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.

$\textbf{(A) } 6 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 84$

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 and Michael will arbitrarily choose seats to sit in a row of ten seats marked $1,2,3,4,5,6,7,8,9,10$ , respectively. Find the probability that Lucas would not sit next to Michael595 AND Michael595 chose an even seat.

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

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 $a+b=0$

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

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 16

The number ways are there to permute $AAAAABBBBBCCCCC$ such that all five letters of exactly one of the three letters in $A,B,C$ are all adjacent is $n$ . Find the remainder when $n$ is divided by $1000$ . (Examples: $AAAAABCBCBCBCBC$ and $ACCBBBBBACACCAA$ )

$\textbf{(A)} ~123 \qquad\textbf{(B)} ~840 \qquad\textbf{(C)} ~906 \qquad\textbf{(D)} ~920\qquad\textbf{(E)}~953$

Problem 17

There exists a polynomial $f(x)=x^2+ax+b$ which $a$ and $b$ are both integers. How many of the following statements are true about all quadratics $f(x)$ ?

1. For every possible $f(x)$ , there are at least $4$ of them such that $|a|=2b$ but two quadratic that $a=-b$ if the such $f(x)$ has all integer roots.

2. For all roots $(r_1)$ of any quadratic in $f(x)$ , there exists infinite number of quadratic $q(x)$ such that $Q(r)=r_2$ if and only if $f(x)$ has all real solutions and all terms of $q(x)$ are real numbers.

3. For any quadratics in $f(x)$ , there exists at least one quadratics such that they shares exactly one of the roots of $f(x)$ and all of the roots are positive integers.

4. Statement $1,2$

5. Statement $2,3$

6. Statement $1,2,3$

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

Problem 18

Among the $n$ roots of the polynomial $x^{24}-1=0$ , there are m values of $x$ such that $x^{10}+x^{5}+x^3+1$ is a real number. Find $m+n$ .

$\textbf{(A)} ~26 \qquad\textbf{(B)} `30 \qquad\textbf{(C)} ~37 \qquad\textbf{(D)} ~47 \qquad\textbf{(E)} ~48$

Problem 19

There exists an increasing sequence of positive integers $a_1,a_2,a_3,a_4,a_5,......$ such that the value of $\frac{13^{21}+1}{168}$ can be expressed $n^{a_1}+n^{a_2}+n^{a_3}+n^{a_4}+n^{a_5}+......+n^{a_{k-1}}+n^{a_k}$ with a remainder of $x$ . which $n$ is a prime number and $a_n$ are integers as small as possible. Find the sum of $2n+a_1+a_2+a_3+a_4+a_5+...+a_k+x$ .

$\textbf{(A)} ~123 \qquad\textbf{(B)} ~124 \qquad\textbf{(C)} ~125 \qquad\textbf{(D)} ~126 \qquad\textbf{(E)} ~127$

Problem 20

Given that $(x+y+z)^3=x^3+y^3+z^3+6591-507x-507y-507z+39xy+39yz+39xz-3xyz$ . Find $MAX(xyz)-MIN(xyz)$ such that all of $x,y,z$ are nonnegative integers. (*Note that $MAX(xyz)$ is the largest possible product of $xyz$ , and $MIN(xyz)$ is the smallest possible product of $xyz$ .)

$\textbf{(A)} ~69 \qquad\textbf{(B)} ~80 \qquad\textbf{(C)} ~102 \qquad\textbf{(D)} ~132 \qquad\textbf{(E)} ~156$

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 $2a+b+2c+d+e$

$\textbf{(A)} ~32 \qquad\textbf{(B)} ~40 \qquad\textbf{(C)} ~48 \qquad\textbf{(D)} ~56 \qquad\textbf{(E)} ~64$

Problem 22

Given that on a complex plane, there is a polar coordinate $z=(1,\frac{\pi}{4})$ . The point $z$ is rotated $90^{\circ}$ clockwise to form the new point $z'$ , $180^{\circ}$ to form $z''$ , and $270^{\circ}$ to form $z'''$ degrees around the origin. Evaluate:

$(z^{2}+z^{4}+z^{8}+z^{16})(z'^{4}+z'^{8}+z'^{16}+z'^{32})(z''^{8}+z''^{16}+z''^{32}+z''^{64})(z'''^{16}+z'''^{32}+z'''^{64}+z'''^{128})$

$\textbf{(A)} ~32+32i \qquad\textbf{(B)} ~32+16\sqrt{2} \qquad\textbf{(C)} ~32-16\sqrt{2}+16i \qquad\textbf{(D)} ~128 \qquad\textbf{(E)} ~256$

Problem 23

Find the sum of last five digits of $\sum_{s=1}^{200} s\sum_{k=4}^{50} {k-1 \choose 3}$ .

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~11 \qquad\textbf{(E)} ~12$

Problem 24

The Terminator is playing a game. He has a deck of card numbered from $1-12$ and he also has two dices. There are three green cards, three blue cards, and six orange cards. Terminator already knew that the three blue cards are $9, 6, 4$ but not necessary in this order, and the three green cards are $8,7,5$ in some order. Terminator will play this game two rounds by choose one card without replacement and roll the two dice. However, at the beginning of the second round, he MUST choose either by reroll one die , re-choose one card WITH replacement or re-choose the card and choose one die to reroll. He wins if in the first rounds his sum of the card number is equal to the sum of the numbers on the dice and in the second round the sum of numbers on the dice is greater than the card number. Given that terminator is a perfectionist and he always optimize his chance of winning. Find the probability that Terminator will replace only his card(He will replace the card and choose one die to reroll IF the probability of replace one card and reroll one die is equal.

$\textbf{(A)} ~\frac{1}{27} \qquad\textbf{(B)} ~\frac{1}{9} \qquad\textbf{(C)} ~\frac{5}{27} \qquad\textbf{(D)} ~\frac{8}{27} \qquad\textbf{(E)} ~\frac{1}{3}$

Problem 25

All the solution of $(z-2-\sqrt{2})^{24}=4096$ are vertices of a polygon. The smallest solution that when express in polar coordinate, has a $y$ value greater than $0$ but smaller than $\frac{\pi}{2}$ can be expressed as $\frac{\sqrt{a}+b+i\sqrt{c}-i+d\sqrt{e}}{f}$ which $a,b,c,d,e,f$ are all not necessarily distinct positive integers, $i$ in this case represents the imagenary number, $i$ , and the fraction is in the most simplified form. Find $a+b+c+d+e+f$ .