Difference between revisions of "2021 GMC 12"
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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) } | + | <math>\textbf{(A) } 6 \qquad\textbf{(B) } 12 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 42 \qquad\textbf{(E) } 84</math> |
==Problem 2== | ==Problem 2== | ||
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The number ways are there to permute <math>AAAAABBBBBCCCCC</math> such that all five letters of exactly one of the three letters in <math>A,B,C</math> are all adjacent is <math>n</math>. Find the remainder when <math>n</math> is divided by <math>1000</math>. (Examples: <math>AAAAABCBCBCBCBC</math> and <math>ACCBBBBBACACCAA</math>) | The number ways are there to permute <math>AAAAABBBBBCCCCC</math> such that all five letters of exactly one of the three letters in <math>A,B,C</math> are all adjacent is <math>n</math>. Find the remainder when <math>n</math> is divided by <math>1000</math>. (Examples: <math>AAAAABCBCBCBCBC</math> and <math>ACCBBBBBACACCAA</math>) | ||
− | <math>\textbf{(A)} ~ | + | <math>\textbf{(A)} ~123 \qquad\textbf{(B)} ~840 \qquad\textbf{(C)} ~906 \qquad\textbf{(D)} ~920\qquad\textbf{(E)}~953 </math> |
==Problem 17== | ==Problem 17== | ||
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==Problem 18== | ==Problem 18== | ||
− | Among the <math>n</math> roots of the polynomial <math>x^{24}-1=0</math>, there are m values of <math>x</math> such that <math>x^{ | + | Among the <math>n</math> roots of the polynomial <math>x^{24}-1=0</math>, there are m values of <math>x</math> such that <math>x^{10}+x^{5}+x^3+1</math> is a real number. Find <math>m+n</math>. |
− | <math>\textbf{(A)} ~ | + | <math>\textbf{(A)} ~26 \qquad\textbf{(B)} `30 \qquad\textbf{(C)} ~37 \qquad\textbf{(D)} ~47 \qquad\textbf{(E)} ~48 </math> |
==Problem 19== | ==Problem 19== | ||
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==Problem 20== | ==Problem 20== | ||
− | Given that <math>(x+y+z)^3=x^3+y^3+z^3+6591-507x-507y-507z+39xy+39yz+39xz-3xyz</math>. Find <math>MAX(xyz)-MIN(xyz)</math>. (*Note that <math>MAX(xyz)</math> is the largest possible product of <math>xyz</math>, and <math>MIN(xyz)</math> is the smallest possible product of <math>xyz</math>.) | + | Given that <math>(x+y+z)^3=x^3+y^3+z^3+6591-507x-507y-507z+39xy+39yz+39xz-3xyz</math>. Find <math>MAX(xyz)-MIN(xyz)</math> such that all of <math>x,y,z</math> are nonnegative integers. (*Note that <math>MAX(xyz)</math> is the largest possible product of <math>xyz</math>, and <math>MIN(xyz)</math> is the smallest possible product of <math>xyz</math>.) |
<math>\textbf{(A)} ~69 \qquad\textbf{(B)} ~80 \qquad\textbf{(C)} ~102 \qquad\textbf{(D)} ~132 \qquad\textbf{(E)} ~156</math> | <math>\textbf{(A)} ~69 \qquad\textbf{(B)} ~80 \qquad\textbf{(C)} ~102 \qquad\textbf{(D)} ~132 \qquad\textbf{(E)} ~156</math> | ||
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<math>(z^{2}+z^{4}+z^{8}+z^{16})(z'^{4}+z'^{8}+z'^{16}+z'^{32})(z''^{8}+z''^{16}+z''^{32}+z''^{64})(z'''^{16}+z'''^{32}+z'''^{64}+z'''^{128})</math> | <math>(z^{2}+z^{4}+z^{8}+z^{16})(z'^{4}+z'^{8}+z'^{16}+z'^{32})(z''^{8}+z''^{16}+z''^{32}+z''^{64})(z'''^{16}+z'''^{32}+z'''^{64}+z'''^{128})</math> | ||
− | <math>\textbf{(A)} ~32+32i \qquad\textbf{(B)} ~32 | + | <math>\textbf{(A)} ~32+32i \qquad\textbf{(B)} ~32+16\sqrt{2} \qquad\textbf{(C)} ~32-16\sqrt{2}+16i \qquad\textbf{(D)} ~128 \qquad\textbf{(E)} ~256</math> |
==Problem 23== | ==Problem 23== | ||
− | Find the sum of last five digits of <cmath>\sum_{ | + | Find the sum of last five digits of <cmath>\sum_{s=1}^{200} s\sum_{k=4}^{50} {k-1 \choose 3}</cmath>. |
<math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~11 \qquad\textbf{(E)} ~12</math> | <math>\textbf{(A)} ~3 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~11 \qquad\textbf{(E)} ~12</math> | ||
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==Problem 25== | ==Problem 25== | ||
− | All the solution of <math>(z-2-\sqrt{2})^{24}=4096</math> are vertices of a polygon. The smallest solution that when express in polar | + | All the solution of <math>(z-2-\sqrt{2})^{24}=4096</math> are vertices of a polygon. The smallest solution that when express in polar coordinate, has a <math>y</math> value greater than <math>0</math> but smaller than <math>\frac{\pi}{2}</math> can be expressed as <math>\frac{\sqrt{a}+b+i\sqrt{c}-i+d\sqrt{e}}{f}</math> which <math>a,b,c,d,e,f</math> are all not necessarily distinct positive integers, <math>i</math> in this case represents the imagenary number, <math>i</math>, and the fraction is in the most simplified form. Find <math>a+b+c+d+e+f</math>. |
+ | |||
+ | <math>\textbf{(A)} ~11 \qquad\textbf{(B)} ~13 \qquad\textbf{(C)} ~15 \qquad\textbf{(D)} ~17 \qquad\textbf{(E)} ~19</math> |
Latest revision as of 17:38, 29 April 2021
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
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 and Michael will arbitrarily choose seats to sit in a row of ten seats marked , respectively. Find the probability that Lucas would not sit next to Michael595 AND Michael595 chose an even seat.
Problem 4
If , find
(Note that
means logarithmic function that has a base of
, and
is the natural logarithm.).
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
If , find the maximum possible value of
such that both of
and
are integers and
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.
Problem 15
In a circle with a radius of , four arcs are drawn inside the circle. Smaller circles are inscribed inside the eye-shape diagrams. Let
denote the area of the greatest circle that can be inscribed inside the unshaded region. and let
denote the total area of unshaded region. Find
Problem 16
The number ways are there to permute such that all five letters of exactly one of the three letters in
are all adjacent is
. Find the remainder when
is divided by
. (Examples:
and
)
Problem 17
There exists a polynomial which
and
are both integers. How many of the following statements are true about all quadratics
?
1. For every possible , there are at least
of them such that
but two quadratic that
if the such
has all integer roots.
2. For all roots of any quadratic in
, there exists infinite number of quadratic
such that
if and only if
has all real solutions and all terms of
are real numbers.
3. For any quadratics in , there exists at least one quadratics such that they shares exactly one of the roots of
and all of the roots are positive integers.
4. Statement
5. Statement
6. Statement
Problem 18
Among the roots of the polynomial
, there are m values of
such that
is a real number. Find
.
Problem 19
There exists an increasing sequence of positive integers such that the value of
can be expressed
with a remainder of
. which
is a prime number and
are integers as small as possible. Find the sum of
.
Problem 20
Given that . Find
such that all of
are nonnegative integers. (*Note that
is the largest possible product of
, and
is the smallest possible product of
.)
Problem 21
The exact value of can be expressed as
which the fraction is in the most simplified form,
and
,
,
,
and
are not necessary distinct positive integers. Find
Problem 22
Given that on a complex plane, there is a polar coordinate . The point
is rotated
clockwise to form the new point
,
to form
, and
to form
degrees around the origin. Evaluate:
Problem 23
Find the sum of last five digits of .
Problem 24
The Terminator is playing a game. He has a deck of card numbered from and he also has two dices. There are three green cards, three blue cards, and six orange cards. Terminator already knew that the three blue cards are
but not necessary in this order, and the three green cards are
in some order. Terminator will play this game two rounds by choose one card without replacement and roll the two dice. However, at the beginning of the second round, he MUST choose either by reroll one die , re-choose one card WITH replacement or re-choose the card and choose one die to reroll. He wins if in the first rounds his sum of the card number is equal to the sum of the numbers on the dice and in the second round the sum of numbers on the dice is greater than the card number. Given that terminator is a perfectionist and he always optimize his chance of winning. Find the probability that Terminator will replace only his card(He will replace the card and choose one die to reroll IF the probability of replace one card and reroll one die is equal.
Problem 25
All the solution of are vertices of a polygon. The smallest solution that when express in polar coordinate, has a
value greater than
but smaller than
can be expressed as
which
are all not necessarily distinct positive integers,
in this case represents the imagenary number,
, and the fraction is in the most simplified form. Find
.