2021 GMC 12

Problem 1

Compute the number of ways to arrange $2$ distinguishable apples and $5$ indistinguishable books such that all five books must be adjacent.

$\textbf{(A) } 12 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 84$

Problem 2

In square $ABCD$ with side length $8$ , point $E$ and $F$ are on side $BC$ and $CD$ respectively, such that $AE$ is perpendicular to $EF$ and $CF=2$ . Find the area enclosed by the quadrilateral $AECF$ .

$\textbf{(A) } 20 \qquad\textbf{(B) } 24 \qquad\textbf{(C) } 28 \qquad\textbf{(D) } 32 \qquad\textbf{(E) } 36$

Problem 3

Lucas wants to choose a seat to sit in a row of ten seats marked $1,2,3,4,5,6,7,8,9,10$ , respectively. Given that Michael595 is an even-preferer and he would sit in any even seat. Find the probability that Lucas would not sit next to Michael595 (because he hates him).

$\textbf{(A) } \frac{1}{5} \qquad\textbf{(B) } \frac{1}{3} \qquad\textbf{(C) } \frac{7}{18} \qquad\textbf{(D) } \frac{2}{5} \qquad\textbf{(E) } \frac{37}{90}$

Problem 4

If $\ln(\ln(x))=e^4$ , find $x$ (Note that $\ln(x)$ means logarithmic function that has a base of $e$ , and $e$ is the natural logarithm.).

$\textbf{(A)} ~e^{e^4} \qquad\textbf{(B)} ~e^{e^{16}} \qquad\textbf{(C)} ~e^{{4e}^e} \qquad\textbf{(D)} ~e^{e^{4^e}} \qquad\textbf{(E)} ~e^{e^{e^4}}$

Problem 5

Let $a_n$ be a sequence of positive integers with $a_0=1$ and $a_1=2$ and $a_n=a_{n-1}\cdot a_{n+1}$ for all integers $n$ such that $n\geq 1$ . Find $a_{2021}+a_{2023}+a_{2025}$ .

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~\frac{7}{2} \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~\frac{9}{2} \qquad\textbf{(E)} ~5$

Problem 6

Compute the remainder when the summation

$\sum_{k=1}^{14} k^3$

is divided by $10000$ .

$\textbf{(A)} ~1025 \qquad\textbf{(B)} ~3025 \qquad\textbf{(C)} ~5025 \qquad\textbf{(D)} ~7025 \qquad\textbf{(E)} ~9025$

Problem 7

$\frac{\sqrt{2}}{3}+\frac{\sqrt{3}}{4}+\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{7}+\frac{\sqrt{2}}{6}+\frac{\sqrt{3}}{8}+\frac{\sqrt{5}}{10}+\frac{\sqrt{6}}{14}+\frac{\sqrt{2}}{12}+\frac{\sqrt{3}}{16}+......$

The answer of this problem can be expressed as $\frac{a\sqrt{b}}{c}+\frac{\sqrt{e}}{f}+\frac{g\sqrt{h}}{j}+\frac{k\sqrt{m}}{n}$ which $a,b,c,d,e,f,g,h,j,k,m,n$ are not necessarily distinct positive integers, and all of $a,b,c,d,e,f,g,h,j,k,m,n$ are not divisible by any square number. Find $a+b+c+d+e+f+g+h+j+k+m+n$ .

$\textbf{(A)} ~39 \qquad\textbf{(B)} ~40 \qquad\textbf{(C)} ~41 \qquad\textbf{(D)} ~42 \qquad\textbf{(E)} ~43$

Problem 8

Let $S_n=a_1,a_2,a_3,a_4,a_5,a_6$ be a $6$ term sequence of positive integers such that $2\cdot a_1=a_2$ , $4\cdot a_2=a_3$ , $8\cdot a_3=a_4$ , $16\cdot a_4=a_5$ , $32\cdot a_5=a_6$ . Find the number of such sequences $S_n$ such that all of $a_1,a_2,a_3,a_4,a_5,a_6<10^{7}$ .

$\textbf{(A)} ~7 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~76 \qquad\textbf{(D)} ~305 \qquad\textbf{(E)} ~306$

Problem 9

Find the largest possible $m+n$ such that $48!+49!+50!$ is divisible by $2^n5^m$

$\textbf{(A)} ~61 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~63 \qquad\textbf{(D)} ~64 \qquad\textbf{(E)} ~132$

Problem 10

Let $p_n$ be the sequence of positive integers such that all of the elements in the sequence only includes either a combination of digits of $2$ and $0$ , or only $2$ . For example: $22222$ and $20202$ are examples, but not $00000$ . The first several terms of the sequence is $2,20,22,200,202,220,222....$ . The $n$ th term of the sequence is $22222$ . What is $n$ ?

$\textbf{(A)} ~30 \qquad\textbf{(B)} ~31 \qquad\textbf{(C)} ~32 \qquad\textbf{(D)} ~33 \qquad\textbf{(E)} ~34$

Problem 11

Given that the two roots of polynomial $x^{2}-ax+\frac{1}{2}$ are $\sec(n)$ and $\csc(n)$ which $n$ represents an angle. Find $a$

$\textbf{(A)} ~\frac{\sqrt{2}}{2} \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~\frac{\sqrt{5}}{2} \qquad\textbf{(D)} ~\sqrt{3} \qquad\textbf{(E)} ~2$

Problem 12

Find the number of nonempty subsets of $\{1,2,3,4,5,6,7,8,9,10\}$ such that the product of all the numbers in the subset is NOT divisible by $16$ .

$\textbf{(A)} ~341 \qquad\textbf{(B)} ~352 \qquad\textbf{(C)} ~415 \qquad\textbf{(D)} ~416 \qquad\textbf{(E)} ~448$

Problem 13

If $\frac{gcd(a,b)}{a+b}=\frac{a^2+b^2}{lcm(a,b)}$ , find the maximum possible value of $ab$ such that both of $a$ and $b$ are integers and $a+b=0$

$\textbf{(A)} ~2741 \qquad\textbf{(B)} ~2742 \qquad\textbf{(C)} ~2743 \qquad\textbf{(D)} ~2744 \qquad\textbf{(E)} ~2745$

Problem 14

A square with side length $2$ is rotated $45^{\circ}$ about its center. The square would externally swept out $4$ identical small regions as it rotates. Find the area of one of the small region.

$\textbf{(A)} ~\frac{\pi}{8} \qquad\textbf{(B)} ~\frac{\pi+\sqrt{2}-4}{4} \qquad\textbf{(C)} ~\frac{\pi+\sqrt{3}+\sqrt{2}-2}{4} \qquad\textbf{(D)} ~\frac{\pi+2\sqrt{3}-3}{4} \qquad\textbf{(E)} ~\frac{\pi+3-2\sqrt{2}}{4}$

Problem 15

In a circle with a radius of $4$ , four arcs are drawn inside the circle. Smaller circles are inscribed inside the eye-shape diagrams. Let $x$ denote the area of the greatest circle that can be inscribed inside the unshaded region. and let $y$ denote the total area of unshaded region. Find $\frac{x}{y}$

$\textbf{(A)} ~\frac{(3-2\sqrt{2})\pi}{4-\pi} \qquad\textbf{(B)} ~\frac{(16-10\sqrt{2})\pi}{32-8\pi} \qquad\textbf{(C)} ~\frac{(2-\sqrt{2})\pi}{8-2\pi} \qquad\textbf{(D)} ~\frac{\pi}{32-8\pi}\qquad$

$\textbf{(E)}~\frac{\pi}{16-4\pi}$

Problem 21

The exact value of $\sqrt{\sqrt{\sqrt{\sqrt{\frac{-1-\sqrt{3}i}{2}}}}}$ can be expressed as $\frac{\sqrt{a}+\sqrt{b}+i\sqrt{c}-i\sqrt{d}}{e}$ which the fraction is in the most simplified form, $a>b, c>d$ and $a$ , $b$ , $c$ , $d$ and $e$ are not necessary distinct positive integers. Find $2a+b+2c+d+e$

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