Difference between revisions of "2021 GMC 12"
(→Problem 1) |
(→Problem 1) |
||
Line 1: | Line 1: | ||
==Problem 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. | |
+ | |||
+ | <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> |
Revision as of 16:50, 26 April 2021
Contents
Problem 1
Compute the number of ways to arrange distinguishable apples and
indistinguishable books such that all five books must be adjacent.
Problem 2
In square with side length
, point
and
are on side
and
respectively, such that
is perpendicular to
and
. Find the area enclosed by the quadrilateral
.
Problem 3
Lucas wants to choose a seat to sit in a row of ten seats marked , 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).
Problem 4
Find the sum of all the solutions of , where
can be any number. The roots may be repeated.
Problem 5
Let be a sequence of positive integers with
and
and
for all integers
such that
. Find
.
Problem 6
Compute the remainder when the summation
is divided by .
Problem 7
The answer of this problem can be expressed as
which
are not necessarily distinct positive integers, and all of
are not divisible by any square number. Find
.
Problem 8
Let be a
term sequence of positive integers such that
,
,
,
,
. Find the number of such sequences
such that all of
.
Problem 9
Find the largest possible such that
is divisible by
Problem 10
Let be the sequence of positive integers such that all of the elements in the sequence only includes either a combination of digits of
and
, or only
. For example:
and
are examples, but not
. The first several terms of the sequence is
. The
th term of the sequence is
. What is
?
Problem 11
Given that the two roots of polynomial are
and
which
represents an angle. Find
Problem 12
Find the number of nonempty subsets of such that the product of all the numbers in the subset is NOT divisible by
.
Problem 13
Given that ,
, and
, find
.
Problem 14
A square with side length is rotated
about its center. The square would externally swept out
identical small regions as it rotates. Find the area of one of the small region.