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Difference between revisions of "2021 GMC 12B"

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[[2021 GMC 10B Problems/Problem 10|Solution]]
  
 
==Problem 10==
 
==Problem 10==

Revision as of 13:13, 7 March 2022

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

10.png

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

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.

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

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 $a+b+c+d$?

30.png

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

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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