2016 AMC 8 — Official Competition Problems (November 2019)
📅 2016 年11月📝 25题选择题⏱ 40分钟🎯 满分25分✅ 含解题思路👥 612 人已练习
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1
第 1 题
时间问题
The longest professional tennis match ever played lasted a total of 11 hours and 5 minutes. How many minutes was this?
💡 解题思路
It is best to split 11 hours and 5 minutes into 2 parts, one of 11 hours and another of 5 minutes. We know that there is $60$ minutes in a hour. Therefore, there are $11 \cdot 60 = 660$ minutes in 11
2
第 2 题
几何·面积
In rectangle ABCD , AB=6 and AD=8 . Point M is the midpoint of \overline{AD} . What is the area of \triangle AMC ? [图]
💡 解题思路
Using the triangle area formula for triangles: $A = \frac{bh}{2},$ where $A$ is the area, $b$ is the base, and $h$ is the height. This equation gives us $A = \frac{4 \cdot 6}{2} = \frac{24}{2} =\boxed
3
第 3 题
统计
Four students take an exam. Three of their scores are 70, 80, and 90 . If the average of their four scores is 70 , then what is the remaining score?
💡 解题思路
Let $r$ be the remaining student's score. We know that the average, 70, is equal to $\frac{70 + 80 + 90 + r}{4}$ . We can use basic algebra to solve for $r$ : \[\frac{70 + 80 + 90 + r}{4} = 70\] \[\fr
4
第 4 题
行程问题
When Cheenu was a boy, he could run 15 miles in 3 hours and 30 minutes. As an old man, he can now walk 10 miles in 4 hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy?
💡 解题思路
When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes $= 3\times60 + 30$ minutes $= 210$ minutes, thus running $\frac{210}{15} = 14$ minutes per mile. Now that he is an old man,
5
第 5 题
数论
The number N is a two-digit number. • When N is divided by 9 , the remainder is 1 . • When N is divided by 10 , the remainder is 3 . What is the remainder when N is divided by 11 ?
💡 解题思路
From the second bullet point, we know that the second digit must be $3$ , for a number divisible by $10$ ends in zero. Since there is a remainder of $1$ when $N$ is divided by $9$ , the multiple of $9
6
第 6 题
坐标几何
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names? [图]
💡 解题思路
We first notice that the median name willl be the $(19+1)/2=10^{\mbox{th}}$ name. The $10^{\mbox{th}}$ name is $\boxed{\textbf{(B)}\ 4}$ .
7
第 7 题
几何·面积
Which of the following numbers is not a perfect square?
💡 解题思路
Our answer must have an odd exponent in order for it to not be a square. Because $4$ is a perfect square, $4^{2019}$ is also a perfect square, so our answer is $\boxed{\textbf{(B) }2^{2017}}$ .
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第 8 题
综合
💡 解题思路
We can group each subtracting pair together: \[(100-98)+(96-94)+(92-90)+ \ldots +(8-6)+(4-2).\] After subtracting, we have: \[2+2+2+\ldots+2+2=2(1+1+1+\ldots+1+1).\] There are $50$ even numbers, there
9
第 9 题
数论
What is the sum of the distinct prime integer divisors of 2016 ?
💡 解题思路
The prime factorization is $2016=2^5\times3^2\times7$ . Since the problem is only asking us for the distinct prime factors, we have $2,3,7$ . Their desired sum is then $\boxed{\textbf{(B) }12}$ .
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第 10 题
综合
💡 解题思路
Let us plug in $(5 * x)=1$ into $3a-b$ . Thus it would be $3(5)-x$ . Now we have $2*(15-x)=1$ . Plugging $2*(15-x)$ into $3a-b$ , we have $6-15+x=1$ . Solving for $x$ we have \[-9+x=1\] \[x=\boxed{\te
11
第 11 题
规律与数列
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 132.
💡 解题思路
We can see that the original number can be written as $10a+b$ , where $a$ represents the tens digit and $b$ represents the units digit. When this number is added to the number obtained by reversing it
12
第 12 题
分数与比例
Jefferson Middle School has the same number of boys and girls. \frac{3}{4} of the girls and \frac{2}{3} of the boys went on a field trip. What fraction of the students on the field trip were girls?
💡 解题思路
Let there be $b$ boys and $g$ girls in the school. We see $g=b$ , which means $\frac{3}{4}b+\frac{2}{3}b=\frac{17}{12}b$ kids went on the trip and $\frac{3}{4}b$ kids are girls. So, the answer is $\fr
13
第 13 题
概率
Two different numbers are randomly selected from the set \{ - 2, -1, 0, 3, 4, 5\} and multiplied together. What is the probability that the product is 0 ?
💡 解题思路
1. Identify the total number of ways to select two different numbers from the set:
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第 14 题
行程问题
Karl's car uses a gallon of gas every 35 miles, and his gas tank holds 14 gallons when it is full. One day, Karl started with a full tank of gas, drove 350 miles, bought 8 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?
💡 解题思路
Since he uses a gallon of gas every $35$ miles, he had used $\frac{350}{35} = 10$ gallons after $350$ miles. Therefore, after the first leg of his trip he had $14 - 10 = 4$ gallons of gas left. Then,
15
第 15 题
综合
What is the largest power of 2 that is a divisor of 13^4 - 11^4 ?
💡 解题思路
First, we use difference of squares on $13^4 - 11^4 = (13^2)^2 - (11^2)^2$ to get $13^4 - 11^4 = (13^2 + 11^2)(13^2 - 11^2)$ . Using difference of squares again and simplifying, we get $(169 + 121)(13
16
第 16 题
综合
Annie and Bonnie are running laps around a 400 -meter oval track. They started together, but Annie has pulled ahead, because she runs 25\% faster than Bonnie. How many laps will Annie have run when she first passes Bonnie?
💡 解题思路
Each lap Bonnie runs, Annie runs another quarter lap, so Bonnie will run four laps before she is overtaken because it takes 4 of $\frac{1}{4}$ of the total lap. to total one lap. When Bonnie runs 4 la
17
第 17 题
规律与数列
An ATM password at Fred's Bank is composed of four digits from 0 to 9 , with repeated digits allowable. If no password may begin with the sequence 9,1,1, then how many passwords are possible?
💡 解题思路
For the first three digits, there are $10^3-1=999$ combinations since $911$ is not allowed. For the final digit, any of the $10$ numbers are allowed. $999 \cdot 10 = 9990 \rightarrow \boxed{\textbf{(D
18
第 18 题
几何·面积
In an All-Area track meet, 216 sprinters enter a 100- meter dash competition. The track has 6 lanes, so only 6 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?
💡 解题思路
From any $n-$ th race, only $\frac{1}{6}$ will continue on. Since we wish to find the total number of races, a column representing the races over time is ideal. Starting with the first race: \[\frac{2
19
第 19 题
规律与数列
The sum of 25 consecutive even integers is 10,000 . What is the largest of these 25 consecutive integers?
💡 解题思路
Let $n$ be the 13th consecutive even integer that's being added up. By doing this, we can see that the sum of all 25 even numbers will simplify to $25n$ since $(n-2k)+\dots+(n-4)+(n-2)+(n)+(n+2)+(n+4)
20
第 20 题
数论
The least common multiple of a and b is 12 , and the least common multiple of b and c is 15 . What is the least possible value of the least common multiple of a and c ?
💡 解题思路
We wish to find possible values of $a$ , $b$ , and $c$ . By finding the greatest common factor of $12$ and $15$ , we can find that $b$ is 3. Moving on to $a$ and $c$ , in order to minimize them, we wi
21
第 21 题
概率
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?
💡 解题思路
We put five chips randomly in order and then pick the chips from the left to the right. To find the number of ways to rearrange the three red chips and two green chips, we solve for $\binom{5}{2} = 10
22
第 22 题
几何·面积
Rectangle DEFA below is a 3 × 4 rectangle with DC=CB=BA=1 . The area of the "bat wings" (shaded area) is [图]
💡 解题思路
The area of trapezoid $CBFE$ is $\frac{1+3}2\cdot 4=8$ . Next, we find the height of each triangle to calculate their area. The two non-colored isosceles triangles are similar, and are in a $3:1$ rati
23
第 23 题
几何·面积
Two congruent circles centered at points A and B each pass through the other circle's center. The line containing both A and B is extended to intersect the circles at points C and D . The circles intersect at two points, one of which is E . What is the degree measure of \angle CED ?
💡 解题思路
Observe that $\triangle{EAB}$ is equilateral (all are radii of congruent circles). Therefore, $m\angle{AEB}=m\angle{EAB}=m\angle{EBA} = 60^{\circ}$ . Since $CD$ is a straight line, we conclude that $m
24
第 24 题
数论
The digits 1 , 2 , 3 , 4 , and 5 are each used once to write a five-digit number PQRST . The three-digit number PQR is divisible by 4 , the three-digit number QRS is divisible by 5 , and the three-digit number RST is divisible by 3 . What is P ?
💡 解题思路
We see that since $QRS$ is divisible by $5$ , $S$ must equal either $0$ or $5$ , but it cannot equal $0$ , so $S=5$ . We notice that since $PQR$ must be even, $R$ must be either $2$ or $4$ . However,
25
第 25 题
几何·面积
A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle? [图] There are many solutions here, and all of them are equally good. For your own benefit, look at all of the solutions, as they employ many unique techniques to get to the final answer.