Difference between revisions of "2016 AMC 8 Problems"
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+ | {{AMC8 Problems|year=2016|}} | ||
==Problem 1== | ==Problem 1== | ||
− | + | The longest professional tennis match lasted a total of 11 hours and 5 minutes. How many minutes is that? | |
− | The longest professional tennis match | ||
− | <math>\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105</math> | + | <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 | [[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>. What is the area of the "bat wings" (shaded region)?<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 | ||
+ | ]] | ||
+ | |||
+ | ==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]] |
Latest revision as of 06:00, 8 July 2024
2016 AMC 8 (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
The longest professional tennis match lasted a total of 11 hours and 5 minutes. How many minutes is that?
Problem 2
In rectangle ,
and
. Point
is the midpoint of
. What is the area of
?
Problem 3
Four students take an exam. Three of their scores are and
. If the average of their four scores is
, then what is the remaining score?
Problem 4
When Cheenu was a boy, he could run miles in
hours and
minutes. As an old man, he can now walk
miles in
hours. How many minutes longer does it take for him to travel a mile now compared to when he was a boy?
Problem 5
The number is a two-digit number.
• When is divided by
, the remainder is
.
• When is divided by
, the remainder is
.
What is the remainder when is divided by
?
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?
Problem 7
Which of the following numbers is not a perfect square?
Problem 8
Find the value of the expression
Problem 9
What is the sum of the distinct prime integer divisors of ?
Problem 10
Suppose that means
What is the value of
if
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
Problem 12
Jefferson Middle School has the same number of boys and girls. of the girls and
of the boys went on a field trip. What fraction of the students on the field trip were girls?
Problem 13
Two different numbers are randomly selected from the set and multiplied together. What is the probability that the product is
?
Problem 14
Karl's car uses a gallon of gas every miles, and his gas tank holds
gallons when it is full. One day, Karl started with a full tank of gas,
drove
miles, bought
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?
Problem 15
What is the largest power of that is a divisor of
?
Problem 16
Annie and Bonnie are running laps around a -meter oval track. They started together, but Annie has pulled ahead because she runs
faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
Problem 17
An ATM password at Fred's Bank is composed of four digits from to
, with repeated digits allowable. If no password may begin with the sequence
then how many passwords are possible?
Problem 18
In an All-Area track meet, sprinters enter a
meter dash competition. The track has
lanes, so only
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?
Problem 19
The sum of consecutive even integers is
. What is the largest of these
consecutive integers?
Problem 20
The least common multiple of and
is
, and the least common multiple of
and
is
. What is the least possible value of the least common multiple of
and
?
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?
Problem 22
Rectangle below is a
rectangle with
. What is the area of the "bat wings" (shaded region)?
Problem 23
Two congruent circles centered at points and
each pass through the other circle's center. The line containing both
and
is extended to intersect the circles at points
and
. The circles intersect at two points, one of which is
. What is the degree measure of
?
Problem 24
The digits ,
,
,
, and
are each used once to write a five-digit number
. The three-digit number
is divisible by
, the three-digit number
is divisible by
, and the three-digit number
is divisible by
. What is
?
Problem 25
A semicircle is inscribed in an isosceles triangle with base and height
so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?
See Also
2016 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by 2015 AMC 8 |
Followed by 2017 AMC 8 | |
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 AJHSME/AMC 8 Problems and Solutions |