2014 AMC 8 — Official Competition Problems (November 2019)
📅 2014 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
综合
Harry and Terry are each told to calculate 8-(2+5) . Harry gets the correct answer. Terry ignores the parentheses and calculates 8-2+5 . If Harry's answer is H and Terry's answer is T , what is H-T ?
💡 解题思路
We have $H=8-7=1$ and $T=8-2+5=11$ . Clearly $1-11=-10$ , so our answer is $\boxed{\textbf{(A)}-10}$ .
2
第 2 题
概率
Paul owes Paula 35 cents and has a pocket full of 5 -cent coins, 10 -cent coins, and 25 -cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her?
💡 解题思路
The fewest amount of coins that can be used is $2$ (a quarter and a dime). The greatest amount is $7$ , if he only uses nickels. Therefore we have $7-2=\boxed{\textbf{(E)}~5}$ .
3
第 3 题
统计
Isabella had a week to read a book for a school assignment. She read an average of 36 pages per day for the first three days and an average of 44 pages per day for the next three days. She then finished the book by reading 10 pages on the last day. How many pages were in the book?
💡 解题思路
Isabella read $3\cdot 36+3\cdot 44$ pages in the first 6 days. Although this can be calculated directly, it is simpler to calculate it as $3\cdot (36+44)=3\cdot 80$ , which gives that she read $240$ p
4
第 4 题
数论
The sum of two prime numbers is 85 . What is the product of these two prime numbers?
💡 解题思路
Since the two prime numbers sum to an odd number, one of them must be even. The only even prime number is $2$ . The other prime number is $85-2=83$ , and the product of these two numbers is $83\cdot2=
5
第 5 题
行程问题
Margie's car can go 32 miles on a gallon of gas, and gas currently costs \4 per gallon. How many miles can Margie drive on \textdollar 20$ worth of gas?
💡 解题思路
Margie can afford $20/4=5$ gallons of gas. She can go $32\cdot5=\boxed{\textbf{(C)}~160}$ miles on this amount of gas.
6
第 6 题
几何·面积
Six rectangles each with a common base width of 2 have lengths of 1, 4, 9, 16, 25 , and 36 . What is the sum of the areas of the six rectangles?
💡 解题思路
The sum of the areas is equal to $2\cdot1+2\cdot4+2\cdot9+2\cdot16+2\cdot25+2\cdot36$ . This is equal to $2(1+4+9+16+25+36)$ , which is equal to $2\cdot91$ . This is equal to our final answer of $\box
7
第 7 题
分数与比例
There are four more girls than boys in Ms. Raub's class of 28 students. What is the ratio of number of girls to the number of boys in her class?
💡 解题思路
Let $g$ being the number of girls in the class. The number of boys in the class is equal to $g-4$ . Since the total number of students is equal to $28$ , we get $g+g-4=28$ . Solving this equation, we
8
第 8 题
应用题
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker \textdollar\underline{1} \underline{A} \underline{2} . What is the missing digit A of this 3 -digit number?
💡 解题思路
Since all the eleven members paid the same amount, that means that the total must be divisible by $11$ . We can do some trial-and-error to get $A=3$ , so our answer is $\boxed{\textbf{(D)}~3}$ ~Sparkl
9
第 9 题
几何·面积
In \bigtriangleup ABC , D is a point on side \overline{AC} such that BD=DC and \angle BCD measures 70^\circ . What is the degree measure of \angle ADB ? [图] https://youtu.be/jLnqUOe0HPE ~Education, the Study of Everything https://www.youtube.com/watch?v=HP-lBKohxhE ~David https://youtu.be/j5KrHM81HZ8 ~savannahsolver https://youtu.be/abSgjn4Qs34?t=3140
💡 解题思路
Using angle chasing is a good way to solve this problem. $BD = DC$ , so $\angle DBC = \angle DCB = 70$ , because it is an isosceles triangle. Then $\angle CDB = 180-(70+70) = 40$ . Since $\angle ADB$
10
第 10 题
行程问题
The first AMC 8 was given in 1985 and it has been given annually since that time. Samantha turned 12 years old the year that she took the seventh AMC 8 . In what year was Samantha born?
💡 解题思路
The seventh AMC 8 would have been given in $1991$ . If Samantha was 12 then, that means she was born 12 years ago, so she was born in $1991-12=1979$ .
11
第 11 题
方程
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks? This is a combonatorics problem. For these problems, you should start by counting the total, then subtracting the ways that don't work. Using word rearrangements, we can find the total number of paths, then find the number of paths through the dangerous intersection using the same method. Then you subtract the first from the second. This is called Complementary Counting -JasonDaGoat https://youtu.be/YCr5GFcmdcI ~Education, the Study of Everything https://www.youtube.com/watch?v=rxrQLNxESW0 ~David https://youtu.be/pWLm41JtiCw ~savannahsolver
💡 解题思路
We can apply complementary counting and count the paths that DO go through the blocked intersection, which is $\dbinom{2}{1}\dbinom{3}{1}=6$ . There are a total of $\dbinom{5}{2}=10$ paths, so there a
12
第 12 题
概率
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
💡 解题思路
For the first celebrity, you have a 1/3 chance of picking the photo. Given you get the picture right, you now have a 1/2 chance of picking the next photo. If you get both of them right, you are guaran
13
第 13 题
方程
If n and m are integers and n^2+m^2 is even, which of the following is impossible? https://youtu.be/jQLIxT5qCTY ~Education, the Study of Everything https://www.youtube.com/watch?v=boXUIcEcAno ~David https://youtu.be/_3n4f0v6B7I ~savannahsolver
💡 解题思路
Since $n^2+m^2$ is even, either both $n^2$ and $m^2$ are even, or they are both odd. Therefore, $n$ and $m$ are either both even or both odd, since the square of an even number is even and the square
14
第 14 题
几何·面积
Rectangle ABCD and right triangle DCE have the same area. They are joined to form a trapezoid, as shown. What is DE ? [图]
💡 解题思路
The area of $\bigtriangleup CDE$ is $\frac{DC\cdot CE}{2}$ . The area of $ABCD$ is $AB\cdot AD=5\cdot 6=30$ , which also must be equal to the area of $\bigtriangleup CDE$ , which, since $DC=5$ , must
15
第 15 题
几何·面积
The circumference of the circle with center O is divided into 12 equal arcs, marked the letters A through L as seen below. What is the number of degrees in the sum of the angles x and y ? [图]
💡 解题思路
The measure of an inscribed angle is half the measure of its corresponding central angle. Since each unit arc is $\frac{1}{12}$ of the circle's circumference, each unit central angle measures $\frac{3
16
第 16 题
综合
The "Middle School Eight" basketball conference has 8 teams. Every season, each team plays every other conference team twice (home and away), and each team also plays 4 games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
💡 解题思路
Within the conference, there are 8 teams, so there are $\dbinom{8}{2}=28$ pairings of teams, and each pair must play two games, for a total of $28\cdot 2=56$ games within the conference.
17
第 17 题
行程问题
George walks 1 mile to school. He leaves home at the same time each day, walks at a steady speed of 3 miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first \frac{1}{2} mile at a speed of only 2 miles per hour. At how many miles per hour must George run the last \frac{1}{2} mile in order to arrive just as school begins today?
💡 解题思路
Note that on a normal day, it takes him $1/3$ hour to get to school. However, today it took $\frac{1/2 \text{ mile}}{2 \text{ mph}}=1/4$ hour to walk the first $1/2$ mile. That means that he has $1/3
18
第 18 题
规律与数列
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely? (A) All are boys (B) All are girls (C) 2 are boys and 2 are girls (D) 3 are the same gender and 1 is of the other gender (E) All of these outcomes are equally likely
💡 解题思路
We'll just start by breaking cases down. The probability of A occurring is $\left(\frac{1}{2}\right)^4 = \frac{1}{16}$ . The probability of B occurring is $\left(\frac{1}{2}\right)^4 = \frac{1}{16}$ .
19
第 19 题
几何·面积
A cube with 3 -inch edges is to be constructed from 27 smaller cubes with 1 -inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3 -inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white? In a 3×3×3 cube, 26 of the cubes are on the outside. We put one white colored cube on the inside, and we put them in the middle, thus having one square per cube. This means that \dfrac{5}{54} of the squares are white so the answer is (A).
💡 解题思路
For the least possible surface area that is white, we should have 1 cube in the center, and the other 5 with only 1 face exposed. This gives 5 square inches of white, surface area. Since the cube has
20
第 20 题
几何·面积
Rectangle ABCD has sides CD=3 and DA=5 . A circle of radius 1 is centered at A , a circle of radius 2 is centered at B , and a circle of radius 3 is centered at C . Which of the following is closest to the area of the region inside the rectangle but outside all three circles? [图] (A) 3.5 (B) 4.0 (C) 4.5 (D) 5.0 (E) 5.5
💡 解题思路
The area in the rectangle but outside the circles is the area of the rectangle minus the area of all three of the quarter circles in the rectangle.
21
第 21 题
数论
The 7 -digit numbers \underline{7} \underline{4} \underline{A} \underline{5} \underline{2} \underline{B} \underline{1} and \underline{3} \underline{2} \underline{6} \underline{A} \underline{B} \underline{4} \underline{C} are each multiples of 3 . Which of the following could be the value of C ?
💡 解题思路
Since both numbers are divisible by 3, the sum of their digits has to be divisible by three. $7 + 4 + 5 + 2 + 1 = 19$ . To be a multiple of $3$ , $A + B$ has to be either $2$ or $5$ or $8$ ... and so
22
第 22 题
方程
A 2 -digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number? https://youtu.be/7an5wU9Q5hk?t=2226 https://youtu.be/AR3Ke23N1I8 ~savannahsolver
💡 解题思路
We can think of the number as $10a+b$ , where a is the tens digit and b is the unit digit. Since the number is equal to the product of the digits ( $ab$ ) plus the sum of the digits ( $a+b$ ), we can
23
第 23 题
数论
Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all 2 -digit primes. Bethany : And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of your two uniform numbers is today's date. What number does Caitlin wear? https://www.youtube.com/watch?v=6S0u_fDjSxc
💡 解题思路
The maximum amount of days any given month can have is $31$ , and the smallest two-digit primes are $11, 13,$ and $17$ . There are a few different sums that can be deduced from the following numbers,
24
第 24 题
方程
One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day? https://youtu.be/VlJzQ-ZNmmk ~savannahsolver https://www.youtube.com/watch?v=TkZvMa30Juo&t=2342s ~ pi_is_3.141592653589793238462643383279502884197
💡 解题思路
In order to maximize the median, we need to make the first half of the numbers as small as possible. Since there are $100$ people, the median will be the average of the $50\text{th}$ and $51\text{st}$
25
第 25 题
几何·面积
A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch? [图] Note: 1 mile = 5280 feet
💡 解题思路
There are two possible interpretations of the problem: that the road as a whole is $40$ feet wide, or that each lane is $40$ feet wide. Both interpretations will arrive at the same result. However, le