2021 GMC 12B

Problem 1

When $30\%$ of $x$ is a positive perfect square integer, what is $min(x)$ such that $x$ is also an integer?

$\textbf{(A)} ~9 \qquad\textbf{(B)} ~12 \qquad\textbf{(C)} ~30 \qquad\textbf{(D)} ~36 \qquad\textbf{(E)} ~120$

Problem 2

In the Great $MICWELL$ civilization, each number digits of a number will be replaced by two times of the digit. For example: $1234$ in $MICWELL$ civilization is $2468$ . Find the number that is equal to $1111+1111$ in $MICWELL$ civilization.

$\textbf{(A)} ~1111 \qquad\textbf{(B)} ~2222 \qquad\textbf{(C)} ~4444 \qquad\textbf{(D)} ~6666 \qquad\textbf{(E)} ~8888$

Problem 3

The expression $\frac{100!+99!}{99!+98!}$ can be written as $\frac{a}{b}$ which $a$ and $b$ are natural numbers and they are relatively prime. Find $a+b$ .

$\textbf{(A)} ~9999 \qquad\textbf{(B)} ~10009 \qquad\textbf{(C)} ~10099 \qquad\textbf{(D)} ~10999 \qquad\textbf{(E)} ~11009$

Problem 4

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

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

Solution

Problem 5

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 $\left \lfloor{x}\right \rfloor$

$\textbf{(A)} ~19 \qquad\textbf{(B)} ~20 \qquad\textbf{(C)} ~25 \qquad\textbf{(D)} ~30 \qquad\textbf{(E)} ~31$

Problem 6

If $x^2+(y-20)^2=400$ and $(x-21)^2+y^2=441$ where $0<x,y$ . The value of $x$ can be expressed as $\frac{a^2+b}{c}$ . Find $\sqrt{b+c}$ .

$\textbf{(A)} ~3\sqrt{5} \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~5\sqrt{2} \qquad\textbf{(D)} ~2\sqrt{13} \qquad\textbf{(E)} ~3\sqrt{6}$

Problem 7

Given a natural number is $12-addictor$ has $12$ divisors and its product of digits is divisible by $12$ , find the number of $12-addictor$ that are less than or equal to $100$ .

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

Problem 8

Bob is standing on the point $(0,0)$ on the Cartesian coordinate plane and he will move to the points $(x+3,y+1)$ or $(x+1,y+3)$ . Find the number of ways he can move such that he eventually reaches $(20,20)$ .

$\textbf{(A)} ~70 \qquad\textbf{(B)} ~92 \qquad\textbf{(C)} ~126 \qquad\textbf{(D)} ~252 \qquad\textbf{(E)} ~253$

Problem 9

What is the remainder when $88!^{{{{(88!-1)}^{(88!-2)}}^{(88!-3)}}^{.....1}}\cdot 1^{2^{3^{4^{.....88!}}}}$ is divided by $89$ ?

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~44 \qquad\textbf{(D)} ~59 \qquad\textbf{(E)} ~88$

Solution

Problem 10

In square $ABCD$ , let $E$ be the midpoint of side $CD$ , and let $F$ and $G$ be reflections of the center of the square across side $BC$ and $AD$ , respectively. Let $H$ be the reflection of $E$ across side $AB$ . Find the ratio between the area of kite $EFGH$ and square $ABCD$ .

$\textbf{(A)} ~2 \qquad\textbf{(B)} ~\frac{5}{2} \qquad\textbf{(C)} ~3 \qquad\textbf{(D)} ~\frac{25}{8} \qquad\textbf{(E)} ~\frac{25}{4}$

Problem 11

How many of the following statement are true for all parallelogram?

Statement 1: All parallelograms are cyclic quadrilaterals.

Statement 2: All cyclic quadrilaterals are parallelograms.

Statement 3: When all of the midpoint are chosen, the resulting figure is a parallelogram.

Statement 4: The length of a diagonal is the product of two adjacent sides.

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

Solution

Problem 12

Let polynomial $f(x)=x^3-3x^2+5x-20$ such that $f(x)$ has three roots $r,s,t$ . Let $q(x)$ be the polynomial with leading coefficient 1 and roots $r+s,s+t,r+t$ . $q(x)$ can be expressed in the form of $x^3+ax^2+bx+c$ . What is $|b|$ ?

$\textbf{(A)} ~13 \qquad\textbf{(B)} ~14 \qquad\textbf{(C)} ~17 \qquad\textbf{(D)} ~20 \qquad\textbf{(E)} ~21$

Problem 13

Let $a$ and $b$ be two legs of a right triangle with hypotenuse $2$ . Find the greatest possible value of $a^3+a^2b+ab^2+b^3$

$\textbf{(A)} ~4\sqrt{2}+2 \qquad\textbf{(B)} ~4\sqrt{3}+4 \qquad\textbf{(C)} ~8\sqrt{2} \qquad\textbf{(D)} ~6\sqrt{3} \qquad\textbf{(E)} ~12$

Solution

Problem 14

Let $ABC$ be an equilateral triangle with side length $2$ , and let $D$ , $E$ and $F$ be the midpoints of side $AB$ , $BC$ , and $AC$ , respectively. Let $G$ be the reflection of $D$ across the point $F$ and let $H$ be the intersection of line segment $AC$ and $EG$ . A circle is constructed with radius $DE$ and center at $D$ . Find the area of pentagon $ABCHG$ that lines outside the circle $D$ .

$\textbf{(A)} ~\frac{3\sqrt{3}}{4}-\frac{\pi}{3} \qquad\textbf{(B)} ~\frac{9\sqrt{3}}{8}-\frac{\pi}{3} \qquad\textbf{(C)} ~\frac{11\sqrt{3}}{8}-\frac{\pi}{3} \qquad\textbf{(D)} ~\frac{3\sqrt{3}}{2}-\frac{\pi}{3} \qquad\textbf{(E)} ~2\sqrt{3}-\frac{\pi}{3}$

Problem 15

Let $n$ be the sum of base $16$ logarithms of the sum of all divisors of $2^{103}$ . Find the last two digits of $n$ .

$\textbf{(A)} ~36 \qquad\textbf{(B)} ~48 \qquad\textbf{(C)} ~56 \qquad\textbf{(D)} ~72 \qquad\textbf{(E)} ~84$

Problem 16

Find the remainder when $3^{18}-1$ is divided by $811$ .

$\textbf{(A)} ~111 \qquad\textbf{(B)} ~142 \qquad\textbf{(C)} ~157 \qquad\textbf{(D)} ~221\qquad\textbf{(E)} ~229$

Problem 17

Let $n=(e^{({\sin(\pi)+\cos(\pi)})\pi})^{e^{\frac{i\pi}{2}}}$ , find the remainder when $\left \lfloor{\sum_{k=0}^{20} 2^{n+k}}\right \rfloor$ is divided by $21$ .

$\textbf{(A)} ~1 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~19\qquad\textbf{(E)} ~20$

Problem 18

In the diagram below, let square with side length $4$ inscribed in the circle. Each new squares are constructed by connecting points that divide the side of the previous square into a ratio of $3:1$ . The new square also forms four right triangular regions. Let $a_n$ be the $n$ th square inside the circle and let $x$ be the sum of the four arcs that are included in the circle but excluded from $a_1$ .

$x+\frac{1}{a_1} \sum_{n=2}^{\infty} \frac{a_n-a_{n+1}}{4}$

can be expressed as $\frac{a}{b}+c\pi-d$ which $gcd(a,b,c,d)=1$ . What is $a+b+c+d$ ?

$\textbf{(A)} ~29 \qquad\textbf{(B)} ~47 \qquad\textbf{(C)} ~50 \qquad\textbf{(D)} ~61\qquad\textbf{(E)} ~69$

Problem 19

What range does $\log_2 y$ lies if $\log_2 xy+\log_{xy} 16=1$ ?

$\textbf{(A)} ~(0,1) \qquad\textbf{(B)} ~(1,2) \qquad\textbf{(C)} ~(2,3) \qquad\textbf{(D)} ~(3,4)\qquad\textbf{(E)} ~No Solution$

Problem 20

Find the remainder when $3^{1624}+7^{1604}$ is divided by $1000$ .

$\textbf{(A)} ~122 \qquad\textbf{(B)} ~322 \qquad\textbf{(C)} ~482 \qquad\textbf{(D)} ~882 \qquad\textbf{(E)} ~922$

Solution

Problem 21

Let $x^2=\sqrt{x^4+\frac{1}{x^2}}-\sqrt{x^4-\frac{1}{x^2}}$ and $y=\sqrt{y^2+\frac{1}{y^4}}-\sqrt{y^2-\frac{1}{y^4}}$ . The value of $xy$ can be expressed as $\frac{a^{\frac{b}{c}}}{d^{-\frac{e}{f}}}$ such that $a,b,c,d,e,f$ are positive integers, $gcd(a,b,d,e)=1$ . Find $a+b+c+d+e+f$ .

$\textbf{(A)} ~23 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~39 \qquad\textbf{(D)} ~41 \qquad\textbf{(E)} ~47$

Problem 22

How many ways are there to choose $4$ balls out of $3$ yellow balls, $2$ black balls and $3$ white balls? (Assume that the balls with same color are indistinguishable.)

$\textbf{(A)} ~10 \qquad\textbf{(B)} ~18 \qquad\textbf{(C)} ~21 \qquad\textbf{(D)} ~35 \qquad\textbf{(E)} ~70$

Problem 23

Let $s=\frac{1+i\sqrt{3}}{2}$ . What is $\sum_{n=2}^{14} s^{n^2-2n+1}\sum_{n=2}^{14} s^{2n-n^2-1}\sum_{n=2}^{14} s^{(-n)^2-2n+1}$

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~\frac{1+i\sqrt{3}}{2} \qquad\textbf{(D)} ~\frac{-1+i\sqrt{3}}{2} \qquad\textbf{(E)} ~\frac{-1-i\sqrt{3}}{2}$

Problem 24

$255\cdot ({26+\sum_{n=0}^{24} \sum_{k=0}^{3+4n} 2^k})$ can be expressed as $a^b+c^d-e$ such that $a,b,c,d,e$ are not necessarily distinct positive integers, $b$ and $d$ are maximized, and $a$ and $c$ and $e$ are minimized. Find $a+b+c+d+e$

$\textbf{(A)} ~220 \qquad\textbf{(B)} ~233 \qquad\textbf{(C)} ~240 \qquad\textbf{(D)} ~245 \qquad\textbf{(E)} ~252$

Problem 25

In an unfair math competition audition, the coach, Mr.George, needs to choose $4$ people outside from $24$ participants, and he will partition the $4$ people into two different groups with each $2$ people and then choose $1$ people to become an temporary coach within the team. Then, the $40$ remaining teammates would either join one of the two teams, or left out and give up. Find the remainder when the total ways of arrangements is divided by $1000$ .

$\textbf{(A)} ~226 \qquad\textbf{(B)} ~244 \qquad\textbf{(C)} ~248 \qquad\textbf{(D)} ~264 \qquad\textbf{(E)} ~352$

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2021 GMC 12B

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25