Difference between revisions of "2021 MECC Mock AMC 10"

(Problem 22)
(Problem 14)
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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>.
 
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>
+
<math>\textbf{(A)} ~ \qquad\textbf{(B)} ~352 \qquad\textbf{(C)} ~415 \qquad\textbf{(D)} ~416 \qquad\textbf{(E)} ~448</math>
  
 
==Problem 15==
 
==Problem 15==

Revision as of 17:37, 22 April 2021

Problem 1

Compute $|2^{2}+2^{1}+2^{0}-3^{1}-3^{2}-3^{3}|$

$\textbf{(A)} ~31 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~33 \qquad\textbf{(D)} ~34 \qquad\textbf{(E)} ~35$

Problem 2

Define a binary operation $a\%b=a^{2}+4ab+4b^{2}$. Find the number of possible ordered pair of positive integers $(a,b)$ such that $a\%b=25$.

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4$

Problem 3

$\sqrt{8+4\sqrt{3}}$ can be expressed as $\sqrt{a}+\sqrt{b}$. Find $a+b$.

$\textbf{(A)} ~6 \qquad\textbf{(B)} ~8 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~12 \qquad\textbf{(E)} ~14$

Problem 4

Compute the number of ways to arrange 2 distinguishable apples and five indistinguishable books.

$\textbf{(A)} ~21 \qquad\textbf{(B)} ~42 \qquad\textbf{(C)} ~63 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126$

Problem 5

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 6

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 7

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 8

Define $x$ the number of real numbers $n$ such that $\frac{(n)(n!)+n(n-1)!}{(n-1)!}$ is a perfect square. Find $x$.

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4$

Problem 9

A unit cube ABCDEFGH is shown below. $A$ is reflected across the plane that contains line $CD$ and line $GH$. Then, it is reflected again across the plane that contains line $BC$ and $FG$. Call the new point $A'$. Find $FA'$.

$\textbf{(A)} ~\sqrt{6} \qquad\textbf{(B)} ~2\sqrt{2} \qquad\textbf{(C)} ~3 \qquad\textbf{(D)} ~2\sqrt{3} \qquad\textbf{(E)} ~4$

4.png

Problem 10

$\frac{\sqrt{2}}{3}+\frac{\sqrt{3}}{4}+\frac{\sqrt{5}}{5}+\frac{\sqrt{6}}{7}+\frac{\sqrt{2}}{6}+\frac{\sqrt{3}}{8}+\frac{\sqrt{5}}{10}+\frac{\sqrt{6}}{14}+\frac{\sqrt{2}}{12}+\frac{\sqrt{3}}{16}+......$

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 11

Find the remainder when $147_{-16}$ expressed in base $10$ is divided by $1000$.

$\textbf{(A)} ~198 \qquad\textbf{(B)} ~199 \qquad\textbf{(C)} ~200 \qquad\textbf{(D)} ~201 \qquad\textbf{(E)} ~202$

Problem 12

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 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.

$\textbf{(A)} ~24 \qquad\textbf{(B)} ~36 \qquad\textbf{(C)} ~48 \qquad\textbf{(D)} ~72 \qquad\textbf{(E)} ~144$

Problem 14

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 $16$.

$\textbf{(A)} ~ \qquad\textbf{(B)} ~352 \qquad\textbf{(C)} ~415 \qquad\textbf{(D)} ~416 \qquad\textbf{(E)} ~448$

Problem 15

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 16

In square $ABCD$, point $E$ and point $F$ are on side $CD$ and $DA$, respectively. Line segments ED and FC are drawn inside the square and they intersects at point $G$. Find the ratio between the sum of the areas of quadrilaterals $FGED+GECB$ and the square $ABCD$.

$\textbf{(A)} ~\frac{1}{2} \qquad\textbf{(B)} ~\frac{13}{24} \qquad\textbf{(C)} ~\frac{7}{12}\qquad\textbf{(D)} ~\frac{2}{3} \qquad\textbf{(E)} ~\frac{3}{4}$

Problem 17

In a circle with a radius of $4$, four arcs are drawn inside the circle. Smaller circles are inscribed inside the eye-shape diagrams. Let $x$ denote the area of the greatest circle that can be inscribed inside the unshaded region. and let $y$ denote the total area of unshaded region. Find $\frac{x}{y}$

18.png

$\textbf{(A)} ~\frac{(3-2\sqrt{2})\pi}{4-\pi} \qquad\textbf{(B)} ~\frac{(16-3\sqrt{2})\pi}{32-8\pi} \qquad\textbf{(C)} ~\frac{(2-\sqrt{2})\pi}{8-2\pi} \qquad\textbf{(D)} ~\frac{\pi}{32-8\pi}\qquad$

$\textbf{(E)}~\frac{\pi}{16-4\pi}$

Problem 18

There exists a polynomial $f(x)=x^2+ax+b$ which $a$ and $b$ are both integers. How many of the following statements are true about all quadratics $f(x)$?


1. For every possible $f(x)$, there are at least $4$ of them such that $|a|=2b$ but two quadratic that $a=-b$ if the such $f(x)$ has all integer roots.


2. For all roots$(r_1)$ of any quadratic in $f(x)$, there exists infinite number of quadratic $q(x)$ such that $Q(r)=r_2$ if and only if $f(x)$ has all real solutions and all terms of $q(x)$ are real numbers.


3. For any quadratics in $f(x)$, there exists at least one quadratics such that they shares exactly one of the roots of $f(x)$ and all of the roots are positive integers.


4. Statement $1,2$


5. Statement $2,3$


6. Statement $1,2,3$

$\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4$

Problem 19

Find the number of positive integers $n$ that are less than or equal to $720$ such that $\frac{n}{720}$ is a four digit terminating decimal $0.abcd$ which $d \neq 0$.

$\textbf{(A)} ~16 \qquad\textbf{(B)} ~40 \qquad\textbf{(C)} ~72 \qquad\textbf{(D)} ~120 \qquad\textbf{(E)} ~240$

Problem 20

In a square with length $2$, 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.

20.png

$\textbf{(A)} ~\frac{4-\sqrt{3}-\pi+\sqrt{6}}{4} \qquad\textbf{(B)} ~\frac{2-\pi+\sqrt{3}}{4} \qquad\textbf{(C)} ~\frac{\pi+6\sqrt{3}}{24} \qquad\textbf{(D)} ~\frac{\pi}{48} \qquad\textbf{(E)} ~\frac{3\sqrt{3}-\pi}{12}$

Problem 21

Josh is playing a game. There are eight cards, each numbered from $2, 4,8,16,32,64,128,256$. Josh would choose $4$ cards arbitrary with replacement. Given that the four numbers are $a,b,c,d$, find the probability that $a\cdot b\cdot c\cdot d$ is a multiple of $8$ but a factor of $512$.

$\textbf{(A)} ~\frac{15}{512} \qquad\textbf{(B)} ~\frac{121}{4096} \qquad\textbf{(C)} ~\frac{31}{1024} \qquad\textbf{(D)} ~\frac{63}{2048} \qquad\textbf{(E)} ~\frac{1}{32}$

Problem 22

There exists an increasing sequence of positive integers $a_1,a_2,a_3,a_4,a_5,......$ such that the value of $\frac{13^{21}+1}{168}$ can be expressed $n^{a_1}+n^{a_2}+n^{a_3}+n^{a_4}+n^{a_5}+......+n^{a_{k-1}}+n^{a_k}$ with a remainder of $x$. which $n$ is a prime number and $a_n$ are integers as small as possible. Find the sum of $2n+a_1+a_2+a_3+a_4+a_5+...+a_k+x$.

$\textbf{(A)} ~123 \qquad\textbf{(B)} ~124 \qquad\textbf{(C)} ~125 \qquad\textbf{(D)} ~126 \qquad\textbf{(E)} ~127$

Problem 23

Given that $(x+y+z)^3=x^3+y^3+z^3+6591-507x-507y-507z+39xy+39yz+39xz-3xyz$. Find $MAX(xyz)-MIN(xyz)$. (*Note that $MAX(xyz)$ is the largest possible product of $xyz$, and $MIN(xyz)$ is the smallest possible product of $xyz$.)

$\textbf{(A)} ~69 \qquad\textbf{(B)} ~80 \qquad\textbf{(C)} ~102 \qquad\textbf{(D)} ~132 \qquad\textbf{(E)} ~156$

Problem 24

Find the sum of last five digits of \[\sum_{k=1}^{200} \sum_{k=4}^{50} {k-1 \choose 3}\].

$\textbf{(A)} ~3 \qquad\textbf{(B)} ~7 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~11 \qquad\textbf{(E)} ~12$

Problem 25

The Terminator is playing a game. He has a deck of card numbered from $1-12$ 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 $9, 6, 4$ but not necessary in this order, and the three green cards are $8,7,5$ 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.

$\textbf{(A)} ~\frac{1}{27} \qquad\textbf{(B)} ~\frac{1}{9} \qquad\textbf{(C)} ~\frac{5}{27} \qquad\textbf{(D)} ~\frac{7}{27} \qquad\textbf{(E)} ~\frac{1}{3}$

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