[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] 


[b]B1.[/b] If the values of two angles in a triangle are $60$ and $75$ degrees respectively, what is the measure of the third angle?


[b]B2.[/b] Square $ABCD$ has side length $1$. What is the area of triangle $ABC$?


[b]B3 / G1.[/b] 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?


[b]B4 / G2.[/b] What is the maximum possible number of sides and diagonals of equal length in a quadrilateral?


[b]B5.[/b] 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?


[b]B6 / G3.[/b] 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?


[b]B7 / G4.[/b] 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 $ADC$.


[b]B8 / G5.[/b] What is the largest $P$ such that any rectangle inscribed in an equilateral triangle of side length $1$ has a perimeter of at least $P$?


[b]G6.[/b] 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 $s$.


[b]G7.[/b] 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 $\angle MBC$?


[b]G8.[/b] 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 $PB$.
[i]The circumcenter of a triangle is the intersection point of the three perpendicular bisectors of the sides.
[/i]


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the largest \( P \) such that any rectangle inscribed in an equilateral triangle of side length \( 1 \) has a perimeter of at least \( P \).

1. **Define the Equilateral Triangle and Rectangle:**
   Let the equilateral triangle be \( \triangle ABC \) with side length \( 1 \). Let the rectangle be \( DEFG \), where \( D \) and \( E \) are on \( BC \) with \( D \) closer to \( B \) and \( E \) closer to \( C \). By symmetry, let \( BD = EC = x \).

2. **Determine the Lengths of Rectangle Sides:**
   Since \( D \) and \( E \) are on \( BC \), the length \( DE \) is given by:
   \[
   DE = BC - BD - EC = 1 - 2x
   \]
   Similarly, \( F \) is on \( AC \) and \( G \) is on \( AB \), where \( FE \) and \( GD \) are perpendicular to \( BC \). 

3. **Use the 30-60-90 Triangle Properties:**
   Since \( \triangle BDG \) is a 30-60-90 triangle and \( BD = x \), we have:
   \[
   GD = x \sqrt{3}
   \]
   Similarly, \( FE = x \sqrt{3} \).

4. **Calculate the Perimeter of the Rectangle:**
   The perimeter \( P \) of the rectangle \( DEFG \) is:
   \[
   P = 2(DE + GD) = 2((1 - 2x) + x \sqrt{3}) = 2(1 - 2x + x \sqrt{3})
   \]

5. **Simplify the Perimeter Expression:**
   \[
   P = 2(1 - 2x + x \sqrt{3}) = 2 - 4x + 2x \sqrt{3}
   \]

6. **Find the Minimum Perimeter:**
   To find the minimum value of \( P \), we need to consider the range of \( x \). Since \( 0 < x < \frac{1}{2} \), we analyze the perimeter function:
   \[
   P = 2 - 4x + 2x \sqrt{3}
   \]
   The minimum value of \( P \) occurs at the endpoints of the interval \( 0 < x < \frac{1}{2} \).

   - When \( x = 0 \):
     \[
     P = 2 - 4(0) + 2(0) \sqrt{3} = 2
     \]
   - When \( x = \frac{1}{2} \):
     \[
     P = 2 - 4\left(\frac{1}{2}\right) + 2\left(\frac{1}{2}\right) \sqrt{3} = 2 - 2 + \sqrt{3} = \sqrt{3}
     \]

   Therefore, the minimum perimeter \( P \) is \( \sqrt{3} \).

The final answer is \( \boxed{\sqrt{3}} \).