Difference between revisions of "2011 AMC 12A Problems"
Redcomet46 (talk | contribs) (→Problem 1) |
m (→Problem 21: I think we should use curly brackets because it is less confusing.) |
||
(65 intermediate revisions by 23 users not shown) | |||
Line 1: | Line 1: | ||
+ | {{AMC12 Problems|year=2011|ab=A}} | ||
== Problem 1 == | == Problem 1 == | ||
+ | A cell phone plan costs <math>\$20</math> dollars each month, plus <math>5</math> cents per text message sent, plus <math>10</math> cents for each minute used over <math>30</math> hours. In January Michelle sent <math>100</math> text messages and talked for <math>30.5</math> hours. How much did she have to pay? | ||
+ | <math> | ||
+ | \textbf{(A)}\ 24.00 \qquad | ||
+ | \textbf{(B)}\ 24.50 \qquad | ||
+ | \textbf{(C)}\ 25.50 \qquad | ||
+ | \textbf{(D)}\ 28.00 \qquad | ||
+ | \textbf{(E)}\ 30.00 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 1|Solution]] | [[2011 AMC 12A Problems/Problem 1|Solution]] | ||
== Problem 2 == | == Problem 2 == | ||
+ | There are <math>5</math> coins placed flat on a table according to the figure. What is the order of the coins from top to bottom? | ||
+ | <asy> | ||
+ | size(100); defaultpen(linewidth(.8pt)+fontsize(8pt)); | ||
+ | draw(arc((0,1), 1.2, 25, 214)); | ||
+ | draw(arc((.951,.309), 1.2, 0, 360)); | ||
+ | draw(arc((.588,-.809), 1.2, 132, 370)); | ||
+ | draw(arc((-.588,-.809), 1.2, 75, 300)); | ||
+ | draw(arc((-.951,.309), 1.2, 96, 228)); | ||
+ | label("$A$",(0,1),NW); label("$B$",(-1.1,.309),NW); label("$C$",(.951,.309),E); label("$D$",(-.588,-.809),W); label("$E$",(.588,-.809),S);</asy> | ||
+ | <math> | ||
+ | \textbf{(A)}\ (C, A, E, D, B) \qquad | ||
+ | \textbf{(B)}\ (C, A, D, E, B) \qquad | ||
+ | \textbf{(C)}\ (C, D, E, A, B) \qquad | ||
+ | \textbf{(D)}\ (C, E, A, D, B) \qquad \\ | ||
+ | \textbf{(E)}\ (C, E, D, A, B) </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 2|Solution]] | [[2011 AMC 12A Problems/Problem 2|Solution]] | ||
− | |||
− | <math> | + | == Problem 3 == |
− | + | A small bottle of shampoo can hold <math>35</math> milliliters of shampoo, whereas a large bottle can hold <math>500</math> milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? | |
− | + | ||
<math> | <math> | ||
− | \textbf{( | + | \textbf{(A)}\ 11 \qquad |
− | \textbf{( | + | \textbf{(B)}\ 12 \qquad |
+ | \textbf{(C)}\ 13 \qquad | ||
+ | \textbf{(D)}\ 14 \qquad | ||
+ | \textbf{(E)}\ 15 </math> | ||
− | |||
[[2011 AMC 12A Problems/Problem 3|Solution]] | [[2011 AMC 12A Problems/Problem 3|Solution]] | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
== Problem 4 == | == Problem 4 == | ||
− | + | At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of <math>12</math>, <math>15</math>, and <math>10</math> minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students? | |
− | At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students? | ||
− | + | <math> | |
+ | \textbf{(A)}\ 12 \qquad | ||
\textbf{(B)}\ \frac{37}{3} \qquad | \textbf{(B)}\ \frac{37}{3} \qquad | ||
− | \textbf{(C)}\ | + | \textbf{(C)}\ \frac{88}{7} \qquad |
− | \textbf{(D)}\ 13 | + | \textbf{(D)}\ 13 \qquad |
− | \textbf{(E)}\ 14</math> | + | \textbf{(E)}\ 14 </math> |
+ | |||
+ | [[2011 AMC 12A Problems/Problem 4|Solution]] | ||
== Problem 5 == | == Problem 5 == | ||
− | + | Last summer <math>30\%</math> of the birds living on Town Lake were geese, <math>25\%</math> were swans, <math>10\%</math> were herons, and <math>35\%</math> were ducks. What percent of the birds that were not swans were geese? | |
− | Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese? | ||
− | + | <math> | |
+ | \textbf{(A)}\ 20 \qquad | ||
\textbf{(B)}\ 30 \qquad | \textbf{(B)}\ 30 \qquad | ||
− | \textbf{(C)}\ | + | \textbf{(C)}\ 40 \qquad |
\textbf{(D)}\ 50 \qquad | \textbf{(D)}\ 50 \qquad | ||
− | \textbf{(E)}\ 60 | + | \textbf{(E)}\ 60</math> |
− | </math> | + | |
+ | [[2011 AMC 12A Problems/Problem 5|Solution]] | ||
== Problem 6 == | == Problem 6 == | ||
− | + | The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was <math>61</math> points. How many free throws did they make? | |
− | The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make? | ||
− | + | <math> | |
+ | \textbf{(A)}\ 13 \qquad | ||
\textbf{(B)}\ 14 \qquad | \textbf{(B)}\ 14 \qquad | ||
\textbf{(C)}\ 15 \qquad | \textbf{(C)}\ 15 \qquad | ||
\textbf{(D)}\ 16 \qquad | \textbf{(D)}\ 16 \qquad | ||
− | \textbf{(E)}\ 17 | + | \textbf{(E)}\ 17 </math> |
− | </math> | + | |
+ | [[2011 AMC 12A Problems/Problem 6|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
+ | A majority of the <math>30</math> students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than <math>1</math>. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was <math>17.71</math>. What was the cost of a pencil in cents? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 7 \qquad | ||
+ | \textbf{(B)}\ 11 \qquad | ||
+ | \textbf{(C)}\ 17 \qquad | ||
+ | \textbf{(D)}\ 23 \qquad | ||
+ | \textbf{(E)}\ 77 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 7|Solution]] | [[2011 AMC 12A Problems/Problem 7|Solution]] | ||
== Problem 8 == | == Problem 8 == | ||
+ | In the eight term sequence <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, <math>E</math>, <math>F</math>, <math>G</math>, <math>H</math>, the value of <math>C</math> is <math>5</math> and the sum of any three consecutive terms is <math>30</math>. What is <math>A+H</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 17 \qquad | ||
+ | \textbf{(B)}\ 18 \qquad | ||
+ | \textbf{(C)}\ 25 \qquad | ||
+ | \textbf{(D)}\ 26 \qquad | ||
+ | \textbf{(E)}\ 43 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 8|Solution]] | [[2011 AMC 12A Problems/Problem 8|Solution]] | ||
== Problem 9 == | == Problem 9 == | ||
+ | At a twins and triplets convention, there were <math>9</math> sets of twins and <math>6</math> sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 324 \qquad | ||
+ | \textbf{(B)}\ 441 \qquad | ||
+ | \textbf{(C)}\ 630 \qquad | ||
+ | \textbf{(D)}\ 648 \qquad | ||
+ | \textbf{(E)}\ 882 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 9|Solution]] | [[2011 AMC 12A Problems/Problem 9|Solution]] | ||
== Problem 10 == | == Problem 10 == | ||
+ | A pair of standard <math>6</math>-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ \frac{1}{36} \qquad | ||
+ | \textbf{(B)}\ \frac{1}{12} \qquad | ||
+ | \textbf{(C)}\ \frac{1}{6} \qquad | ||
+ | \textbf{(D)}\ \frac{1}{4} \qquad | ||
+ | \textbf{(E)}\ \frac{5}{18} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 10|Solution]] | [[2011 AMC 12A Problems/Problem 10|Solution]] | ||
== Problem 11 == | == Problem 11 == | ||
+ | Circles <math>A, B,</math> and <math>C</math> each have radius 1. Circles <math>A</math> and <math>B</math> share one point of tangency. Circle <math>C</math> has a point of tangency with the midpoint of <math>\overline{AB}.</math> What is the area inside circle <math>C</math> but outside circle <math>A</math> and circle <math>B?</math> | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 3 - \frac{\pi}{2} \qquad | ||
+ | \textbf{(B)}\ \frac{\pi}{2} \qquad | ||
+ | \textbf{(C)}\ 2 \qquad | ||
+ | \textbf{(D)}\ \frac{3\pi}{4} \qquad | ||
+ | \textbf{(E)}\ 1+\frac{\pi}{2} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 11|Solution]] | [[2011 AMC 12A Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | A power boat and a raft both left dock <math>A</math> on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock <math>B</math> downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock <math>A.</math> How many hours did it take the power boat to go from <math>A</math> to <math>B?</math> | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 3 \qquad | ||
+ | \textbf{(B)}\ 3.5 \qquad | ||
+ | \textbf{(C)}\ 4 \qquad | ||
+ | \textbf{(D)}\ 4.5 \qquad | ||
+ | \textbf{(E)}\ 5 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 12|Solution]] | [[2011 AMC 12A Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | Triangle <math>ABC</math> has side-lengths <math>AB = 12, BC = 24,</math> and <math>AC = 18.</math> The line through the incenter of <math>\triangle ABC</math> parallel to <math>\overline{BC}</math> intersects <math>\overline{AB}</math> at <math>M</math> and <math>\overline{AC}</math> at <math>N.</math> What is the perimeter of <math>\triangle AMN?</math> | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 27 \qquad | ||
+ | \textbf{(B)}\ 30 \qquad | ||
+ | \textbf{(C)}\ 33 \qquad | ||
+ | \textbf{(D)}\ 36 \qquad | ||
+ | \textbf{(E)}\ 42 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 13|Solution]] | [[2011 AMC 12A Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | Suppose <math>a</math> and <math>b</math> are single-digit positive integers chosen independently and at random. What is the probability that the point <math>(a,b)</math> lies above the parabola <math>y=ax^2-bx</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ \frac{11}{81} \qquad | ||
+ | \textbf{(B)}\ \frac{13}{81} \qquad | ||
+ | \textbf{(C)}\ \frac{5}{27} \qquad | ||
+ | \textbf{(D)}\ \frac{17}{81} \qquad | ||
+ | \textbf{(E)}\ \frac{19}{81} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 14|Solution]] | [[2011 AMC 12A Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | The circular base of a hemisphere of radius <math>2</math> rests on the base of a square pyramid of height <math>6</math>. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 3\sqrt{2} \qquad | ||
+ | \textbf{(B)}\ \frac{13}{3} \qquad | ||
+ | \textbf{(C)}\ 4\sqrt{2} \qquad | ||
+ | \textbf{(D)}\ 6 \qquad | ||
+ | \textbf{(E)}\ \frac{13}{2} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 15|Solution]] | [[2011 AMC 12A Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
+ | Each vertex of convex polygon <math>ABCDE</math> is to be assigned a color. There are <math>6</math> colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 2520 \qquad | ||
+ | \textbf{(B)}\ 2880 \qquad | ||
+ | \textbf{(C)}\ 3120 \qquad | ||
+ | \textbf{(D)}\ 3250 \qquad | ||
+ | \textbf{(E)}\ 3750 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 16|Solution]] | [[2011 AMC 12A Problems/Problem 16|Solution]] | ||
== Problem 17 == | == Problem 17 == | ||
+ | Circles with radii <math>1</math>, <math>2</math>, and <math>3</math> are mutually externally tangent. What is the area of the triangle determined by the points of tangency? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ \frac{3}{5} \qquad | ||
+ | \textbf{(B)}\ \frac{4}{5} \qquad | ||
+ | \textbf{(C)}\ 1 \qquad | ||
+ | \textbf{(D)}\ \frac{6}{5} \qquad | ||
+ | \textbf{(E)}\ \frac{4}{3} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 17|Solution]] | [[2011 AMC 12A Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
+ | Suppose that <math>\left|x+y\right|+\left|x-y\right|=2</math>. What is the maximum possible value of <math>x^2-6x+y^2</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 5 \qquad | ||
+ | \textbf{(B)}\ 6 \qquad | ||
+ | \textbf{(C)}\ 7 \qquad | ||
+ | \textbf{(D)}\ 8 \qquad | ||
+ | \textbf{(E)}\ 9 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 18|Solution]] | [[2011 AMC 12A Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
+ | At a competition with <math>N</math> players, the number of players given elite status is equal to <math>2^{1+\lfloor \log_{2} (N-1) \rfloor}-N</math>. Suppose that <math>19</math> players are given elite status. What is the sum of the two smallest possible values of <math>N</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 38 \qquad | ||
+ | \textbf{(B)}\ 90 \qquad | ||
+ | \textbf{(C)}\ 154 \qquad | ||
+ | \textbf{(D)}\ 406 \qquad | ||
+ | \textbf{(E)}\ 1024 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 19|Solution]] | [[2011 AMC 12A Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
+ | Let <math>f(x)=ax^2+bx+c</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are integers. Suppose that <math>f(1)=0</math>, <math>50<f(7)<60</math>, <math>70<f(8)<80</math>, <math>5000k<f(100)<5000(k+1)</math> for some integer <math>k</math>. What is <math>k</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 1 \qquad | ||
+ | \textbf{(B)}\ 2 \qquad | ||
+ | \textbf{(C)}\ 3 \qquad | ||
+ | \textbf{(D)}\ 4 \qquad | ||
+ | \textbf{(E)}\ 5 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 20|Solution]] | [[2011 AMC 12A Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
+ | Let <math>f_{1}(x)=\sqrt{1-x}</math>, and for integers <math>n \geq 2</math>, let <math>f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})</math>. If <math>N</math> is the largest value of <math>n</math> for which the domain of <math>f_{n}</math> is nonempty, the domain of <math>f_{N}</math> is <math>\{ c\}</math>. What is <math>N+c</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ -226 \qquad | ||
+ | \textbf{(B)}\ -144 \qquad | ||
+ | \textbf{(C)}\ -20 \qquad | ||
+ | \textbf{(D)}\ 20 \qquad | ||
+ | \textbf{(E)}\ 144 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 21|Solution]] | [[2011 AMC 12A Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
+ | Let <math>R</math> be a unit square region and <math>n \geq 4</math> an integer. A point <math>X</math> in the interior of <math>R</math> is called ''n-ray partitional'' if there are <math>n</math> rays emanating from <math>X</math> that divide <math>R</math> into <math>n</math> triangles of equal area. How many points are <math>100</math>-ray partitional but not <math>60</math>-ray partitional? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 1500 \qquad | ||
+ | \textbf{(B)}\ 1560 \qquad | ||
+ | \textbf{(C)}\ 2320 \qquad | ||
+ | \textbf{(D)}\ 2480 \qquad | ||
+ | \textbf{(E)}\ 2500 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 22|Solution]] | [[2011 AMC 12A Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
+ | Let <math>f(z)= \frac{z+a}{z+b}</math> and <math>g(z)=f(f(z))</math>, where <math>a</math> and <math>b</math> are complex numbers. Suppose that <math>\left| a \right| = 1</math> and <math>g(g(z))=z</math> for all <math>z</math> for which <math>g(g(z))</math> is defined. What is the difference between the largest and smallest possible values of <math>\left| b \right|</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 0 \qquad | ||
+ | \textbf{(B)}\ \sqrt{2}-1 \qquad | ||
+ | \textbf{(C)}\ \sqrt{3}-1 \qquad | ||
+ | \textbf{(D)}\ 1 \qquad | ||
+ | \textbf{(E)}\ 2 </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 23|Solution]] | [[2011 AMC 12A Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
+ | Consider all quadrilaterals <math>ABCD</math> such that <math>AB=14</math>, <math>BC=9</math>, <math>CD=7</math>, and <math>DA=12</math>. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ \sqrt{15} \qquad | ||
+ | \textbf{(B)}\ \sqrt{21} \qquad | ||
+ | \textbf{(C)}\ 2\sqrt{6} \qquad | ||
+ | \textbf{(D)}\ 5 \qquad | ||
+ | \textbf{(E)}\ 2\sqrt{7} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 24|Solution]] | [[2011 AMC 12A Problems/Problem 24|Solution]] | ||
== Problem 25 == | == Problem 25 == | ||
+ | Triangle <math>ABC</math> has <math>\angle BAC = 60^{\circ}</math>, <math>\angle CBA \leq 90^{\circ}</math>, <math>BC=1</math>, and <math>AC \geq AB</math>. Let <math>H</math>, <math>I</math>, and <math>O</math> be the orthocenter, incenter, and circumcenter of <math>\triangle ABC</math>, respectively. Assume that the area of pentagon <math>BCOIH</math> is the maximum possible. What is <math>\angle CBA</math>? | ||
+ | |||
+ | <math> | ||
+ | \textbf{(A)}\ 60^{\circ} \qquad | ||
+ | \textbf{(B)}\ 72^{\circ} \qquad | ||
+ | \textbf{(C)}\ 75^{\circ} \qquad | ||
+ | \textbf{(D)}\ 80^{\circ} \qquad | ||
+ | \textbf{(E)}\ 90^{\circ} </math> | ||
+ | |||
[[2011 AMC 12A Problems/Problem 25|Solution]] | [[2011 AMC 12A Problems/Problem 25|Solution]] | ||
+ | |||
+ | |||
+ | ==See also== | ||
+ | |||
+ | {{AMC12 box|year=2011|ab=A|before=[[2010 AMC 12B Problems]]|after=[[2011 AMC 12B Problems]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 13:52, 3 July 2021
2011 AMC 12A (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
A cell phone plan costs dollars each month, plus
cents per text message sent, plus
cents for each minute used over
hours. In January Michelle sent
text messages and talked for
hours. How much did she have to pay?
Problem 2
There are coins placed flat on a table according to the figure. What is the order of the coins from top to bottom?
Problem 3
A small bottle of shampoo can hold milliliters of shampoo, whereas a large bottle can hold
milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
Problem 4
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of ,
, and
minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
Problem 5
Last summer of the birds living on Town Lake were geese,
were swans,
were herons, and
were ducks. What percent of the birds that were not swans were geese?
Problem 6
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was points. How many free throws did they make?
Problem 7
A majority of the students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than
. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was
. What was the cost of a pencil in cents?
Problem 8
In the eight term sequence ,
,
,
,
,
,
,
, the value of
is
and the sum of any three consecutive terms is
. What is
?
Problem 9
At a twins and triplets convention, there were sets of twins and
sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
Problem 10
A pair of standard -sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
Problem 11
Circles and
each have radius 1. Circles
and
share one point of tangency. Circle
has a point of tangency with the midpoint of
What is the area inside circle
but outside circle
and circle
Problem 12
A power boat and a raft both left dock on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock
downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock
How many hours did it take the power boat to go from
to
Problem 13
Triangle has side-lengths
and
The line through the incenter of
parallel to
intersects
at
and
at
What is the perimeter of
Problem 14
Suppose and
are single-digit positive integers chosen independently and at random. What is the probability that the point
lies above the parabola
?
Problem 15
The circular base of a hemisphere of radius rests on the base of a square pyramid of height
. The hemisphere is tangent to the other four faces of the pyramid. What is the edge-length of the base of the pyramid?
Problem 16
Each vertex of convex polygon is to be assigned a color. There are
colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
Problem 17
Circles with radii ,
, and
are mutually externally tangent. What is the area of the triangle determined by the points of tangency?
Problem 18
Suppose that . What is the maximum possible value of
?
Problem 19
At a competition with players, the number of players given elite status is equal to
. Suppose that
players are given elite status. What is the sum of the two smallest possible values of
?
Problem 20
Let , where
,
, and
are integers. Suppose that
,
,
,
for some integer
. What is
?
Problem 21
Let , and for integers
, let
. If
is the largest value of
for which the domain of
is nonempty, the domain of
is
. What is
?
Problem 22
Let be a unit square region and
an integer. A point
in the interior of
is called n-ray partitional if there are
rays emanating from
that divide
into
triangles of equal area. How many points are
-ray partitional but not
-ray partitional?
Problem 23
Let and
, where
and
are complex numbers. Suppose that
and
for all
for which
is defined. What is the difference between the largest and smallest possible values of
?
Problem 24
Consider all quadrilaterals such that
,
,
, and
. What is the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral?
Problem 25
Triangle has
,
,
, and
. Let
,
, and
be the orthocenter, incenter, and circumcenter of
, respectively. Assume that the area of pentagon
is the maximum possible. What is
?
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2010 AMC 12B Problems |
Followed by 2011 AMC 12B Problems |
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 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.