Difference between revisions of "2018 AMC 12A Problems"
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==Problem 12== | ==Problem 12== | ||
+ | |||
+ | Let <math>S</math> be a set of 6 integers taken from <math>\{1,2,\dots,12\}</math> with the property that if <math>a</math> and <math>b</math> are elements of <math>S</math> with <math>a<b</math>, then <math>b</math> is not a multiple of <math>a</math>. What is the least possible value of an element in <math>S?</math> | ||
+ | |||
+ | <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 7</math> | ||
[[2018 AMC 12A Problems/Problem 12|Solution]] | [[2018 AMC 12A Problems/Problem 12|Solution]] | ||
==Problem 13== | ==Problem 13== | ||
+ | |||
+ | How many nonnegative integers can be written in the form <cmath>a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,</cmath> | ||
+ | where <math>a_i\in \{-1,0,1\}</math> for <math>0\le i \le 7</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 512 \qquad | ||
+ | \textbf{(B) } 729 \qquad | ||
+ | \textbf{(C) } 1094 \qquad | ||
+ | \textbf{(D) } 3281 \qquad | ||
+ | \textbf{(E) } 59,048 </math> | ||
[[2018 AMC 12A Problems/Problem 13|Solution]] | [[2018 AMC 12A Problems/Problem 13|Solution]] | ||
==Problem 14== | ==Problem 14== | ||
+ | |||
+ | The solutions to the equation <math>\log_{3x} 4 = \log_{2x} 8</math>, where <math>x</math> is a positive real number other than <math>\tfrac{1}{3}</math> or <math>\tfrac{1}{2}</math>, can be written as <math>\tfrac {p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p + q</math>? | ||
+ | |||
+ | <math>\textbf{(A) } 5 \qquad | ||
+ | \textbf{(B) } 13 \qquad | ||
+ | \textbf{(C) } 17 \qquad | ||
+ | \textbf{(D) } 31 \qquad | ||
+ | \textbf{(E) } 35 </math> | ||
[[2018 AMC 12A Problems/Problem 14|Solution]] | [[2018 AMC 12A Problems/Problem 14|Solution]] | ||
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[[2018 AMC 12A Problems/Problem 15|Solution]] | [[2018 AMC 12A Problems/Problem 15|Solution]] | ||
==Problem 16== | ==Problem 16== | ||
+ | |||
+ | Which of the following describes the set of values of <math>a</math> for which the curves <math>x^2+y^2=a^2</math> and <math>y=x^2-a</math> in the real <math>xy</math>-plane intersect at exactly <math>3</math> points? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }a=\frac14 \qquad | ||
+ | \textbf{(B) }\frac14 < a < \frac12 \qquad | ||
+ | \textbf{(C) }a>\frac14 \qquad | ||
+ | \textbf{(D) }a=\frac12 \qquad | ||
+ | \textbf{(E) }a>\frac12 \qquad | ||
+ | </math> | ||
[[2018 AMC 12A Problems/Problem 16|Solution]] | [[2018 AMC 12A Problems/Problem 16|Solution]] | ||
==Problem 17== | ==Problem 17== | ||
+ | |||
+ | Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square <math>S</math> so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from <math>S</math> to the hypotenuse is 2 units. What fraction of the field is planted? | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(4,0)--(0,3)--(0,0)); | ||
+ | draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0)); | ||
+ | fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray); | ||
+ | label("$4$", (2,0), N); | ||
+ | label("$3$", (0,1.5), E); | ||
+ | label("$2$", (.8,1), E); | ||
+ | label("$S$", (0,0), NE); | ||
+ | draw((0.3,0.3)--(1.4,1.9), dashed); | ||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75} </math> | ||
[[2018 AMC 12A Problems/Problem 17|Solution]] | [[2018 AMC 12A Problems/Problem 17|Solution]] | ||
==Problem 18== | ==Problem 18== | ||
+ | |||
+ | Triangle <math>ABC</math> with <math>AB=50</math> and <math>AC=10</math> has area <math>120</math>. Let <math>D</math> be the midpoint of <math>\overline{AB}</math>, and let <math>E</math> be the midpoint of <math>\overline{AC}</math>. The angle bisector of <math>\angle BAC</math> intersects <math>\overline{DE}</math> and <math>\overline{BC}</math> at <math>F</math> and <math>G</math>, respectively. What is the area of quadrilateral <math>FDBG</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }60 \qquad | ||
+ | \textbf{(B) }65 \qquad | ||
+ | \textbf{(C) }70 \qquad | ||
+ | \textbf{(D) }75 \qquad | ||
+ | \textbf{(E) }80 \qquad | ||
+ | </math> | ||
[[2018 AMC 12A Problems/Problem 18|Solution]] | [[2018 AMC 12A Problems/Problem 18|Solution]] | ||
==Problem 19== | ==Problem 19== | ||
+ | |||
+ | Let <math>A</math> be the set of positive integers that have no prime factors other than <math>2</math>, <math>3</math>, or <math>5</math>. The infinite sum <cmath>\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots</cmath>of the reciprocals of the elements of <math>A</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n</math>? | ||
+ | |||
+ | <math>\textbf{(A)} \text{ 16} \qquad \textbf{(B)} \text{ 17} \qquad \textbf{(C)} \text{ 19} \qquad \textbf{(D)} \text{ 23} \qquad \textbf{(E)} \text{ 36}</math> | ||
[[2018 AMC 12A Problems/Problem 19|Solution]] | [[2018 AMC 12A Problems/Problem 19|Solution]] | ||
+ | |||
==Problem 20== | ==Problem 20== | ||
+ | |||
+ | Triangle <math>ABC</math> is an isosceles right triangle with <math>AB=AC=3</math>. Let <math>M</math> be the midpoint of hypotenuse <math>\overline{BC}</math>. Points <math>I</math> and <math>E</math> lie on sides <math>\overline{AC}</math> and <math>\overline{AB}</math>, respectively, so that <math>AI>AE</math> and <math>AIME</math> is a cyclic quadrilateral. Given that triangle <math>EMI</math> has area <math>2</math>, the length <math>CI</math> can be written as <math>\frac{a-\sqrt{b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers and <math>b</math> is not divisible by the square of any prime. What is the value of <math>a+b+c</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A) }9 \qquad | ||
+ | \textbf{(B) }10 \qquad | ||
+ | \textbf{(C) }11 \qquad | ||
+ | \textbf{(D) }12 \qquad | ||
+ | \textbf{(E) }13 \qquad | ||
+ | </math> | ||
[[2018 AMC 12A Problems/Problem 20|Solution]] | [[2018 AMC 12A Problems/Problem 20|Solution]] | ||
+ | |||
==Problem 21== | ==Problem 21== | ||
+ | |||
[[2018 AMC 12A Problems/Problem 21|Solution]] | [[2018 AMC 12A Problems/Problem 21|Solution]] | ||
+ | |||
==Problem 22== | ==Problem 22== | ||
[[2018 AMC 12A Problems/Problem 22|Solution]] | [[2018 AMC 12A Problems/Problem 22|Solution]] | ||
+ | |||
==Problem 23== | ==Problem 23== | ||
[[2018 AMC 12A Problems/Problem 23|Solution]] | [[2018 AMC 12A Problems/Problem 23|Solution]] | ||
+ | |||
==Problem 24== | ==Problem 24== | ||
[[2018 AMC 12A Problems/Problem 24|Solution]] | [[2018 AMC 12A Problems/Problem 24|Solution]] | ||
+ | |||
==Problem 25== | ==Problem 25== | ||
[[2018 AMC 12A Problems/Problem 25|Solution]] | [[2018 AMC 12A Problems/Problem 25|Solution]] |
Revision as of 23:07, 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?
Problem 12
Let be a set of 6 integers taken from with the property that if and are elements of with , then is not a multiple of . What is the least possible value of an element in
Problem 13
How many nonnegative integers can be written in the form where for ?
Problem 14
The solutions to the equation , where is a positive real number other than or , can be written as where and are relatively prime positive integers. What is ?
Problem 15
Problem 16
Which of the following describes the set of values of for which the curves and in the real -plane intersect at exactly points?
Problem 17
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from to the hypotenuse is 2 units. What fraction of the field is planted?
Problem 18
Triangle with and has area . Let be the midpoint of , and let be the midpoint of . The angle bisector of intersects and at and , respectively. What is the area of quadrilateral ?
Problem 19
Let be the set of positive integers that have no prime factors other than , , or . The infinite sum of the reciprocals of the elements of can be expressed as , where and are relatively prime positive integers. What is ?
Problem 20
Triangle is an isosceles right triangle with . Let be the midpoint of hypotenuse . Points and lie on sides and , respectively, so that and is a cyclic quadrilateral. Given that triangle has area , the length can be written as , where , , and are positive integers and is not divisible by the square of any prime. What is the value of ?