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]] | ||
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==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]] | ||
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==Problem 10== | ==Problem 10== | ||
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+ | How many ordered pairs of real numbers <math>(x,y)</math> satisfy the following system of equations? | ||
+ | <cmath>x+3y=3</cmath> | ||
+ | <cmath>\big||x|-|y|\big|=1</cmath> | ||
+ | <math>\textbf{(A) } 1 \qquad | ||
+ | \textbf{(B) } 2 \qquad | ||
+ | \textbf{(C) } 3 \qquad | ||
+ | \textbf{(D) } 4 \qquad | ||
+ | \textbf{(E) } 8 </math> | ||
[[2018 AMC 12A Problems/Problem 10|Solution]] | [[2018 AMC 12A Problems/Problem 10|Solution]] | ||
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==Problem 11== | ==Problem 11== | ||
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+ | 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); | ||
+ | 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> | ||
+ | <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]] | ||
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==Problem 12== | ==Problem 12== | ||
Revision as of 23:03, 8 February 2018
Contents
[hide]- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
Problem 1
A large urn contains balls, of which 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 ? (No red balls are to be removed.)
Problem 2
While exploring a cave, Carl comes across a collection of -pound rocks worth each, -pound rocks worth each, and -pound rocks worth each. There are at least of each size. He can carry at most pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
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.)
Problem 4
Problem 5
What is the sum of all possible values of for which the polynomials and have a root in common?
Problem 6
For positive integers and such that , both the mean and the median of the set are equal to . What is ?
Problem 7
Problem 8
Problem 9
Which of the following describes the largest subset of values of within the closed interval for which for every between and , inclusive?
Problem 10
How many ordered pairs of real numbers satisfy the following system of equations?
Problem 11
A paper triangle with sides of lengths 3,4, and 5 inches, as shown, is folded so that point falls on point . What is the length in inches of the crease?