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 $2a+3^b=4^{ab}$ ?

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

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 $y^2+(y-21)^2=441$ 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 $89$ ?

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

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.

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

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 side lengthes of a right triangle with hypotenuse $2$ . Find the greatest possible value of $a^3+a^2b+ab^2+b^3$

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

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

$\textbf{(A)} ~1 \qquad\textbf{(B)} ~11 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~19\qquad\textbf{(E)} ~20$

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 $a_1$ .