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Difference between revisions of "2021 MECC Mock AMC 10"

(Created page with "==Problem 1== Compute <math>|2^{2}+2^{1}+2^{0}-3^{1}-3^{2}-3^{3}|</math> <math>\textbf{(A)} ~31 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~33 \qquad\textbf{(D)} ~34 \qquad\te...")
 
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==Problem 1== Compute <math>|2^{2}+2^{1}+2^{0}-3^{1}-3^{2}-3^{3}|</math>
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==Problem 1==  
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Compute <math>|2^{2}+2^{1}+2^{0}-3^{1}-3^{2}-3^{3}|</math>
  
 
<math>\textbf{(A)} ~31 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~33 \qquad\textbf{(D)} ~34 \qquad\textbf{(E)} ~35 </math>
 
<math>\textbf{(A)} ~31 \qquad\textbf{(B)} ~32 \qquad\textbf{(C)} ~33 \qquad\textbf{(D)} ~34 \qquad\textbf{(E)} ~35 </math>
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[[2021 April MIMC 10 Problems/Problem 1|Solution]]
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==Problem 2==
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Define a binary operation <math>a\%b=a^{2}+4ab+4b^{2}</math>. Find the number of possible ordered pair of positive integers <math>(a,b)</math> such that <math>a\%b=25</math>.
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<math>\textbf{(A)} ~0 \qquad\textbf{(B)} ~1 \qquad\textbf{(C)} ~2 \qquad\textbf{(D)} ~3 \qquad\textbf{(E)} ~4 </math>
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[[2021 April MIMC 10 Problems/Problem 2|Solution]]
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==Problem 3==
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<math>\sqrt{8+4\sqrt{3}}</math> can be expressed as <math>\sqrt{a}+\sqrt{b}</math>. Find <math>a+b</math>.
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<math>\textbf{(A)} ~6 \qquad\textbf{(B)} ~8 \qquad\textbf{(C)} ~10 \qquad\textbf{(D)} ~12 \qquad\textbf{(E)} ~14 </math>
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[[2021 April MIMC 10 Problems/Problem 3|Solution]]
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==Problem 4==
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Compute the number of ways to arrange 2 distinguishable apples and five indistinguishable books.
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<math>\textbf{(A)} ~21 \qquad\textbf{(B)} ~42 \qquad\textbf{(C)} ~63 \qquad\textbf{(D)} ~84 \qquad\textbf{(E)} ~126 </math>
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[[2021 April MIMC 10 Problems/Problem 4|Solution]]
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==Problem 5==
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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.
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<math>\textbf{(A)} ~24 \qquad\textbf{(B)} ~36 \qquad\textbf{(C)} ~48 \qquad\textbf{(D)} ~72 \qquad\textbf{(E)} ~144 </math>
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[[2021 April MIMC 10 Problems/Problem 5|Solution]]
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==Problem 6==
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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>.

Revision as of 23:41, 20 April 2021

Problem 1

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

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

Solution

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 $a\%b=25$.

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

Solution

Problem 3

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

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

Solution

Problem 4

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

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

Solution

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.

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

Solution

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 $a_{2021}+a_{2023}+a_{2025}$.

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