Difference between revisions of "2021 MECC Mock AMC 10"
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==Problem 4== | ==Problem 4== | ||
− | Compute the number of ways to arrange 2 distinguishable apples and five indistinguishable books. | + | Compute the number of ways to arrange 2 distinguishable apples and five indistinguishable books such that the five books must be all 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 5== | ==Problem 5== | ||
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<math>\textbf{(A)} ~\sqrt{6} \qquad\textbf{(B)} ~2\sqrt{2} \qquad\textbf{(C)} ~3 \qquad\textbf{(D)} ~2\sqrt{3} \qquad\textbf{(E)} ~4</math> | <math>\textbf{(A)} ~\sqrt{6} \qquad\textbf{(B)} ~2\sqrt{2} \qquad\textbf{(C)} ~3 \qquad\textbf{(D)} ~2\sqrt{3} \qquad\textbf{(E)} ~4</math> | ||
− | [[File:4.png]] | + | [[File:4.png|300px]] |
==Problem 10== | ==Problem 10== | ||
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==Problem 16== | ==Problem 16== | ||
− | + | In square <math>ABCD</math>, point <math>E</math> and point <math>F</math> are the midpoints of side <math>CD</math> and <math>DA</math>, respectively. Line segments AE and FC are drawn inside the square and they intersects at point <math>G</math>. Find the ratio between the sum of the areas of quadrilaterals <math>FGED+GECB</math> and the square <math>ABCD</math>. | |
− | <math>\textbf{(A)} ~1 \qquad\textbf{(B)} ~\ | + | <math>\textbf{(A)} ~\frac{1}{2} \qquad\textbf{(B)} ~\frac{13}{24} \qquad\textbf{(C)} ~\frac{7}{12}\qquad\textbf{(D)} ~\frac{2}{3} \qquad\textbf{(E)} ~\frac{3}{4}</math> |
==Problem 17== | ==Problem 17== | ||
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[[File:18.png]] | [[File:18.png]] | ||
− | <math>\textbf{(A)} ~\frac{(3-2\sqrt{2})\pi}{4-\pi} \qquad\textbf{(B)} ~\frac{16- | + | <math>\textbf{(A)} ~\frac{(3-2\sqrt{2})\pi}{4-\pi} \qquad\textbf{(B)} ~\frac{(16-10\sqrt{2})\pi}{32-8\pi} \qquad\textbf{(C)} ~\frac{(2-\sqrt{2})\pi}{8-2\pi} \qquad\textbf{(D)} ~\frac{\pi}{32-8\pi}\qquad</math> |
<math>\textbf{(E)}~\frac{\pi}{16-4\pi} </math> | <math>\textbf{(E)}~\frac{\pi}{16-4\pi} </math> | ||
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In a square with length <math>2</math>, two overlapping quarter circle centered at two of the vertices of the square is drawn. Find the ratio of the shaded region to the area of the entire square. | In a square with length <math>2</math>, two overlapping quarter circle centered at two of the vertices of the square is drawn. Find the ratio of the shaded region to the area of the entire square. | ||
− | [[File: | + | [[File:20.png|250px]] |
<math>\textbf{(A)} ~\frac{4-\sqrt{3}-\pi+\sqrt{6}}{4} \qquad\textbf{(B)} ~\frac{2-\pi+\sqrt{3}}{4} \qquad\textbf{(C)} ~\frac{\pi+6\sqrt{3}}{24} \qquad\textbf{(D)} ~\frac{\pi}{48} \qquad\textbf{(E)} ~\frac{3\sqrt{3}-\pi}{12}</math> | <math>\textbf{(A)} ~\frac{4-\sqrt{3}-\pi+\sqrt{6}}{4} \qquad\textbf{(B)} ~\frac{2-\pi+\sqrt{3}}{4} \qquad\textbf{(C)} ~\frac{\pi+6\sqrt{3}}{24} \qquad\textbf{(D)} ~\frac{\pi}{48} \qquad\textbf{(E)} ~\frac{3\sqrt{3}-\pi}{12}</math> | ||
==Problem 21== | ==Problem 21== | ||
− | + | Josh is playing a game. There are eight cards, each numbered from <math>2, 4,8,16,32,64,128,256</math>. Josh would choose <math>4</math> cards arbitrary with replacement. Given that the four numbers are <math>a,b,c,d</math>, find the probability that <math>a\cdot b\cdot c\cdot d</math> is a multiple of <math>8</math> but a factor of <math>512</math>. | |
− | <math>\textbf{(A)} ~ | + | <math>\textbf{(A)} ~\frac{15}{512} \qquad\textbf{(B)} ~\frac{121}{4096} \qquad\textbf{(C)} ~\frac{31}{1024} \qquad\textbf{(D)} ~\frac{63}{2048} \qquad\textbf{(E)} ~\frac{1}{32}</math> |
==Problem 22== | ==Problem 22== | ||
− | There exists an increasing sequence of positive integers <math>a_1,a_2,a_3,a_4,a_5,......</math> such that the value of <math>\frac{13^{21}+1}{168}</math> can be expressed <math>n^{a_1}+n^{a_2}+n^{a_3}+n^{a_4}+n^{a_5}+......+n^{a_{k-1}}+n^{a_k} | + | There exists an increasing sequence of positive integers <math>a_1,a_2,a_3,a_4,a_5,......</math> such that the value of <math>\frac{13^{21}+1}{168}</math> can be expressed <math>n^{a_1}+n^{a_2}+n^{a_3}+n^{a_4}+n^{a_5}+......+n^{a_{k-1}}+n^{a_k}</math> with a remainder of <math>x</math>. which <math>n</math> is a prime number and <math>a_n</math> are integers as small as possible. Find the sum of <math>2n+a_1+a_2+a_3+a_4+a_5+...+a_k+x</math>. |
<math>\textbf{(A)} ~123 \qquad\textbf{(B)} ~124 \qquad\textbf{(C)} ~125 \qquad\textbf{(D)} ~126 \qquad\textbf{(E)} ~127</math> | <math>\textbf{(A)} ~123 \qquad\textbf{(B)} ~124 \qquad\textbf{(C)} ~125 \qquad\textbf{(D)} ~126 \qquad\textbf{(E)} ~127</math> | ||
==Problem 23== | ==Problem 23== | ||
− | Given that <math>(x+y+z)^3=x^3+y^3+z^3+6591-507x-507y-507z+ | + | 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> | ||
==Problem 24== | ==Problem 24== | ||
− | 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> | ||
==Problem 25== | ==Problem 25== | ||
− | The Terminator is playing a game. He has a deck of card numbered from <math>1-12</math> 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 <math>9, 6, 4</math> but not necessary in this order, and the three green cards are <math>8,7,5</math> 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 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. | + | The Terminator is playing a game. He has a deck of card numbered from <math>1-12</math> 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 <math>9, 6, 4</math> but not necessary in this order, and the three green cards are <math>8,7,5</math> 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. |
− | <math>\textbf{(A)} ~\frac{1}{27} \qquad\textbf{(B)} ~\frac{1}{9} \qquad\textbf{(C)} ~\frac{5}{27} \qquad\textbf{(D)} ~\frac{ | + | <math>\textbf{(A)} ~\frac{1}{27} \qquad\textbf{(B)} ~\frac{1}{9} \qquad\textbf{(C)} ~\frac{5}{27} \qquad\textbf{(D)} ~\frac{8}{27} \qquad\textbf{(E)} ~\frac{1}{3} </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
Problem 2
Define a binary operation . Find the number of possible ordered pair of positive integers
such that
.
Problem 3
can be expressed as
. Find
.
Problem 4
Compute the number of ways to arrange 2 distinguishable apples and five indistinguishable books such that the five books must be all adjacent.
Problem 5
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 6
Let be a sequence of positive integers with
and
and
for all integers
such that
. Find
.
Problem 7
Find the sum of all the solutions of , where
can be any number. The roots may be repeated.
Problem 8
Define the number of real numbers
such that
is a perfect square. Find
.
Problem 9
A unit cube ABCDEFGH is shown below. is reflected across the plane that contains line
and line
. Then, it is reflected again across the plane that contains line
and
. Call the new point
. Find
.
Problem 10
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 11
Find the remainder when expressed in base
is divided by
.
Problem 12
Let be a
term sequence of positive integers such that
,
,
,
,
. Find the number of such sequences
such that all of
.
Problem 13
Galieo, Neton, Timiel, Fidgety and Jay are participants of a game in soccer. Their coach, Mr.Tom, will allocate them into two INDISTINGUISHABLE groups for practice purpose(People in the teams are interchangable). Given that the coach will not put Galieo and Timiel into the same team because they just had a fight. Find the number of ways the coach can put them into two such groups.
Problem 14
Find the number of nonempty subsets of such that the product of all the numbers in the subset is NOT divisible by
.
Problem 15
Given that ,
, and
, find
.
Problem 16
In square , point
and point
are the midpoints of side
and
, respectively. Line segments AE and FC are drawn inside the square and they intersects at point
. Find the ratio between the sum of the areas of quadrilaterals
and the square
.
Problem 17
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 18
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 19
Find the number of positive integers that are less than or equal to
such that
is a four digit terminating decimal
which
.
Problem 20
In a square with length , two overlapping quarter circle centered at two of the vertices of the square is drawn. Find the ratio of the shaded region to the area of the entire square.
Problem 21
Josh is playing a game. There are eight cards, each numbered from . Josh would choose
cards arbitrary with replacement. Given that the four numbers are
, find the probability that
is a multiple of
but a factor of
.
Problem 22
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 23
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 24
Find the sum of last five digits of .
Problem 25
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.