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Difference between revisions of "2018 AMC 12A Problems"

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[[2018 AMC 12A  Problems/Problem 1|Solution]]
 
[[2018 AMC 12A  Problems/Problem 1|Solution]]
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==Problem 2==
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While exploring a cave, Carl comes across a collection of <math>5</math>-pound rocks worth <math>\$14</math> each, <math>4</math>-pound rocks worth <math>\$11</math> each, and <math>1</math>-pound rocks worth <math>\$2</math> each. There are at least <math>20</math> of each size. He can carry at most <math>18</math> pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
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<math>\textbf{(A) } 48 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 51 \qquad \textbf{(E) } 52 </math>
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[[2018 AMC 12A  Problems/Problem 2|Solution]]
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==Problem 3==
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How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
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<math>\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24</math>
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[[2018 AMC 12A  Problems/Problem 3|Solution]]
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==Problem 4==
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[[2018 AMC 12A  Problems/Problem 4|Solution]]
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==Problem 5==
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What is the sum of all possible values of <math>k</math> for which the polynomials <math>x^2 - 3x + 2</math> and <math>x^2 - 5x + k</math> have a root in common?
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<math>\textbf{(A) }3 \qquad\textbf{(B) }4 \qquad\textbf{(C) }5 \qquad\textbf{(D) }6 \qquad\textbf{(E) }10 \qquad</math>
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[[2018 AMC 12A  Problems/Problem 5|Solution]]
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==Problem 6==
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For positive integers <math>m</math> and <math>n</math> such that <math>m+10<n+1</math>, both the mean and the median of the set <math>\{m, m+4, m+10, n+1, n+2, 2n\}</math> are equal to <math>n</math>. What is <math>m+n</math>?
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<math>\textbf{(A)}20\qquad\textbf{(B)}21\qquad\textbf{(C)}22\qquad\textbf{(D)}23\qquad\textbf{(E)}24</math>
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[[2018 AMC 12A  Problems/Problem 6|Solution]]
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==Problem 7==
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[[2018 AMC 12A  Problems/Problem 17|Solution]]
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==Problem 8==
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[[2018 AMC 12A  Problems/Problem 8|Solution]]
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==Problem 9==
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[[2018 AMC 12A  Problems/Problem 9|Solution]]
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==Problem 10==
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[[2018 AMC 12A  Problems/Problem 10|Solution]]
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==Problem 11==
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[[2018 AMC 12A  Problems/Problem 11|Solution]]
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==Problem 12==
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[[2018 AMC 12A  Problems/Problem 12|Solution]]
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==Problem 13==
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[[2018 AMC 12A  Problems/Problem 13|Solution]]
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==Problem 14==
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[[2018 AMC 12A  Problems/Problem 14|Solution]]
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==Problem 15==
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[[2018 AMC 12A  Problems/Problem 15|Solution]]
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==Problem 16==
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[[2018 AMC 12A  Problems/Problem 16|Solution]]
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==Problem 17==
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[[2018 AMC 12A  Problems/Problem 17|Solution]]
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==Problem 18==
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[[2018 AMC 12A  Problems/Problem 18|Solution]]
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==Problem 19==
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[[2018 AMC 12A  Problems/Problem 19|Solution]]
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==Problem 20==
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[[2018 AMC 12A  Problems/Problem 20|Solution]]
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==Problem 21==
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[[2018 AMC 12A  Problems/Problem 21|Solution]]
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==Problem 22==
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[[2018 AMC 12A  Problems/Problem 22|Solution]]
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==Problem 23==
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[[2018 AMC 12A  Problems/Problem 23|Solution]]
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==Problem 24==
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[[2018 AMC 12A  Problems/Problem 24|Solution]]
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==Problem 25==
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[[2018 AMC 12A  Problems/Problem 25|Solution]]
  
 
==Problem 2==
 
==Problem 2==

Revision as of 22:59, 8 February 2018

Problem 1

A large urn contains $100$ balls, of which $36 \%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \%$? (No red balls are to be removed.)

$\textbf{(A)}\ 28 \qquad\textbf{(B)}\ 32 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 64$

Solution

Problem 2

While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $$14$ each, $4$-pound rocks worth $$11$ each, and $1$-pound rocks worth $$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

$\textbf{(A) } 48 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 51 \qquad \textbf{(E) } 52$

Solution

Problem 3

How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

Solution

Problem 4

Solution

Problem 5

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?

$\textbf{(A) }3 \qquad\textbf{(B) }4 \qquad\textbf{(C) }5 \qquad\textbf{(D) }6 \qquad\textbf{(E) }10 \qquad$

Solution

Problem 6

For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?

$\textbf{(A)}20\qquad\textbf{(B)}21\qquad\textbf{(C)}22\qquad\textbf{(D)}23\qquad\textbf{(E)}24$

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

Problem 2

While exploring a cave, Carl comes across a collection of $5$-pound rocks worth $$14$ each, $4$-pound rocks worth $$11$ each, and $1$-pound rocks worth $$2$ each. There are at least $20$ of each size. He can carry at most $18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

$\textbf{(A) } 48 \qquad \textbf{(B) } 49 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 51 \qquad \textbf{(E) } 52$

Solution

Problem 3

How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)

$\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

Solution

Problem 4

Solution

Problem 5

What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?

$\textbf{(A) }3 \qquad\textbf{(B) }4 \qquad\textbf{(C) }5 \qquad\textbf{(D) }6 \qquad\textbf{(E) }10 \qquad$

Solution

Problem 6

For positive integers $m$ and $n$ such that $m+10<n+1$, both the mean and the median of the set $\{m, m+4, m+10, n+1, n+2, 2n\}$ are equal to $n$. What is $m+n$?

$\textbf{(A)}20\qquad\textbf{(B)}21\qquad\textbf{(C)}22\qquad\textbf{(D)}23\qquad\textbf{(E)}24$

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

Problem 2

Solution

Problem 3

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

Solution

Problem 18

Solution

Problem 19

Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

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