Difference between revisions of "2018 AMC 12B Problems"
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+ | {{AMC12 Problems|year=2018|ab=B}} | ||
+ | |||
== Problem 1 == | == Problem 1 == | ||
− | Kate bakes 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? | + | Kate bakes a <math>20</math>-inch by <math>18</math>-inch pan of cornbread. The cornbread is cut into pieces that measure <math>2</math> inches by <math>2</math> inches. How many pieces of cornbread does the pan contain? |
<math>\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360</math> | <math>\textbf{(A) } 90 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 180 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 360</math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
− | Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? | + | Sam drove <math>96</math> miles in <math>90</math> minutes. His average speed during the first <math>30</math> minutes was <math>60</math> mph (miles per hour), and his average speed during the second <math>30</math> minutes was <math>65</math> mph. What was his average speed, in mph, during the last <math>30</math> minutes? |
− | <math>\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68</math> | + | <math> |
+ | \textbf{(A) } 64 \qquad | ||
+ | \textbf{(B) } 65 \qquad | ||
+ | \textbf{(C) } 66 \qquad | ||
+ | \textbf{(D) } 67 \qquad | ||
+ | \textbf{(E) } 68 | ||
+ | </math> | ||
− | + | [[2018 AMC 12B Problems/Problem 2|Solution]] | |
− | + | ==Problem 3== | |
− | <math> | + | A line with slope <math>2</math> intersects a line with slope <math>6</math> at the point <math>(40,30)</math>. What is the distance between the <math>x</math>-intercepts of these two lines? |
− | \textbf{(A) } | + | |
− | \textbf{(B) } | + | <math>\textbf{(A) } 5 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 50</math> |
− | \textbf{(C) } | + | |
− | \textbf{(D) } | + | [[2018 AMC 12B Problems/Problem 3|Solution]] |
− | \textbf{(E) } | ||
− | </math> | ||
== Problem 4 == | == Problem 4 == | ||
− | A | + | A circle has a chord of length <math>10</math>, and the distance from the center of the circle to the chord is <math>5</math>. What is the area of the circle? |
<math> | <math> | ||
− | \textbf{(A) } | + | \textbf{(A) }25\pi \qquad |
− | \textbf{(B) } | + | \textbf{(B) }50\pi \qquad |
− | \textbf{(C) } | + | \textbf{(C) }75\pi \qquad |
− | \textbf{(D) } | + | \textbf{(D) }100\pi \qquad |
− | \textbf{(E) } | + | \textbf{(E) }125\pi \qquad |
</math> | </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
Line 39: | Line 49: | ||
How many subsets of <math>\{2,3,4,5,6,7,8,9\}</math> contain at least one prime number? | How many subsets of <math>\{2,3,4,5,6,7,8,9\}</math> contain at least one prime number? | ||
− | <math>\textbf{(A)} | + | <math> |
+ | \textbf{(A) } 128 \qquad | ||
+ | \textbf{(B) } 192 \qquad | ||
+ | \textbf{(C) } 224 \qquad | ||
+ | \textbf{(D) } 240 \qquad | ||
+ | \textbf{(E) } 256 | ||
+ | </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | Suppose <math>S</math> cans of soda can be purchased from a vending machine for <math>Q</math> quarters. Which of the following expressions describes the number of cans of soda that can be purchased for <math>D</math> dollars, where 1 dollar is worth 4 quarters? | + | Suppose <math>S</math> cans of soda can be purchased from a vending machine for <math>Q</math> quarters. Which of the following expressions describes the number of cans of soda that can be purchased for <math>D</math> dollars, where <math>1</math> dollar is worth <math>4</math> quarters? |
− | <math>\textbf{(A)} \frac{4DQ}{S} \qquad \textbf{(B)} \frac{4DS}{Q} \qquad \textbf{(C)} \frac{4Q}{DS} \qquad \textbf{(D)} \frac{DQ}{4S} \qquad \textbf{(E)} \frac{DS}{4Q}</math> | + | <math>\textbf{(A) } \frac{4DQ}{S} \qquad \textbf{(B) } \frac{4DS}{Q} \qquad \textbf{(C) } \frac{4Q}{DS} \qquad \textbf{(D) } \frac{DQ}{4S} \qquad \textbf{(E) } \frac{DS}{4Q}</math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
− | What is the value of | + | What is the value of <cmath> \log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27? </cmath> |
− | <cmath> \log_37\cdot\log_59\cdot\log_711\cdot\log_913\cdots\log_{21}25\cdot\log_{23}27? </cmath> | + | <math>\textbf{(A) } 3 \qquad \textbf{(B) } 3\log_{7}23 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 10 </math> |
− | + | [[2018 AMC 12B Problems/Problem 7|Solution]] | |
== Problem 8 == | == Problem 8 == | ||
− | Line segment <math>\overline{AB}</math> is a diameter of a circle with <math>AB = 24</math>. Point <math>C</math>, not equal to <math>A</math> or <math>B</math>, lies on the circle. As point <math>C</math> moves around the circle, the centroid (center of mass) of | + | Line segment <math>\overline{AB}</math> is a diameter of a circle with <math>AB = 24</math>. Point <math>C</math>, not equal to <math>A</math> or <math>B</math>, lies on the circle. As point <math>C</math> moves around the circle, the centroid (center of mass) of <math>\triangle ABC</math> traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? |
<math>\textbf{(A) } 25 \qquad \textbf{(B) } 38 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 75 </math> | <math>\textbf{(A) } 25 \qquad \textbf{(B) } 38 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 75 </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
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<cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? </cmath> | <cmath> \sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) ? </cmath> | ||
− | <math> \textbf{(A) }100,100 \qquad | + | <math> \textbf{(A) }100{,}100 \qquad |
− | \textbf{(B) }500,500\qquad | + | \textbf{(B) }500{,}500\qquad |
− | \textbf{(C) }505,000 \qquad | + | \textbf{(C) }505{,}000 \qquad |
− | \textbf{(D) }1,001,000 \qquad | + | \textbf{(D) }1{,}001{,}000 \qquad |
− | \textbf{(E) }1,010,000 \qquad </math> | + | \textbf{(E) }1{,}010{,}000 \qquad </math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
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\textbf{(D) }225 \qquad | \textbf{(D) }225 \qquad | ||
\textbf{(E) }234 \qquad </math> | \textbf{(E) }234 \qquad </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
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<math>\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h </math> | <math>\textbf{(A) } 2(w+h)^2 \qquad \textbf{(B) } \frac{(w+h)^2}2 \qquad \textbf{(C) } 2w^2+4wh \qquad \textbf{(D) } 2w^2 \qquad \textbf{(E) } w^2h </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
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Side <math>\overline{AB}</math> of <math>\triangle ABC</math> has length <math>10</math>. The bisector of angle <math>A</math> meets <math>\overline{BC}</math> at <math>D</math>, and <math>CD = 3</math>. The set of all possible values of <math>AC</math> is an open interval <math>(m,n)</math>. What is <math>m+n</math>? | Side <math>\overline{AB}</math> of <math>\triangle ABC</math> has length <math>10</math>. The bisector of angle <math>A</math> meets <math>\overline{BC}</math> at <math>D</math>, and <math>CD = 3</math>. The set of all possible values of <math>AC</math> is an open interval <math>(m,n)</math>. What is <math>m+n</math>? | ||
− | < | + | <math>\textbf{(A) }16 \qquad |
\textbf{(B) }17 \qquad | \textbf{(B) }17 \qquad | ||
\textbf{(C) }18 \qquad | \textbf{(C) }18 \qquad | ||
\textbf{(D) }19 \qquad | \textbf{(D) }19 \qquad | ||
− | \textbf{(E) }20 \qquad</ | + | \textbf{(E) }20 \qquad</math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
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<math>\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}</math> | <math>\textbf{(A) }100\sqrt{2}\qquad\textbf{(B) }100\sqrt{3}\qquad\textbf{(C) }200\qquad\textbf{(D) }200\sqrt{2}\qquad\textbf{(E) }200\sqrt{3}</math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
− | Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? | + | Joey and Chloe and their daughter Zoe all have the same birthday. Joey is <math>1</math> year older than Chloe, and Zoe is exactly <math>1</math> year old today. Today is the first of the <math>9</math> birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? |
− | <math>\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 </math> | + | <math> |
+ | \textbf{(A) }7 \qquad | ||
+ | \textbf{(B) }8 \qquad | ||
+ | \textbf{(C) }9 \qquad | ||
+ | \textbf{(D) }10 \qquad | ||
+ | \textbf{(E) }11 \qquad | ||
+ | </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
− | How many odd positive 3-digit integers are divisible by 3 but do not contain the digit 3? | + | How many odd positive <math>3</math>-digit integers are divisible by <math>3</math> but do not contain the digit <math>3</math>? |
<math>\textbf{(A) } 96 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 98 \qquad \textbf{(D) } 102 \qquad \textbf{(E) } 120 </math> | <math>\textbf{(A) } 96 \qquad \textbf{(B) } 97 \qquad \textbf{(C) } 98 \qquad \textbf{(D) } 102 \qquad \textbf{(E) } 120 </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
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<math>\textbf{(A) } \frac{1}{6}\sqrt{6} \qquad \textbf{(B) } \frac{3}{2}\sqrt{2}-\frac{3}{2} \qquad \textbf{(C) } 2\sqrt3-3\sqrt2 \qquad \textbf{(D) } \frac{1}{2}\sqrt{2} \qquad \textbf{(E) } \sqrt 3-1</math> | <math>\textbf{(A) } \frac{1}{6}\sqrt{6} \qquad \textbf{(B) } \frac{3}{2}\sqrt{2}-\frac{3}{2} \qquad \textbf{(C) } 2\sqrt3-3\sqrt2 \qquad \textbf{(D) } \frac{1}{2}\sqrt{2} \qquad \textbf{(E) } \sqrt 3-1</math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 16|Solution]] | ||
== Problem 17 == | == Problem 17 == | ||
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<math>\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 </math> | <math>\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
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A function <math>f</math> is defined recursively by <math>f(1)=f(2)=1</math> and <cmath>f(n)=f(n-1)-f(n-2)+n</cmath>for all integers <math>n \geq 3</math>. What is <math>f(2018)</math>? | A function <math>f</math> is defined recursively by <math>f(1)=f(2)=1</math> and <cmath>f(n)=f(n-1)-f(n-2)+n</cmath>for all integers <math>n \geq 3</math>. What is <math>f(2018)</math>? | ||
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 2016 \qquad \textbf{(B) } 2017 \qquad \textbf{(C) } 2018 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2020</math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
− | Mary chose an even <math>4</math>-digit number <math>n</math>. She wrote down all the divisors of <math>n</math> in increasing order from left to right: <math>1,2, | + | Mary chose an even <math>4</math>-digit number <math>n</math>. She wrote down all the divisors of <math>n</math> in increasing order from left to right: <math>1,2,\ldots,\dfrac{n}{2},n</math>. At some moment Mary wrote <math>323</math> as a divisor of <math>n</math>. What is the smallest possible value of the next divisor written to the right of <math>323</math>? |
<math>\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646</math> | <math>\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646</math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
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Let <math>ABCDEF</math> be a regular hexagon with side length <math>1</math>. Denote by <math>X</math>, <math>Y</math>, and <math>Z</math> the midpoints of sides <math>\overline {AB}</math>, <math>\overline{CD}</math>, and <math>\overline{EF}</math>, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of <math>\triangle ACE</math> and <math>\triangle XYZ</math>? | Let <math>ABCDEF</math> be a regular hexagon with side length <math>1</math>. Denote by <math>X</math>, <math>Y</math>, and <math>Z</math> the midpoints of sides <math>\overline {AB}</math>, <math>\overline{CD}</math>, and <math>\overline{EF}</math>, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of <math>\triangle ACE</math> and <math>\triangle XYZ</math>? | ||
− | <math>\textbf{(A)} \frac {3}{8}\sqrt{3} \qquad \textbf{(B)} \frac {7}{16}\sqrt{3} \qquad \textbf{(C)} \frac {15}{32}\sqrt{3} \qquad \textbf{(D)} \frac {1}{2}\sqrt{3} \qquad \textbf{(E)} \frac {9}{16}\sqrt{3} | + | <math>\textbf{(A)}\ \frac {3}{8}\sqrt{3} \qquad \textbf{(B)}\ \frac {7}{16}\sqrt{3} \qquad \textbf{(C)}\ \frac {15}{32}\sqrt{3} \qquad \textbf{(D)}\ \frac {1}{2}\sqrt{3} \qquad \textbf{(E)}\ \frac {9}{16}\sqrt{3} </math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
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In <math>\triangle{ABC}</math> with side lengths <math>AB = 13</math>, <math>AC = 12</math>, and <math>BC = 5</math>, let <math>O</math> and <math>I</math> denote the circumcenter and incenter, respectively. A circle with center <math>M</math> is tangent to the legs <math>AC</math> and <math>BC</math> and to the circumcircle of <math>\triangle{ABC}</math>. What is the area of <math>\triangle{MOI}</math>? | In <math>\triangle{ABC}</math> with side lengths <math>AB = 13</math>, <math>AC = 12</math>, and <math>BC = 5</math>, let <math>O</math> and <math>I</math> denote the circumcenter and incenter, respectively. A circle with center <math>M</math> is tangent to the legs <math>AC</math> and <math>BC</math> and to the circumcircle of <math>\triangle{ABC}</math>. What is the area of <math>\triangle{MOI}</math>? | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ \frac52\qquad\textbf{(B)}\ \frac{11}{4}\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ \frac{13}{4}\qquad\textbf{(E)}\ \frac72</math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
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<math>\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 </math> | <math>\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 </math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
− | Ajay is standing at point <math>A</math> near Pontianak, Indonesia, <math>0^\circ</math> latitude and <math>110^\circ \text{ E}</math> longitude. Billy is standing at point <math>B</math> near Big Baldy Mountain, Idaho, USA, <math>45^\circ \text{ N}</math> latitude and <math>115^\circ \text{ W}</math> longitude. Assume that Earth is a perfect sphere with center <math>C</math> | + | Ajay is standing at point <math>A</math> near Pontianak, Indonesia, <math>0^\circ</math> latitude and <math>110^\circ \text{ E}</math> longitude. Billy is standing at point <math>B</math> near Big Baldy Mountain, Idaho, USA, <math>45^\circ \text{ N}</math> latitude and <math>115^\circ \text{ W}</math> longitude. Assume that Earth is a perfect sphere with center <math>C.</math> What is the degree measure of <math>\angle ACB?</math> |
− | < | + | <math>\textbf{(A) }105 \qquad |
\textbf{(B) }112\frac{1}{2} \qquad | \textbf{(B) }112\frac{1}{2} \qquad | ||
\textbf{(C) }120 \qquad | \textbf{(C) }120 \qquad | ||
\textbf{(D) }135 \qquad | \textbf{(D) }135 \qquad | ||
− | \textbf{(E) }150 \qquad</ | + | \textbf{(E) }150 \qquad</math> |
+ | |||
+ | [[2018 AMC 12B Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
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<math>\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201</math> | <math>\textbf{(A) } 197 \qquad \textbf{(B) } 198 \qquad \textbf{(C) } 199 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 201</math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 24|Solution]] | ||
== Problem 25 == | == Problem 25 == | ||
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<math>\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554</math> | <math>\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554</math> | ||
+ | |||
+ | [[2018 AMC 12B Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See also== | ||
+ | {{AMC12 box|year=2018|ab=B|before=[[2018 AMC 12A Problems]]|after=[[2019 AMC 12A Problems]]}} | ||
+ | {{MAA Notice}} |
Latest revision as of 15:36, 28 October 2021
2018 AMC 12B (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 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
- 26 See also
Problem 1
Kate bakes a -inch by
-inch pan of cornbread. The cornbread is cut into pieces that measure
inches by
inches. How many pieces of cornbread does the pan contain?
Problem 2
Sam drove miles in
minutes. His average speed during the first
minutes was
mph (miles per hour), and his average speed during the second
minutes was
mph. What was his average speed, in mph, during the last
minutes?
Problem 3
A line with slope intersects a line with slope
at the point
. What is the distance between the
-intercepts of these two lines?
Problem 4
A circle has a chord of length , and the distance from the center of the circle to the chord is
. What is the area of the circle?
Problem 5
How many subsets of contain at least one prime number?
Problem 6
Suppose cans of soda can be purchased from a vending machine for
quarters. Which of the following expressions describes the number of cans of soda that can be purchased for
dollars, where
dollar is worth
quarters?
Problem 7
What is the value of
Problem 8
Line segment is a diameter of a circle with
. Point
, not equal to
or
, lies on the circle. As point
moves around the circle, the centroid (center of mass) of
traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
Problem 9
What is
Problem 10
A list of positive integers has a unique mode, which occurs exactly
times. What is the least number of distinct values that can occur in the list?
Problem 11
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point in the figure on the right. The box has base length
and height
. What is the area of the sheet of wrapping paper?
Problem 12
Side of
has length
. The bisector of angle
meets
at
, and
. The set of all possible values of
is an open interval
. What is
?
Problem 13
Square has side length
. Point
lies inside the square so that
and
. The centroids of
,
,
, and
are the vertices of a convex quadrilateral. What is the area of that quadrilateral?
Problem 14
Joey and Chloe and their daughter Zoe all have the same birthday. Joey is year older than Chloe, and Zoe is exactly
year old today. Today is the first of the
birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age?
Problem 15
How many odd positive -digit integers are divisible by
but do not contain the digit
?
Problem 16
The solutions to the equation are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled
and
. What is the least possible area of
Problem 17
Let and
be positive integers such that
and
is as small as possible. What is
?
Problem 18
A function is defined recursively by
and
for all integers
. What is
?
Problem 19
Mary chose an even -digit number
. She wrote down all the divisors of
in increasing order from left to right:
. At some moment Mary wrote
as a divisor of
. What is the smallest possible value of the next divisor written to the right of
?
Problem 20
Let be a regular hexagon with side length
. Denote by
,
, and
the midpoints of sides
,
, and
, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of
and
?
Problem 21
In with side lengths
,
, and
, let
and
denote the circumcenter and incenter, respectively. A circle with center
is tangent to the legs
and
and to the circumcircle of
. What is the area of
?
Problem 22
Consider polynomials of degree at most
, each of whose coefficients is an element of
. How many such polynomials satisfy
?
Problem 23
Ajay is standing at point near Pontianak, Indonesia,
latitude and
longitude. Billy is standing at point
near Big Baldy Mountain, Idaho, USA,
latitude and
longitude. Assume that Earth is a perfect sphere with center
What is the degree measure of
Problem 24
Let denote the greatest integer less than or equal to
. How many real numbers
satisfy the equation
?
Problem 25
Circles ,
, and
each have radius
and are placed in the plane so that each circle is externally tangent to the other two. Points
,
, and
lie on
,
, and
respectively such that
and line
is tangent to
for each
, where
. See the figure below. The area of
can be written in the form
for positive integers
and
. What is
?
See also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by 2018 AMC 12A Problems |
Followed by 2019 AMC 12A Problems |
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All AMC 12 Problems and Solutions |
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