2021 GMC 10B

Problem 1

What is $5!+3!-4!-2!+1!+0!?$

$\textbf{(A)} ~99 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~101 \qquad\textbf{(D)} ~102 \qquad\textbf{(E)} ~103$

Problem 2

The radius of a circle that has an area of $\frac{\pi}{\sqrt{2}}$ is $r$ . Find $r^{2}$

$\textbf{(A)} ~\frac{1}{2} \qquad\textbf{(B)} ~\frac{\sqrt{2}}{2} \qquad\textbf{(C)} ~\sqrt{2} \qquad\textbf{(D)} ~2 \qquad\textbf{(E)} ~4$

Problem 3

What is the sum of the digits of the largest prime that divides $32160$ ?

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~9 \qquad\textbf{(D)} ~11\qquad\textbf{(E)} ~13$

Problem 4

Ary wants to go to the park at afternoon. he walked to halfway, and he's pretty hungry. Therefore, he searched on his phone and found that exactly on the halfway between his house and the park there is a restaurant. After he eats, he continues to walk, however, he stops at the $\frac{3}{4}$ way between the restaurant and park to take a break. Let $x$ be the length that he need to walk to reach the park, and $y$ be the distance between his house and the park. Find $\frac{x}{y}$

$\textbf{(A)} ~\frac{3}{8} \qquad\textbf{(B)} ~\frac{7}{16} \qquad\textbf{(C)} ~\frac{9}{16} \qquad\textbf{(D)} ~\frac{2}{3}\qquad\textbf{(E)} ~\frac{3}{4}$

Problem 5

An octagon has four given vertices $(-1,0), (-1,2), (3,0)$ , $(3,2)$ ,and it partially covers all the four quadrants. Let $a_n$ be the area of the portion of the octagon that lies in the $n$ th quadrant. Find $\frac{a_1\cdot a_4}{a_2\cdot a_3}$

$\textbf{(A)} ~2 \qquad\textbf{(B)} ~\frac{12}{5} \qquad\textbf{(C)} ~\frac{13}{5} \qquad\textbf{(D)} ~\frac{8}{3}\qquad\textbf{(E)} ~3$

Problem 6

6. How many possible ordered pairs of nonnegative integers $(a,b,c)$ are there such that $2a+3^b=4^{abc}$ ?

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3\qquad\textbf{(E)} ~4$

Problem 7

7. In the diagram below, 9 squares with side length $2$ grid has 16 circles with radius of $\frac{1}{2}$ such that all circles have vertices of the square as center. Assume that the diagram continues on forever. Given that the area of the circle is $x\%$ of the entire infinite diagram, find $\left \lfloor{x}\right \rfloor$

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