View source for 2016 AMC 8 Problems

{{AMC8 Problems|year=2016|}}
==Problem 1==

The longest professional tennis match lasted a total of 11 hours and 5 minutes. How many minutes is that?

<math>\textbf{(A) } 605 \qquad\textbf{(B) } 655\qquad\textbf{(C) } 665\qquad\textbf{(D) } 1005\qquad \textbf{(E) } 1105</math>

[[2016 AMC 8 Problems/Problem 1|Solution
]]

==Problem 2==

In rectangle <math>ABCD</math>, <math>AB=6</math> and <math>AD=8</math>.  Point <math>M</math> is the midpoint of <math>\overline{AD}</math>.  What is the area of <math>\triangle AMC</math>?

<asy>draw((0,4)--(0,0)--(6,0)--(6,8)--(0,8)--(0,4)--(6,8)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (6, 0), SE);
label("$C$", (6,8), NE);
label("$D$", (0, 8), NW);
label("$M$", (0, 4), W);
label("$4$", (0, 2), W);
label("$6$", (3, 0), S);</asy>

<math>\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24</math>

[[2016 AMC 8 Problems/Problem 2|Solution
]]

==Problem 3==

Four students take an exam. Three of their scores are <math>70, 80,</math> and <math>90</math>. If the average of their four scores is <math>70</math>, then what is the remaining score?

<math>\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70</math>

[[2016 AMC 8 Problems/Problem 3|Solution
]]

==Problem 4==

When Cheenu was a boy, he could run <math>15</math> miles in <math>3</math> hours and <math>30</math> minutes. As an old man, he can now walk <math>10</math> miles in <math>4</math> hours. How many minutes longer does it take for him to travel a mile now compared to when he was a boy?

<math>\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30</math>

[[2016 AMC 8 Problems/Problem 4|Solution
]]

==Problem 5==

The number <math>N</math> is a two-digit number.

• When <math>N</math> is divided by <math>9</math>, the remainder is <math>1</math>.

• When <math>N</math> is divided by <math>10</math>, the remainder is <math>3</math>.

What is the remainder when <math>N</math> is divided by <math>11</math>?


<math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7</math>

[[2016 AMC 8 Problems/Problem 5|Solution
]]

==Problem 6==
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?
<asy>
unitsize(0.9cm);
draw((-0.5,0)--(10,0), linewidth(1.5));
draw((-0.5,1)--(10,1));
draw((-0.5,2)--(10,2));
draw((-0.5,3)--(10,3));
draw((-0.5,4)--(10,4));
draw((-0.5,5)--(10,5));
draw((-0.5,6)--(10,6));
draw((-0.5,7)--(10,7));
label("frequency",(-0.5,8));
label("0", (-1, 0));
label("1", (-1, 1));
label("2", (-1, 2));
label("3", (-1, 3));
label("4", (-1, 4));
label("5", (-1, 5));
label("6", (-1, 6));
label("7", (-1, 7));
filldraw((0,0)--(0,7)--(1,7)--(1,0)--cycle, black);
filldraw((2,0)--(2,3)--(3,3)--(3,0)--cycle, black);
filldraw((4,0)--(4,1)--(5,1)--(5,0)--cycle, black);
filldraw((6,0)--(6,4)--(7,4)--(7,0)--cycle, black);
filldraw((8,0)--(8,4)--(9,4)--(9,0)--cycle, black);
label("3", (0.5, -0.5));
label("4", (2.5, -0.5));
label("5", (4.5, -0.5));
label("6", (6.5, -0.5));
label("7", (8.5, -0.5));
label("name length", (4.5, -1));
</asy>

<math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7</math>

[[2016 AMC 8 Problems/Problem 6|Solution
]]

==Problem 7==

Which of the following numbers is not a perfect square?

<math>\textbf{(A) }1^{2016}\qquad\textbf{(B) }2^{2017}\qquad\textbf{(C) }3^{2018}\qquad\textbf{(D) }4^{2019}\qquad \textbf{(E) }5^{2020}</math>

[[2016 AMC 8 Problems/Problem 7|Solution
]]

==Problem 8==

Find the value of the expression
<cmath>100-98+96-94+92-90+\cdots+8-6+4-2.</cmath><math>\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }50\qquad\textbf{(D) }80\qquad \textbf{(E) }100</math>

[[2016 AMC 8 Problems/Problem 8|Solution
]]

==Problem 9==

What is the sum of the distinct prime integer divisors of <math>2016</math>?

<math>\textbf{(A) }9\qquad\textbf{(B) }12\qquad\textbf{(C) }16\qquad\textbf{(D) }49\qquad \textbf{(E) }63</math>

[[2016 AMC 8 Problems/Problem 9|Solution
]]

==Problem 10==

Suppose that <math>a * b</math> means <math>3a-b.</math> What is the value of <math>x</math> if
<cmath>2 * (5 * x)=1</cmath>
<math>\textbf{(A) }\frac{1}{10} \qquad\textbf{(B) }2\qquad\textbf{(C) }\frac{10}{3} \qquad\textbf{(D) }10\qquad \textbf{(E) }14.</math>

[[2016 AMC 8 Problems/Problem 10|Solution
]]

==Problem 11==

Determine how many two-digit numbers satisfy the following property: when the number is added to the number obtained by reversing its digits, the sum is <math>132.</math>

<math>\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }12</math>

[[2016 AMC 8 Problems/Problem 11|Solution
]]

==Problem 12==

Jefferson Middle School has the same number of boys and girls. <math>\frac{3}{4}</math> of the girls and <math>\frac{2}{3}</math>
of the boys went on a field trip. What fraction of the students on the field trip were girls?

<math>\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{9}{17}\qquad\textbf{(C) }\frac{7}{13}\qquad\textbf{(D) }\frac{2}{3}\qquad \textbf{(E) }\frac{14}{15}</math>

[[2016 AMC 8 Problems/Problem 12|Solution
]]

==Problem 13==

Two different numbers are randomly selected from the set <math>\{ - 2, -1, 0, 3, 4, 5\}</math> and multiplied together. What is the probability that the product is <math>0</math>?

<math>\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}</math>

[[2016 AMC 8 Problems/Problem 13|Solution
]]

==Problem 14==

Karl's car uses a gallon of gas every <math>35</math> miles, and his gas tank holds <math>14</math> gallons when it is full. One day, Karl started with a full tank of gas, 
drove <math>350</math> miles, bought <math>8</math> gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles did Karl drive that day? 

<math>\textbf{(A)}\mbox{ }525\qquad\textbf{(B)}\mbox{ }560\qquad\textbf{(C)}\mbox{ }595\qquad\textbf{(D)}\mbox{ }665\qquad\textbf{(E)}\mbox{ }735</math>

[[2016 AMC 8 Problems/Problem 14|Solution
]]

==Problem 15==

What is the largest power of <math>2</math> that is a divisor of <math>13^4 - 11^4</math>?

<math>\textbf{(A)}\mbox{ }8\qquad \textbf{(B)}\mbox{ }16\qquad \textbf{(C)}\mbox{ }32\qquad \textbf{(D)}\mbox{ }64\qquad \textbf{(E)}\mbox{ }128</math>

[[2016 AMC 8 Problems/Problem 15|Solution
]]

==Problem 16==

Annie and Bonnie are running laps around a <math>400</math>-meter oval track. They started together, but Annie has pulled ahead because she runs <math>25\%</math> faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?

<math>\textbf{(A) }1\dfrac{1}{4}\qquad\textbf{(B) }3\dfrac{1}{3}\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }25</math>

[[2016 AMC 8 Problems/Problem 16|Solution
]]

==Problem 17==

An ATM password at Fred's Bank is composed of four digits from <math>0</math> to <math>9</math>, with repeated digits allowable. If no password may begin with the sequence <math>9,1,1,</math> then how many passwords are possible?

<math>\textbf{(A)}\mbox{ }30\qquad\textbf{(B)}\mbox{ }7290\qquad\textbf{(C)}\mbox{ }9000\qquad\textbf{(D)}\mbox{ }9990\qquad\textbf{(E)}\mbox{ }9999</math>

[[2016 AMC 8 Problems/Problem 17|Solution
]]

==Problem 18==

In an All-Area track meet, <math>216</math> sprinters enter a <math>100-</math>meter dash competition. The track has <math>6</math> lanes, so only <math>6</math> sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races are needed to determine the champion sprinter?

<math>\textbf{(A)}\mbox{ }36\qquad\textbf{(B)}\mbox{ }42\qquad\textbf{(C)}\mbox{ }43\qquad\textbf{(D)}\mbox{ }60\qquad\textbf{(E)}\mbox{ }72</math>

[[2016 AMC 8 Problems/Problem 18|Solution
]]

==Problem 19==

The sum of <math>25</math> consecutive even integers is <math>10,000</math>. What is the largest of these <math>25</math> consecutive integers?

<math>\textbf{(A)}\mbox{ }360\qquad\textbf{(B)}\mbox{ }388\qquad\textbf{(C)}\mbox{ }412\qquad\textbf{(D)}\mbox{ }416\qquad\textbf{(E)}\mbox{ }424</math>

[[2016 AMC 8 Problems/Problem 19|Solution
]]

==Problem 20==

The least common multiple of <math>a</math> and <math>b</math> is <math>12</math>, and the least common multiple of <math>b</math> and <math>c</math> is <math>15</math>. What is the least possible value of the least common multiple of <math>a</math> and <math>c</math>?

<math>\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180</math>

[[2016 AMC 8 Problems/Problem 20|Solution
]]

==Problem 21==

A top hat contains 3 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 3 of the reds are drawn or until both green chips are drawn. What is the probability that the 3 reds are drawn?

<math>\textbf{(A) }\dfrac{3}{10}\qquad\textbf{(B) }\dfrac{2}{5}\qquad\textbf{(C) }\dfrac{1}{2}\qquad\textbf{(D) }\dfrac{3}{5}\qquad \textbf{(E) }\dfrac{7}{10}</math>

[[2016 AMC 8 Problems/Problem 21|Solution
]]

==Problem 22==
Rectangle <math>DEFA</math> below is a <math>3 \times 4</math> rectangle with <math>DC=CB=BA=1</math>. The area of the "bat wings" (shaded area) is
<asy>
draw((0,0)--(3,0)--(3,4)--(0,4)--(0,0)--(2,4)--(3,0));
draw((3,0)--(1,4)--(0,0));
fill((0,0)--(1,4)--(1.5,3)--cycle, black);
fill((3,0)--(2,4)--(1.5,3)--cycle, black);
label("$A$",(3.05,4.2));
label("$B$",(2,4.2));
label("$C$",(1,4.2));
label("$D$",(0,4.2));
label("$E$", (0,-0.2));
label("$F$", (3,-0.2));
</asy>

<math>\textbf{(A) }2\qquad\textbf{(B) }2 \frac{1}{2}\qquad\textbf{(C) }3\qquad\textbf{(D) }3 \frac{1}{2}\qquad \textbf{(E) }4</math>

[[2016 AMC 8 Problems/Problem 22|Solution
]]

==Problem 23==

Two congruent circles centered at points <math>A</math> and <math>B</math> each pass through the other circle's center. The line containing both <math>A</math> and <math>B</math> is extended to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree measure of <math>\angle CED</math>?

<math>\textbf{(A) }90\qquad\textbf{(B) }105\qquad\textbf{(C) }120\qquad\textbf{(D) }135\qquad \textbf{(E) }150</math>

[[2016 AMC 8 Problems/Problem 23|Solution
]]

==Problem 24==

The digits <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, and <math>5</math> are each used once to write a five-digit number <math>PQRST</math>. The three-digit number <math>PQR</math> is divisible by <math>4</math>, the three-digit number <math>QRS</math> is divisible by <math>5</math>, and the three-digit number <math>RST</math> is divisible by <math>3</math>. What is <math>P</math>?

<math>\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5</math>

[[2016 AMC 8 Problems/Problem 24|Solution
]]

==Problem 25== 

A semicircle is inscribed in an isosceles triangle with base <math>16</math> and height <math>15</math> so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

<asy>draw((0,0)--(8,15)--(16,0)--(0,0));
draw(arc((8,0),7.0588,0,180));</asy>

<math>\textbf{(A) }4 \sqrt{3}\qquad\textbf{(B) } \dfrac{120}{17}\qquad\textbf{(C) }10\qquad\textbf{(D) }\dfrac{17\sqrt{2}}{2}\qquad \textbf{(E)} \dfrac{17\sqrt{3}}{2}</math>

[[2016 AMC 8 Problems/Problem 25|Solution
]]

==Video Solution for Problem 25 (CREATIVE THINKING + ANALYSIS!!!)==
https://youtu.be/I-O_DqXmM4Y

~Education, the Study of Everything

==See Also==
{{AMC8 box|year=2016|before=[[2015 AMC 8 Problems|2015 AMC 8]]|after=[[2017 AMC 8 Problems|2017 AMC 8]]}}
* [[AMC 8]]
* [[AMC 8 Problems and Solutions]]
* [[Mathematics competition resources|Mathematics Competition Resources]]