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Difference between revisions of "2021 GMC 10B"

(→‎Problem 24)
(→‎Problem 25)
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==Problem 25==
 
==Problem 25==
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Given the three polynomials:
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Let <math>ABC</math> be an equilateral triangle. Draw a circle such that arc <math>BC</math> inside triangle <math>ABC</math> is a 60 degrees arc. Let <math>O</math> be the center of the circle, and extend side <math>AB</math> and <math>AC</math>. Construct new lines <math>DE</math> and <math>DF</math> such that <math>DE</math> and <math>DF</math> are tangent to circle at point <math>E</math> and <math>F</math>, respectively, and they intersects line <math>AB</math> and <math>AC</math> at points <math>G</math> and <math>H</math>, respectively. <math>OE</math> and <math>OF</math> are perpendicular and they are both radius. Find the area of kite <math>AGDH</math>.
−
 
 
−
1. <math>f(x)=x^{3}+ax^{2}-16x+b</math>
 
−
 
 
−
2. <math>g(x)=x^2+6x+d</math>
 
−
 
 
−
3. <math>h(x)=x^4+zx^3+kx^2+lx+100=0</math>
 
−
 
 
−
For which <math>f(x)</math> has roots <math>r_1, r_2, r_3</math>, <math>g(x)</math> has roots <math>r_1, r_2</math>, and <math>h(x)</math> has roots <math>r_1, r_2, r_3, r_4</math>. Also, <math>a,b,d,z,x,l,r_1,r_2,r_3</math> and <math>r_4</math> are all real numbers. Find <math>z</math>
 
  
 
<math>\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC</math>
 
<math>\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC</math>

Revision as of 18:49, 2 May 2021

Problem 1

What is $5!-4!+3!-2!+1!-0!?$

$\textbf{(A)} ~99 \qquad\textbf{(B)} ~100 \qquad\textbf{(C)} ~101 \qquad\textbf{(D)} ~102 \qquad\textbf{(E)} ~103$

Problem 2

The radius of a circle that has an area of $\frac{\pi}{\sqrt{2}}$ is $r$. Find $r^{2}$

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

Problem 3

What is the sum of the digits of the largest prime that divides $32160$?

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~9 \qquad\textbf{(D)} ~11\qquad\textbf{(E)} ~13$

Problem 4

Ary wants to go to the park at afternoon. he walked to halfway, and he's pretty hungry. Therefore, he searched on his phone and found that exactly on the halfway between his house and the park there is a restaurant. After he eats, he continues to walk, however, he stops at the $\frac{3}{4}$ way between the restaurant and park to take a break. Let $x$ be the length that he need to walk to reach the park, and $y$ be the distance between his house and the park. Find $\frac{x}{y}$

$\textbf{(A)} ~\frac{3}{16} \qquad\textbf{(B)} ~\frac{7}{16} \qquad\textbf{(C)} ~\frac{9}{16} \qquad\textbf{(D)} ~\frac{2}{3}\qquad\textbf{(E)} ~\frac{3}{4}$

Problem 5

An octagon with diagonal length $\sqrt{2}$ and other 4 length $2$ has four given vertices $(-1,0), (-1,2), (3,0)$, $(3,2)$ ,and it partially covers all the four quadrants. Let $a_n$ be the area of the portion of the octagon that lies in the $n$th quadrant. Find $\frac{a_1\cdot a_4}{a_2\cdot a_3}$

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~13\qquad\textbf{(D)} ~17\qquad\textbf{(E)} ~21$

Problem 6

6. 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 7

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 8

A three digit natural number is $Alternative$ if it has two even digits and one odd digit as its number digits. Find the number of alternative positive integers.

$\textbf{(A)} ~400 \qquad\textbf{(B)} ~450 \qquad\textbf{(C)} ~455 \qquad\textbf{(D)} ~543 \qquad\textbf{(E)} ~550$

Problem 9

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 10

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 11

Two real numbers $x,y$ such that $-4\leq x\leq y\leq 4$ are chosen at random. What is the probability that $|x+y|=|x|+|y|$?

$\textbf{(A)} ~\frac{1}{4} \qquad\textbf{(B)} ~\frac{25}{64} \qquad\textbf{(C)} ~\frac{1}{2} \qquad\textbf{(D)} ~\frac{9}{16} \qquad\textbf{(E)} ~\frac{25}{32}$

Problem 12

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 13

Let $f$ be the positive integer and $g(fn)$ be the sum of digits when $f$ is expressed in base $n$. Find $f$ such that $g(f(9)$ has the greatest possible value and $f\leq 2021$.

$\textbf{(A)} ~1376 \qquad\textbf{(B)} ~1457 \qquad\textbf{(C)} ~1458 \qquad\textbf{(D)} ~1548 \qquad\textbf{(E)} ~2021$

Problem 14

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 15

Given that a number is $n-motivator$ if the last 2 digits are the last two digits of $n$ and it is divisible by $n$. How many $20-motivators$ are there below $10,000$? Example: $6020,20$.

$\textbf{(A)} ~108 \qquad\textbf{(B)} ~109 \qquad\textbf{(C)} ~110 \qquad\textbf{(D)} ~111\qquad\textbf{(E)} ~112$

Problem 16

Keel is choosing classes. His chose Algebra 2, US History, Honor Geometry, English, Advanced Spanish, PE, Math olympiad prep, and Honor Science. He can arrange the eight classes in any order of 9 class periods, and the fifth period is always lunch. Find the number of ways Keel can arrange classes such that none of his math classes are the last period before lunch, nor first period after lunch and last period of the day.

$\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC$

Problem 17

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 $D$.

$\textbf{(A)} ~\frac{3\sqrt{3}}{4}-\frac{\pi}{3} \qquad\textbf{(B)} ~\frac{9\sqrt{3}}{8}-\frac{\pi}{3} \qquad\textbf{(C)} ~\frac{11\sqrt{3}}{8}-\frac{\pi}{3} \qquad\textbf{(D)} ~\frac{6\sqrt{3}}{4}-\frac{\pi}{3} \qquad\textbf{(E)} ~2\sqrt{3}-\frac{\pi}{3}$

Problem 18

Let $f(n)$ be the largest possible power of $2$ that divides $n$. Find $f((3^2-3)(4^2-4)(5^2-5)(6^2-6)(7^2-7)(8^2-8)...(99^2-99)(100^2-100))$.

$\textbf{(A)} ~191 \qquad\textbf{(B)} ~192 \qquad\textbf{(C)} ~193 \qquad\textbf{(D)} ~198\qquad\textbf{(E)} ~199$

Problem 19

Sigre won a national competition of Mathocontition in an infinite-populated world and each of the person in such world has exactly 5 best friends. He would disseminate such honor by first day telling all of his best friends, and at the same day each of the best friends would tell 1 of their best friends. Each of the five people would then tell one of their five best friends, and in the same day all of the five best friends of the five people would tell another one of their five best friends and so on. Let $n$ be the number of people that got informed on the end of the $70$th day, find the remainder when $n$ is divided by $9$.

$\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~8$

Problem 20

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 21

How many ordered pairs $(a,b,c,d)$ satisfies $abcd=120$?

$\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC$

Problem 22

James wrote all the positive divisors of $250$ on pieces of paper and randomly choose $5$ pieces with replacement. Find the probability that $2|a^5+b^5+c^5+d^5+e^5$.

$\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC$

Problem 23

In the game of "Infinite war", James need to put 5 different portal: no 1, no 2, no 3, no 4 and no 5 into 3 different boxes $A,B,C$ such that no boxes can be empty, and then he would choose to transfigure himself temporarily into light or shadow to transfer through the portal into three different locations, and then transfigure back into his initial composition. The four locations that he's able to transfer to are Experiment room, Weapon house, Poison gas station and food house. Given that no 1 and no 3 can only go to food house, find the probability that he would go to experiment room by jumping into a portal inside box $C$ and through transfiguration of light.

$\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC$

Problem 24

\[255\cdot ({26+\sum_{n=1}^{24} \sum_{k=0}^{3+4n} 2^k})\] 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

Let $ABC$ be an equilateral triangle. Draw a circle such that arc $BC$ inside triangle $ABC$ is a 60 degrees arc. Let $O$ be the center of the circle, and extend side $AB$ and $AC$. Construct new lines $DE$ and $DF$ such that $DE$ and $DF$ are tangent to circle at point $E$ and $F$, respectively, and they intersects line $AB$ and $AC$ at points $G$ and $H$, respectively. $OE$ and $OF$ are perpendicular and they are both radius. Find the area of kite $AGDH$.

$\textbf{(A)} ~AC \qquad\textbf{(B)} ~AC \qquad\textbf{(C)} ~AC \qquad\textbf{(D)} ~AC \qquad\textbf{(E)} ~AC$

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