99 & 0.791 & 0.272 \\
+ 0.385 & 0.68 & 0.98 \\
+ 0.502 & 0.873 & 0.597 \\
+ 0.107 & 0.034 & 0.734 \\
+ 0.134 & 0.182 & 0.934 \\
+ 0.92 & 0.455 & 0.961 \\
+ 0.434 & 0.667 & 0.105 \\
+ 0.541 & 0.143 & 0.783 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.19$ 
+Solid Angle: $3.17$ 
+Surface Area: $1.98$

pretraining/mathematica/geometry/solids/54129.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.785 & 0.916 & 0.37 \\
+ 0.353 & 0.07 & 0.124 \\
+ 0.543 & 0. & 0.357 \\
+ 0.239 & 0.542 & 0.711 \\
+ 0.149 & 0.139 & 0.757 \\
+ 0.84 & 0.447 & 0.04 \\
+ 0.479 & 0.304 & 0.875 \\
+ 0.647 & 0.092 & 0.269 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.48$ 
+Surface Area: $1.52$ 
+Volume: $0.11$

pretraining/mathematica/geometry/solids/54821.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.519 & 0.753 & 0.771 \\
+ 0.572 & 0.919 & 0.65 \\
+ 0.613 & 0.567 & 0.084 \\
+ 0.807 & 0.502 & 0.405 \\
+ 0.442 & 0.549 & 0.192 \\
+ 0.785 & 0.699 & 0.794 \\
+ 0.276 & 0.001 & 0.283 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.06$ 
+Surface Area: $0.96$ 
+Solid Angle: $2.15$

pretraining/mathematica/geometry/solids/55706.txt

@@ -0,0 +1,67 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0 & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & 0 & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{4} \sqrt{34+\frac{62}{\sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{4} \sqrt{26+\frac{58}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ 0 & \sqrt{\frac{5}{2}+\frac{11}{2 \sqrt{5}}} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{4} \sqrt{2-\frac{2}{\sqrt{5}}} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ 0 & -\sqrt{\frac{5}{2}+\frac{11}{2 \sqrt{5}}} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \left(-3-\sqrt{5}\right) & \sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & \sqrt{\frac{17}{8}+\frac{31}{8 \sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(5+\sqrt{5}\right) & -\sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & \sqrt{\frac{17}{8}+\frac{31}{8 \sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & 0 & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(5+\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & -\sqrt{1+\frac{2}{\sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{10} \left(5-\sqrt{5}\right)} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} & \sqrt{\frac{9}{4}+\frac{9}{2 \sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{17}{8}+\frac{31}{8 \sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ \frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(-5-\sqrt{5}\right) & \sqrt{\frac{5}{8}+\frac{11}{8 \sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{2} & \frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{\sqrt{5}-3} & -\sqrt{\frac{13}{8}+\frac{29}{8 \sqrt{5}}} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ -\frac{1}{2} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -1-\frac{\sqrt{5}}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \left(-3-\sqrt{5}\right) & \frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ \frac{1}{4} \left(-5-\sqrt{5}\right) & -\frac{1}{4} \sqrt{10+\frac{22}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \left(-3-\sqrt{5}\right) & -\frac{1}{2} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & 0 & -\frac{1}{2} \sqrt{5+2 \sqrt{5}} \\
+ -\frac{1}{2} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & -\sqrt{1+\frac{2}{\sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ 0 & -\sqrt{\frac{1}{10} \left(5+\sqrt{5}\right)} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -1-\frac{\sqrt{5}}{2} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} \\
+ -\frac{1}{2} & \sqrt{\frac{1}{4}+\frac{1}{2 \sqrt{5}}} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \sqrt{1+\frac{2}{\sqrt{5}}} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+ -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} \sqrt{1+\frac{2}{\sqrt{5}}} & \frac{1}{2} \sqrt{1-\frac{2}{\sqrt{5}}} \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & -\frac{1}{4} \sqrt{2-\frac{2}{\sqrt{5}}} & -\frac{3}{2} \sqrt{1+\frac{2}{\sqrt{5}}} \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$120$

pretraining/mathematica/geometry/solids/58129.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.211 & 0.209 & 0.1 \\
+ 0.341 & 0.807 & 0.219 \\
+ 0.905 & 0.621 & 0.772 \\
+ 0.962 & 0.485 & 0.553 \\
+ 0.738 & 0.198 & 0.243 \\
+ 0.017 & 0.235 & 0.765 \\
+ 0.887 & 0.166 & 0.595 \\
+ 0.91 & 0.008 & 0.273 \\
+ 0.597 & 0.348 & 0.791 \\
+ 0.645 & 0.211 & 0.713 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.84$ 
+Solid Angle: $1.48$ 
+Volume: $0.17$

pretraining/mathematica/geometry/solids/60429.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.178 & 0.698 & 0.609 \\
+ 0.837 & 0.749 & 0.909 \\
+ 0.503 & 0.035 & 0.029 \\
+ 0.356 & 0.118 & 0.011 \\
+ 0.581 & 0.533 & 0.097 \\
+ 0.554 & 0.859 & 0.062 \\
+ 0.925 & 0.994 & 0.467 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.13$ 
+Volume: $0.12$ 
+Surface Area: $1.6$

pretraining/mathematica/geometry/solids/6543.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.547 & 0.704 & 0.114 \\
+ 0.668 & 0.857 & 0.448 \\
+ 0.404 & 0.83 & 0.8 \\
+ 0.284 & 0.132 & 0.153 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.49$ 
+Surface Area: $0.6$ 
+Volume: $0.01$

pretraining/mathematica/geometry/solids/67606.txt

@@ -0,0 +1,99 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -3.521 & 0. & -1.761 \\
+ -2.849 & -2.07 & 1.761 \\
+ -2.849 & 2.07 & 1.761 \\
+ -2.176 & 0. & 0.416 \\
+ -1.761 & -1.279 & -0.416 \\
+ -1.761 & 1.279 & -0.416 \\
+ -1.46 & -0.354 & -0.186 \\
+ -1.46 & 0.354 & -0.186 \\
+ -1.389 & -0.572 & 0.186 \\
+ -1.389 & 0.572 & 0.186 \\
+ -1.345 & 0. & 1.761 \\
+ -1.274 & -0.219 & 0.787 \\
+ -1.274 & 0.219 & 0.787 \\
+ -1.159 & -0.572 & -0.787 \\
+ -1.159 & 0.572 & -0.787 \\
+ -1.088 & -3.349 & -1.761 \\
+ -1.088 & -0.791 & -1.761 \\
+ -1.088 & 0.791 & -1.761 \\
+ -1.088 & 3.349 & -1.761 \\
+ -0.973 & -1.144 & 0.186 \\
+ -0.973 & 0. & -1.159 \\
+ -0.973 & 1.144 & 0.186 \\
+ -0.902 & -0.926 & -0.787 \\
+ -0.902 & 0.926 & -0.787 \\
+ -0.787 & -1.279 & -0.186 \\
+ -0.787 & -0.572 & 1.159 \\
+ -0.787 & 0.572 & 1.159 \\
+ -0.787 & 1.279 & -0.186 \\
+ -0.672 & -2.07 & 0.416 \\
+ -0.672 & 2.07 & 0.416 \\
+ -0.602 & -1.144 & 0.787 \\
+ -0.602 & 0. & -1.389 \\
+ -0.602 & 1.144 & 0.787 \\
+ -0.487 & -0.354 & 1.389 \\
+ -0.487 & 0.354 & 1.389 \\
+ -0.416 & -1.279 & 1.761 \\
+ -0.416 & 1.279 & 1.761 \\
+ -0.301 & -0.926 & -1.159 \\
+ -0.301 & 0.926 & -1.159 \\
+ -0.186 & -1.279 & 0.787 \\
+ -0.186 & -0.572 & -1.389 \\
+ -0.186 & 0.572 & -1.389 \\
+ -0.186 & 1.279 & 0.787 \\
+ -0.115 & -1.498 & -0.186 \\
+ -0.115 & 1.498 & -0.186 \\
+ 0. & 0. & -3.937 \\
+ 0. & 0. & 3.937 \\
+ 0.115 & -1.498 & 0.186 \\
+ 0.115 & 1.498 & 0.186 \\
+ 0.186 & -1.279 & -0.787 \\
+ 0.186 & -0.572 & 1.389 \\
+ 0.186 & 0.572 & 1.389 \\
+ 0.186 & 1.279 & -0.787 \\
+ 0.301 & -0.926 & 1.159 \\
+ 0.301 & 0.926 & 1.159 \\
+ 0.416 & -1.279 & -1.761 \\
+ 0.416 & 1.279 & -1.761 \\
+ 0.487 & -0.354 & -1.389 \\
+ 0.487 & 0.354 & -1.389 \\
+ 0.602 & -1.144 & -0.787 \\
+ 0.602 & 0. & 1.389 \\
+ 0.602 & 1.144 & -0.787 \\
+ 0.672 & -2.07 & -0.416 \\
+ 0.672 & 2.07 & -0.416 \\
+ 0.787 & -1.279 & 0.186 \\
+ 0.787 & -0.572 & -1.159 \\
+ 0.787 & 0.572 & -1.159 \\
+ 0.787 & 1.279 & 0.186 \\
+ 0.902 & -0.926 & 0.787 \\
+ 0.902 & 0.926 & 0.787 \\
+ 0.973 & -1.144 & -0.186 \\
+ 0.973 & 0. & 1.159 \\
+ 0.973 & 1.144 & -0.186 \\
+ 1.088 & -3.349 & 1.761 \\
+ 1.088 & -0.791 & 1.761 \\
+ 1.088 & 0.791 & 1.761 \\
+ 1.088 & 3.349 & 1.761 \\
+ 1.159 & -0.572 & 0.787 \\
+ 1.159 & 0.572 & 0.787 \\
+ 1.274 & -0.219 & -0.787 \\
+ 1.274 & 0.219 & -0.787 \\
+ 1.345 & 0. & -1.761 \\
+ 1.389 & -0.572 & -0.186 \\
+ 1.389 & 0.572 & -0.186 \\
+ 1.46 & -0.354 & 0.186 \\
+ 1.46 & 0.354 & 0.186 \\
+ 1.761 & -1.279 & 0.416 \\
+ 1.761 & 1.279 & 0.416 \\
+ 2.176 & 0. & -0.416 \\
+ 2.849 & -2.07 & -1.761 \\
+ 2.849 & 2.07 & -1.761 \\
+ 3.521 & 0. & 1.761 \\
+\end{array}
+\right)$. Determine the Incenter.
+Answer:
+$\{0.,0.,0.\}$

pretraining/mathematica/geometry/solids/69074.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.034 & 0.715 & 0.417 \\
+ 0.22 & 0.157 & 0.942 \\
+ 0.54 & 0.185 & 0.799 \\
+ 0.08 & 0.314 & 0.472 \\
+ 0.895 & 0.611 & 0.227 \\
+ 0.954 & 0.634 & 0.735 \\
+ 0.495 & 0.035 & 0.285 \\
+ 0.858 & 0.821 & 0.33 \\
+ 0.107 & 0.796 & 0.718 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.2$ 
+Surface Area: $1.96$ 
+Solid Angle: $1.77$

pretraining/mathematica/geometry/solids/6947.txt

@@ -0,0 +1,30 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ 0. & 0. & 1.401 \\
+ 0. & 0. & -1.401 \\
+ 0.178 & -1.309 & 0.467 \\
+ 0.178 & 1.309 & 0.467 \\
+ 0.467 & -0.809 & -1.044 \\
+ 0.467 & 0.809 & -1.044 \\
+ 1.044 & -0.809 & 0.467 \\
+ 1.044 & 0.809 & 0.467 \\
+ -1.223 & -0.5 & 0.467 \\
+ -1.223 & 0.5 & 0.467 \\
+ 1.223 & -0.5 & -0.467 \\
+ 1.223 & 0.5 & -0.467 \\
+ -0.934 & 0. & -1.044 \\
+ -0.467 & -0.809 & 1.044 \\
+ -0.467 & 0.809 & 1.044 \\
+ 0.934 & 0. & 1.044 \\
+ -1.044 & -0.809 & -0.467 \\
+ -1.044 & 0.809 & -0.467 \\
+ -0.995 & 0. & 1.303 \\
+ -0.178 & -1.309 & -0.467 \\
+ -0.178 & 1.309 & -0.467 \\
+ 0.805 & 1.394 & -0.308 \\
+ 0.805 & -1.394 & -0.308 \\
+\end{array}
+\right)$. Determine the EdgeCount.
+Answer:
+$45.$

pretraining/mathematica/geometry/solids/69803.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.695 & 0.837 & 0.07 \\
+ 0.983 & 0.925 & 0.296 \\
+ 0.626 & 0.044 & 0.873 \\
+ 0.649 & 0.818 & 0.815 \\
+ 0.57 & 0.958 & 0.036 \\
+ 0.477 & 0.669 & 0.189 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $3.27$ 
+Surface Area: $1.14$ 
+Volume: $0.06$

pretraining/mathematica/geometry/solids/69860.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.443 & 0.557 & 0.302 \\
+ 0.33 & 0.142 & 0.882 \\
+ 0.434 & 0.643 & 0.784 \\
+ 0.439 & 0.825 & 0.299 \\
+ 0.468 & 0.75 & 0.891 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.3$ 
+Surface Area: $0.53$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/7050.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.82 & 0.067 & 0.325 \\
+ 0.549 & 0.376 & 0.975 \\
+ 0.87 & 0.699 & 0.891 \\
+ 0.671 & 0.503 & 0.528 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.13$ 
+Volume: $0.01$ 
+Surface Area: $0.48$

pretraining/mathematica/geometry/solids/70762.txt

@@ -0,0 +1,19 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.394 & 0.563 & 0.394 \\
+ 0.703 & 0.757 & 0.508 \\
+ 0.598 & 0.896 & 0.595 \\
+ 0.28 & 0.376 & 0.527 \\
+ 0.15 & 0.822 & 0.718 \\
+ 0.492 & 0.813 & 0.875 \\
+ 0.272 & 0.89 & 0.038 \\
+ 0.403 & 0.457 & 0.638 \\
+ 0.012 & 0.707 & 0.159 \\
+ 0.028 & 0.554 & 0.172 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.09$ 
+Solid Angle: $5.46$ 
+Volume: $0.07$

pretraining/mathematica/geometry/solids/71187.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.259 & 0.422 & 0.045 \\
+ 0.98 & 0.556 & 0.519 \\
+ 0.424 & 0.879 & 0.126 \\
+ 0.388 & 0.492 & 0.72 \\
+ 0.337 & 0.05 & 0.555 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.06$ 
+Surface Area: $1.1$ 
+Solid Angle: $0.8$

pretraining/mathematica/geometry/solids/77069.txt

@@ -0,0 +1,15 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.631 & 0.07 & 0.229 \\
+ 0.446 & 0.486 & 0.118 \\
+ 0.683 & 0.712 & 0.17 \\
+ 0.246 & 0.339 & 0.668 \\
+ 0.979 & 0.535 & 0.119 \\
+ 0.691 & 0.3 & 0.629 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $1.28$ 
+Surface Area: $0.88$ 
+Volume: $0.05$

pretraining/mathematica/geometry/solids/80224.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.264 & 0.997 & 0.564 \\
+ 0.434 & 0.704 & 0.501 \\
+ 0.307 & 0.152 & 0.12 \\
+ 0.046 & 0.573 & 0.912 \\
+ 0.771 & 0.633 & 0.769 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.84$ 
+Volume: $0.06$ 
+Surface Area: $1.06$

pretraining/mathematica/geometry/solids/80512.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.728 & 0.669 & 0.786 \\
+ 0.824 & 0.56 & 0.358 \\
+ 0.089 & 0.561 & 0.159 \\
+ 0.359 & 0.778 & 0.239 \\
+ 0.027 & 0.988 & 0.927 \\
+ 0.262 & 0.651 & 0.094 \\
+ 0.804 & 0.04 & 0.775 \\
+ 0.584 & 0.082 & 0.992 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.96$ 
+Volume: $0.17$ 
+Solid Angle: $2.79$

pretraining/mathematica/geometry/solids/83016.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.21 & 0. & 0.129 \\
+ 0.287 & 0.689 & 0.497 \\
+ 0.313 & 0.602 & 0.963 \\
+ 0.647 & 0.433 & 0.142 \\
+ 0.584 & 0.798 & 0.696 \\
+ 0.625 & 0.445 & 0.926 \\
+ 0.039 & 0.088 & 0.841 \\
+ 0.201 & 0.083 & 0.018 \\
+ 0.727 & 0.571 & 0.134 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.14$ 
+Solid Angle: $1.78$ 
+Surface Area: $1.75$

pretraining/mathematica/geometry/solids/8404.txt

@@ -0,0 +1,21 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & 0 \\
+ -\sqrt{\frac{2}{3}} & 0 & -\frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\
+ -\sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & 0 \\
+ 0 & -\sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ 0 & -\sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ 0 & 0 & -\frac{2}{\sqrt{3}} \\
+ 0 & 0 & \frac{2}{\sqrt{3}} \\
+ 0 & \sqrt{\frac{2}{3}} & -\frac{1}{\sqrt{3}} \\
+ 0 & \sqrt{\frac{2}{3}} & \frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & -\sqrt{\frac{2}{3}} & 0 \\
+ \sqrt{\frac{2}{3}} & 0 & -\frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & 0 & \frac{1}{\sqrt{3}} \\
+ \sqrt{\frac{2}{3}} & \sqrt{\frac{2}{3}} & 0 \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$\frac{16}{3 \sqrt{3}}$

pretraining/mathematica/geometry/solids/90245.txt

@@ -0,0 +1,67 @@
+Problem:
+A polyhedron has vertex coordinates $\left(
+\begin{array}{ccc}
+ -\frac{1}{2} & -\frac{1}{2} & -1-\frac{\sqrt{5}}{2} \\
+ -\frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) \\
+ -\frac{1}{2} & \frac{1}{2} & -1-\frac{\sqrt{5}}{2} \\
+ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) \\
+ -\frac{1}{2} & -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} \\
+ 0 & -\frac{3}{4}-\frac{13}{4 \sqrt{5}} & \frac{1}{20} \left(5+\sqrt{5}\right) \\
+ -\frac{1}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & 1+\frac{2}{\sqrt{5}} & \frac{1}{20} \left(15+\sqrt{5}\right) \\
+ 0 & \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(-5-\sqrt{5}\right) \\
+ -\frac{1}{2} & -\frac{1}{2}-\frac{2}{\sqrt{5}} & 1+\frac{3}{2 \sqrt{5}} \\
+ 0 & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-5-\sqrt{5}\right) \\
+ \frac{1}{2} & \frac{1}{2}+\frac{2}{\sqrt{5}} & 1+\frac{3}{2 \sqrt{5}} \\
+ \frac{1}{2} & -\frac{1}{2} & -1-\frac{\sqrt{5}}{2} \\
+ \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) \\
+ \frac{1}{2} & \frac{1}{2} & -1-\frac{\sqrt{5}}{2} \\
+ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) \\
+ \frac{1}{2} & -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & -1-\frac{2}{\sqrt{5}} & \frac{1}{20} \left(15+\sqrt{5}\right) \\
+ \frac{1}{2} & \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} \\
+ 0 & \frac{3}{4}+\frac{13}{4 \sqrt{5}} & \frac{1}{20} \left(5+\sqrt{5}\right) \\
+ \frac{1}{4} \left(-5-\sqrt{5}\right) & 0 & \frac{1}{\sqrt{5}-3} \\
+ \frac{1}{4} \left(-5-\sqrt{5}\right) & 0 & \frac{1}{4} \left(3+\sqrt{5}\right) \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & -1-\frac{2}{\sqrt{5}} & \frac{1}{20} \left(15+\sqrt{5}\right) \\
+ \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} \\
+ -\frac{1}{2} & \frac{1}{2}+\frac{2}{\sqrt{5}} & 1+\frac{3}{2 \sqrt{5}} \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(1+\sqrt{5}\right) \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(1+\sqrt{5}\right) \\
+ -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} & -\frac{1}{2} \\
+ -1-\frac{\sqrt{5}}{2} & -\frac{1}{2} & \frac{1}{2} \\
+ -1-\frac{\sqrt{5}}{2} & \frac{1}{2} & -\frac{1}{2} \\
+ -1-\frac{\sqrt{5}}{2} & \frac{1}{2} & \frac{1}{2} \\
+ \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(-5-\sqrt{5}\right) & 0 \\
+ \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\
+ \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\
+ \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(5+\sqrt{5}\right) & 0 \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} \\
+ \frac{1}{2} & -\frac{1}{2}-\frac{2}{\sqrt{5}} & 1+\frac{3}{2 \sqrt{5}} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} \\
+ \frac{1}{4} \left(1+\sqrt{5}\right) & 1+\frac{2}{\sqrt{5}} & \frac{1}{20} \left(15+\sqrt{5}\right) \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{\sqrt{5}-3} & \frac{1}{4} \left(1+\sqrt{5}\right) \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{2} \left(1+\sqrt{5}\right) & \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(1+\sqrt{5}\right) \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} & -\frac{1}{2} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & -\frac{1}{2} & \frac{1}{2} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & \frac{1}{2} & -\frac{1}{2} \\
+ \frac{1}{2} \left(2+\sqrt{5}\right) & \frac{1}{2} & \frac{1}{2} \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-5-\sqrt{5}\right) & 0 \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(-1-\sqrt{5}\right) \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(1+\sqrt{5}\right) & \frac{1}{2} \left(1+\sqrt{5}\right) \\
+ \frac{1}{4} \left(3+\sqrt{5}\right) & \frac{1}{4} \left(5+\sqrt{5}\right) & 0 \\
+ \frac{1}{4} \left(5+\sqrt{5}\right) & 0 & \frac{1}{\sqrt{5}-3} \\
+ \frac{1}{4} \left(5+\sqrt{5}\right) & 0 & \frac{1}{4} \left(3+\sqrt{5}\right) \\
+\end{array}
+\right)$. Determine the Volume.
+Answer:
+$20+\frac{29 \sqrt{5}}{3}$

pretraining/mathematica/geometry/solids/90387.txt

@@ -0,0 +1,13 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.465 & 0.274 & 0.655 \\
+ 0.202 & 0.106 & 0.067 \\
+ 0.763 & 0.37 & 0.308 \\
+ 0.346 & 0.166 & 0.476 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Solid Angle: $0.11$ 
+Surface Area: $0.33$ 
+Volume: $0.$

pretraining/mathematica/geometry/solids/93420.txt

@@ -0,0 +1,18 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.882 & 0.754 & 0.5 \\
+ 0.768 & 0.267 & 0.612 \\
+ 0.087 & 0.091 & 0.477 \\
+ 0.106 & 0.031 & 0.624 \\
+ 0.754 & 0.435 & 0.274 \\
+ 0.187 & 0.088 & 0.13 \\
+ 0.127 & 0.995 & 0.348 \\
+ 0.199 & 0.001 & 0.781 \\
+ 0.691 & 0.011 & 0.157 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.94$ 
+Volume: $0.17$ 
+Solid Angle: $0.93$

pretraining/mathematica/geometry/solids/94880.txt

@@ -0,0 +1,17 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.443 & 0.18 & 0.776 \\
+ 0.053 & 0.558 & 0.483 \\
+ 0.418 & 0.48 & 0.881 \\
+ 0.934 & 0.102 & 0.723 \\
+ 0.952 & 0.53 & 0.939 \\
+ 0.265 & 0.902 & 0.744 \\
+ 0.783 & 0.768 & 0.971 \\
+ 0.59 & 0.987 & 0.832 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Volume: $0.06$ 
+Solid Angle: $1.86$ 
+Surface Area: $1.29$

pretraining/mathematica/geometry/solids/96125.txt

@@ -0,0 +1,16 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.97 & 0.094 & 0.294 \\
+ 0.34 & 0.297 & 0.646 \\
+ 0.413 & 0.636 & 0.758 \\
+ 0.749 & 0.143 & 0.522 \\
+ 0.593 & 0.068 & 0.201 \\
+ 0.885 & 0.129 & 0.159 \\
+ 0.379 & 0.391 & 0.506 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $0.62$ 
+Solid Angle: $0.83$ 
+Volume: $0.03$

pretraining/mathematica/geometry/solids/97386.txt

@@ -0,0 +1,14 @@
+Problem:
+A polyhedron has vertices with the coordinates $\left(
+\begin{array}{ccc}
+ 0.271 & 0.005 & 0.912 \\
+ 0.119 & 0.296 & 0.018 \\
+ 0.298 & 0.942 & 0.372 \\
+ 0.447 & 0.151 & 0.495 \\
+ 0.767 & 0.183 & 0.932 \\
+\end{array}
+\right)$. Estimate the polyhedron's surface area, volume, and the solid angle at the first listed point p spanned by edges with common point p.
+Answer:
+Surface Area: $1.29$ 
+Volume: $0.06$ 
+Solid Angle: $0.72$

pretraining/mathematica/number_theory/relatively_prime/10562.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{-862,-749\}$.
+Answer:
+$\text{True}$

pretraining/mathematica/number_theory/relatively_prime/12944.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{-338,981\}$.
+Answer:
+$\text{True}$

pretraining/mathematica/number_theory/relatively_prime/14118.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{-423,29,54\}$.
+Answer:
+$\text{False}$

pretraining/mathematica/number_theory/relatively_prime/14590.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{673,760,-304\}$.
+Answer:
+$\text{False}$

pretraining/mathematica/number_theory/relatively_prime/14889.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{-113,100\}$.
+Answer:
+$\text{True}$

pretraining/mathematica/number_theory/relatively_prime/17101.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{334,988\}$.
+Answer:
+$\text{False}$

pretraining/mathematica/number_theory/relatively_prime/18194.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{30,393\}$.
+Answer:
+$\text{False}$

pretraining/mathematica/number_theory/relatively_prime/20569.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{649,675\}$.
+Answer:
+$\text{True}$

pretraining/mathematica/number_theory/relatively_prime/22099.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{-750,253,-265\}$.
+Answer:
+$\text{False}$

pretraining/mathematica/number_theory/relatively_prime/24439.txt

@@ -0,0 +1,4 @@
+Problem:
+Are the following numbers relatively prime (coprime)? $\{571,-704\}$.
+Answer:
+$\text{True}$