Difference between revisions of "2021 GMC 12"

Revision as of 15:50, 26 April 2021

Problem 1

Compute the number of ways to arrange $2$ distinguishable apples and $5$ indistinguishable books such that all five books must be adjacent.

$\textbf{(A) } 12 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 84$

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 wants to choose a seat to sit in a row of ten seats marked $1,2,3,4,5,6,7,8,9,10$ , respectively. Given that Michael595 is an even-preferer and he would sit in any even seat. Find the probability that Lucas would not sit next to Michael595 (because he hates him).

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

Problem 4

Find the sum of all the solutions of $x^{3}+9x-8=k+2x$ , where $k$ can be any number. The roots may be repeated.

$\textbf{(A)} ~-11 \qquad\textbf{(B)} ~-7 \qquad\textbf{(C)} ~0 \qquad\textbf{(D)} ~7 \qquad\textbf{(E)} ~11$

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

Given that $x+y=8$ , $x^2y^2+x^2+y^2=99$ , and $x<y$ , find $x^{16}+y^3+x^2y^4$ .

$\textbf{(A)} ~2741 \qquad\textbf{(B)} ~2742 \qquad\textbf{(C)} ~2743 \qquad\textbf{(D)} ~2744 \qquad\textbf{(E)} ~2745$

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.

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

@@ Line 1: / Line 1: @@
 ==Problem 1==
-. Compute the number of ways to arrange <math>2</math> distinguishable apples and <math>5</math> indistinguishable books such that all five books must be adjacent.
+Compute the number of ways to arrange <math>2</math> distinguishable apples and <math>5</math> indistinguishable books such that all five books must be adjacent.
+<math>\textbf{(A) } 12 \qquad\textbf{(B) } 21 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 84</math>
+==Problem 2==
+In square <math>ABCD</math> with side length <math>8</math>, point <math>E</math> and <math>F</math> are on side <math>BC</math> and <math>CD</math> respectively, such that <math>AE</math> is perpendicular to <math>EF</math> and <math>CF=2</math>. Find the area enclosed by the quadrilateral <math>AECF</math>.
+<math>\textbf{(A) } 20 \qquad\textbf{(B) } 24 \qquad\textbf{(C) } 28 \qquad\textbf{(D) } 32 \qquad\textbf{(E) } 36</math>
+==Problem 3==
+Lucas wants to choose a seat to sit in a row of ten seats marked <math>1,2,3,4,5,6,7,8,9,10</math>, respectively. Given that Michael595 is an even-preferer and he would sit in any even seat. Find the probability that Lucas would not sit next to Michael595 (because he hates him).
+<math>\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}</math>
+==Problem 4==
+Find the sum of all the solutions of <math>x^{3}+9x-8=k+2x</math>, where <math>k</math> can be any number. The roots may be repeated.
+<math>\textbf{(A)} ~-11 \qquad\textbf{(B)} ~-7 \qquad\textbf{(C)} ~0 \qquad\textbf{(D)} ~7 \qquad\textbf{(E)} ~11</math>
+==Problem 5==
+Let <math>a_n</math> be a sequence of positive integers with <math>a_0=1</math> and <math>a_1=2</math> and <math>a_n=a_{n-1}\cdot a_{n+1}</math> for all integers <math>n</math> such that <math>n\geq 1</math>. Find <math>a_{2021}+a_{2023}+a_{2025}</math>.
+<math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~\frac{7}{2} \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~\frac{9}{2} \qquad\textbf{(E)} ~5</math>
+==Problem 6==
+Compute the remainder when the summation
+<cmath>\sum_{k=1}^{14} k^3</cmath>
+is divided by <math>10000</math>.
+<math>\textbf{(A)} ~1025 \qquad\textbf{(B)} ~3025 \qquad\textbf{(C)} ~5025 \qquad\textbf{(D)} ~7025 \qquad\textbf{(E)} ~9025</math>
+==Problem 7==
+<math>\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}+......</math>
+The answer of this problem can be expressed as
+<math>\frac{a\sqrt{b}}{c}+\frac{\sqrt{e}}{f}+\frac{g\sqrt{h}}{j}+\frac{k\sqrt{m}}{n}</math> which <math>a,b,c,d,e,f,g,h,j,k,m,n</math> are not necessarily distinct positive integers, and all of <math>a,b,c,d,e,f,g,h,j,k,m,n</math> are not divisible by any square number. Find <math>a+b+c+d+e+f+g+h+j+k+m+n</math>.
+<math>\textbf{(A)} ~39 \qquad\textbf{(B)} ~40 \qquad\textbf{(C)} ~41 \qquad\textbf{(D)} ~42 \qquad\textbf{(E)} ~43</math>
+==Problem 8==
+Let <math>S_n=a_1,a_2,a_3,a_4,a_5,a_6</math> be a <math>6</math> term sequence of positive integers such that <math>2\cdot a_1=a_2</math>,<math>4\cdot a_2=a_3</math>, <math>8\cdot a_3=a_4</math>, <math>16\cdot a_4=a_5</math>, <math>32\cdot a_5=a_6</math>. Find the number of such sequences <math>S_n</math> such that all of <math>a_1,a_2,a_3,a_4,a_5,a_6<10^{7}</math>.
+<math>\textbf{(A)} ~7 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~76 \qquad\textbf{(D)} ~305 \qquad\textbf{(E)} ~306</math>
+==Problem 9==
+Find the largest possible <math>m+n</math> such that <math>48!+49!+50!</math> is divisible by <math>2^n5^m</math>
+<math>\textbf{(A)} ~61 \qquad\textbf{(B)} ~62 \qquad\textbf{(C)} ~63 \qquad\textbf{(D)} ~64 \qquad\textbf{(E)} ~132</math>
+==Problem 10==
+Let <math>p_n</math> be the sequence of positive integers such that all of the elements in the sequence only includes either a combination of digits of <math>2</math> and <math>0</math>, or only <math>2</math>. For example: <math>22222</math> and <math>20202</math> are examples, but not <math>00000</math>. The first several terms of the sequence is <math>2,20,22,200,202,220,222....</math>. The <math>n</math>th term of the sequence is <math>22222</math>. What is <math>n</math>?
+<math>\textbf{(A)} ~30 \qquad\textbf{(B)} ~31 \qquad\textbf{(C)} ~32 \qquad\textbf{(D)} ~33 \qquad\textbf{(E)} ~34</math>
+==Problem 11==
+Given that the two roots of polynomial <math>x^{2}-ax+\frac{1}{2}</math> are <math>\sec(n)</math> and <math>\csc(n)</math> which <math>n</math> represents an angle. Find <math>a</math>
+<math>\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</math>
+==Problem 12==
+Find the number of nonempty subsets of <math>\{1,2,3,4,5,6,7,8,9,10\}</math> such that the product of all the numbers in the subset is NOT divisible by <math>16</math>.
+<math>\textbf{(A)} ~341 \qquad\textbf{(B)} ~352 \qquad\textbf{(C)} ~415 \qquad\textbf{(D)} ~416 \qquad\textbf{(E)} ~448</math>
+==Problem 13==
+Given that <math>x+y=8</math>, <math>x^2y^2+x^2+y^2=99</math>, and <math>x<y</math>, find <math>x^{16}+y^3+x^2y^4</math>.
+<math>\textbf{(A)} ~2741 \qquad\textbf{(B)} ~2742 \qquad\textbf{(C)} ~2743 \qquad\textbf{(D)} ~2744 \qquad\textbf{(E)} ~2745</math>
+==Problem 14==
+A square with side length <math>2</math> is rotated <math>45^{\circ}</math> about its center. The square would externally swept out <math>4</math> identical small regions as it rotates. Find the area of one of the small region.
+<math>\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}</math>

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