2021 MECC Mock AMC 10

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

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 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 $FA'$ .

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

Problem 10

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 11

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 12

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 13

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 14

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

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 $a+b+c+d+e+f+g+h+j+k+m+n$ .

$\textbf{(A)} ~39 \qquad\textbf{(B)} ~40 \qquad\textbf{(C)} ~41 \qquad\textbf{(D)} ~42 \qquad\textbf{(E)} ~43$

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