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Difference between revisions of "2018 AMC 12A Problems"
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[[2018 AMC 12A  Problems/Problem 6|Solution]]
 
[[2018 AMC 12A  Problems/Problem 6|Solution]]
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==Problem 7==
 
==Problem 7==
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[[2018 AMC 12A  Problems/Problem 17|Solution]]
 
[[2018 AMC 12A  Problems/Problem 17|Solution]]
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==Problem 8==
 
==Problem 8==
  
 
[[2018 AMC 12A  Problems/Problem 8|Solution]]
 
[[2018 AMC 12A  Problems/Problem 8|Solution]]
 +
 
==Problem 9==
 
==Problem 9==
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 +
Which of the following describes the largest subset of values of <math>y</math> within the closed interval <math>[0,\pi]</math> for which
 +
<cmath>\sin(x+y)\leq \sin(x)+\sin(y)</cmath>for every <math>x</math> between <math>0</math> and <math>\pi</math>, inclusive?
 +
<cmath>\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi </cmath>
  
 
[[2018 AMC 12A  Problems/Problem 9|Solution]]
 
[[2018 AMC 12A  Problems/Problem 9|Solution]]
 +
 
==Problem 10==
 
==Problem 10==
 +
 +
How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations?
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<cmath>x+3y=3</cmath>
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<cmath>\big||x|-|y|\big|=1</cmath>
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<math>\textbf{(A) } 1 \qquad
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\textbf{(B) } 2 \qquad
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\textbf{(C) } 3 \qquad
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\textbf{(D) } 4 \qquad
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\textbf{(E) } 8 </math>
  
 
[[2018 AMC 12A  Problems/Problem 10|Solution]]
 
[[2018 AMC 12A  Problems/Problem 10|Solution]]
 +
 
==Problem 11==
 
==Problem 11==
 +
 +
A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point <math>A</math> falls on point <math>B</math>. What is the length in inches of the crease?
 +
<asy>
 +
draw((0,0)--(4,0)--(4,3)--(0,0));
 +
label("$A$", (0,0), SW);
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label("$B$", (4,3), NE);
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label("$C$", (4,0), SE);
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label("$4$", (2,0), S);
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label("$3$", (4,1.5), E);
 +
label("$5$", (2,1.5), NW);
 +
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray);
 +
</asy>
 +
<math>\textbf{(A) }  1+\frac12 \sqrt2  \qquad        \textbf{(B) }  \sqrt3  \qquad    \textbf{(C) }  \frac74  \qquad  \textbf{(D) }  \frac{15}{8} \qquad  \textbf{(E) }  2 </math>
  
 
[[2018 AMC 12A  Problems/Problem 11|Solution]]
 
[[2018 AMC 12A  Problems/Problem 11|Solution]]
 +
 
==Problem 12==
 
==Problem 12==
  

Revision as of 00:03, 9 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

Which of the following describes the largest subset of values of $y$ within the closed interval $[0,\pi]$ for which \[\sin(x+y)\leq \sin(x)+\sin(y)\]for every $x$ between $0$ and $\pi$, inclusive? \[\textbf{(A) } y=0 \qquad \textbf{(B) } 0\leq y\leq \frac{\pi}{4} \qquad \textbf{(C) } 0\leq y\leq \frac{\pi}{2} \qquad \textbf{(D) } 0\leq y\leq \frac{3\pi}{4} \qquad \textbf{(E) } 0\leq y\leq \pi\]

Solution

Problem 10

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \[x+3y=3\] \[\big||x|-|y|\big|=1\] $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8$

Solution

Problem 11

A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? [asy] draw((0,0)--(4,0)--(4,3)--(0,0)); label("$A$", (0,0), SW); label("$B$", (4,3), NE); label("$C$", (4,0), SE); label("$4$", (2,0), S); label("$3$", (4,1.5), E); label("$5$", (2,1.5), NW); fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray); [/asy] $\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2$

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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