
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
<details><summary>C stands for Cantor, G stands for Gauss</summary>they had two problem sets under those two names</details> **C1.** A circle has circumference  $6\pi$ . Find the area of this circle.**C2 / G2.** Points  $A$ ,  $B$ , and  $C$  are on a line such that  $AB = 6$  and  $BC = 11$ . Find all possible values of  $AC$ .**C3.** A trapezoid has area  $84$  and one base of length  $5$ . If the height is  $12$ , what is the length of the other base?**C4 / G1.**  $27$  cubes of side length 1 are arranged to form a  $3 \times 3 \times 3$  cube. If the corner  $1 \times 1 \times 1$  cubes are removed, what fraction of the volume of the big cube is left?**C5.** There is a  $50$ -foot tall wall and a  $300$ -foot tall guard tower  $50$  feet from the wall. What is the minimum  $a$  such that a flat “ $X$ ” drawn on the ground  $a$  feet from the side of the wall opposite the guard tower is visible from the top of the guard tower?**C6.** Steven’s pizzeria makes pizzas in the shape of equilateral triangles. If a pizza with side length 8 inches will feed 2 people, how many people will a pizza of side length of 16 inches feed?**C7 / G3.** Consider rectangle  $ABCD$ , with  $1 = AB < BC$ . The angle bisector of  $\angle DAB$  intersects  $\overline{BC}$  at  $E$  and  $\overline{DC}$  at  $F$ . If  $FE = FD$ , find  $BC$ .**C8 / G6.**  $\vartriangle ABC$ . is a right triangle with  $\angle A = 90^o$ . Square  $ADEF$  is drawn, with  $D$  on  $\overline{AB}$ ,  $F$  on  $\overline{AC}$ , and  $E$  inside  $\vartriangle ABC$ . Point  $G$  is chosen on  $\overline{BC}$  such that  $EG$  is perpendicular to  $BC$ . Additionally,  $DE = EG$ . Given that  $\angle C = 20^o$ , find the measure of  $\angle BEG$ .**G4.** Consider a lamp in the shape of a hollow cylinder with the circular faces removed with height  $48$  cm and radius  $7$  cm. A point source of light is situated at the center of the lamp. The lamp is held so that the bottom of the lamp is at a height  $48$  cm above an infinite flat sheet of paper. What is the area of the illuminated region on the flat sheet of paper, in  $cm^2$ ?
![Image](https://cdn.artofproblemsolving.com/attachments/c/6/6e5497a67ae5ff5a7bff7007834a4271ce3ca7.png)**G5.** There exist two triangles  $ABC$  such that  $AB = 13$ ,  $BC = 12\sqrt2$ , and  $\angle C = 45^o$ . Find the positive difference between their areas.**G7.** Let  $ABC$  be an equilateral triangle with side length  $2$ . Let the circle with diameter  $AB$  be  $\Gamma$ . Consider the two tangents from  $C$  to  $\Gamma$ , and let the tangency point closer to  $A$  be  $D$ . Find the area of  $\angle CAD$ .**G8.** Let  $ABC$  be a triangle with  $\angle A = 60^o$ ,  $AB = 37$ ,  $AC = 41$ . Let  $H$  and  $O$  be the orthocenter and circumcenter of  $ABC$ , respectively. Find  $OH$ . 
*The orthocenter of a triangle is the intersection point of the three altitudes. The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.*


PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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