
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>B stands for Bernoulli, G stands for Germain</summary>they had two problem sets under those two names</details> **B1.** If the values of two angles in a triangle are  $60$  and  $75$  degrees respectively, what is the measure of the third angle?**B2.** Square  $ABCD$  has side length  $1$ . What is the area of triangle  $ABC$ ?**B3 / G1.** An equilateral triangle and a square have the same perimeter. If the side length of the equilateral triangle is  $8$ , what is the square’s side length?**B4 / G2.** What is the maximum possible number of sides and diagonals of equal length in a quadrilateral?**B5.** A square of side length  $4$  is put within a circle such that all  $4$  corners lie on the circle. What is the diameter of the circle?**B6 / G3.** Patrick is rafting directly across a river  $20$  meters across at a speed of  $5$  m/s. The river flows in a direction perpendicular to Patrick’s direction at a rate of  $12$  m/s. When Patrick reaches the shore on the other end of the river, what is the total distance he has traveled?**B7 / G4.** Quadrilateral  $ABCD$  has side lengths  $AB = 7$ ,  $BC = 15$ ,  $CD = 20$ , and  $DA = 24$ . It has a diagonal length of  $BD = 25$ . Find the measure, in degrees, of the sum of angles  $ABC$  and  $ADC$ .**B8 / G5.** What is the largest  $P$  such that any rectangle inscribed in an equilateral triangle of side length  $1$  has a perimeter of at least  $P$ ?**G6.** A circle is inscribed in an equilateral triangle with side length  $s$ . Points  $A$ , $B$ , $C$ , $D$ , $E$ , $F$  lie on the triangle such that line segments  $AB$ ,  $CD$ , and  $EF$  are parallel to a side of the triangle, and tangent to the circle. If the area of hexagon  $ABCDEF = \frac{9\sqrt3}{2}$  , find  $s$ .**G7.** Let  $\vartriangle ABC$  be such that  $\angle A = 105^o$ ,  $\angle B = 45^o$ ,  $\angle C = 30^o$ . Let  $M$  be the midpoint of  $AC$ . What is  $\angle MBC$ ?**G8.** Points  $A$ ,  $B$ , and  $C$  lie on a circle centered at  $O$  with radius  $10$ . Let the circumcenter of  $\vartriangle AOC$  be  $P$ . If  $AB = 16$ , find the minimum value of  $PB$ .
*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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