Difference between revisions of "2016 AMC 8 Problems"
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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>? | 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>? | ||
− | tbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24 | + | tbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24<math> |
[[2016 AMC 8 Problems/Problem 2|Solution | [[2016 AMC 8 Problems/Problem 2|Solution | ||
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==Problem 3== | ==Problem 3== | ||
− | Four students take an exam. Three of their scores are <math>70, 80,< | + | 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>\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 | [[2016 AMC 8 Problems/Problem 3|Solution | ||
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==Problem 4== | ==Problem 4== | ||
− | When Cheenu was a boy he could run <math>15< | + | 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>\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 | [[2016 AMC 8 Problems/Problem 4|Solution | ||
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==Problem 5== | ==Problem 5== | ||
− | The number <math>N< | + | The number </math>N<math> is a two-digit number. |
− | • When <math>N< | + | • When </math>N<math> is divided by </math>9<math>, the remainder is </math>1<math>. |
− | • When <math>N< | + | • When </math>N<math> is divided by </math>10<math>, the remainder is </math>3<math>. |
− | What is the remainder when <math>N< | + | 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>\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 | [[2016 AMC 8 Problems/Problem 5|Solution | ||
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==Problem 6== | ==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? | The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? | ||
− | <math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7 | + | </math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$ |
<asy> | <asy> | ||
unitsize(0.9cm); | unitsize(0.9cm); |
Revision as of 00:36, 1 March 2020
Contents
Problem 1
The longest professional tennis match lasted a total of 11 hours and 5 minutes. How many minutes was that?
Problem 2
In rectangle ,
and
. Point
is the midpoint of
. What is the area of
?
tbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24$[[2016 AMC 8 Problems/Problem 2|Solution
]]
==Problem 3==
Four students take an exam. Three of their scores are$ (Error compiling LaTeX. Unknown error_msg)70, 80,90
70
\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$[[2016 AMC 8 Problems/Problem 3|Solution
]]
==Problem 4==
When Cheenu was a boy he could run$ (Error compiling LaTeX. Unknown error_msg)153
30
10
4
\textbf{(A) }6\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad \textbf{(E) }30$[[2016 AMC 8 Problems/Problem 4|Solution
]]
==Problem 5==
The number$ (Error compiling LaTeX. Unknown error_msg)N$is a two-digit number.
• When$ (Error compiling LaTeX. Unknown error_msg)N9
1$.
• When$ (Error compiling LaTeX. Unknown error_msg)N10
3$.
What is the remainder when$ (Error compiling LaTeX. Unknown error_msg)N11
\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7$[[2016 AMC 8 Problems/Problem 5|Solution
]]
==Problem 6==
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?$ (Error compiling LaTeX. Unknown error_msg)\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$
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 area)?
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?
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.