2014 AMC 10B — Official Competition Problems (February 2014)
📅 2014 B 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
概率
Leah has 13 coins, all of which are pennies and nickels. If she had one more nickel than she has now, then she would have the same number of pennies and nickels. In cents, how much are Leah's coins worth? \textbf {(A) } 33 \textbf {(B) } 35 \textbf {(C) } 37 \textbf {(D) } 39 \textbf {(E) } 41
💡 解题思路
If Leah has $1$ more nickel, she has $14$ total coins. Because she has the same number of nickels and pennies, she has $7$ nickels and $7$ pennies. This is after the nickel has been added, so we must
We can synchronously multiply ${2^3}$ to the expresions both above and below the fraction bar. Thus, \[\frac{2^3+2^3}{2^{-3}+2^{-3}}\\=\frac{2^6+2^6}{1+1}\\={2^6}.\] Hence, the fraction equals to $\bo
3
第 3 题
综合
Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? \textbf {(A) } 30 \textbf {(B) } \frac{400}{11} \textbf {(C) } \frac{75}{2} \textbf {(D) } 40 \textbf {(E) } \frac{300}{7}
💡 解题思路
Let the total distance be $x$ . We have $\dfrac{x}{3} + 20 + \dfrac{x}{5} = x$ , or $\dfrac{8x}{15} + 20 = x$ . Subtracting $\dfrac{8x}{15}$ from both sides gives us $20 = \dfrac{7x}{15}$ . Multiplyin
4
第 4 题
行程问题
Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is how many times as expensive as a banana? \textbf {(A) } \frac{3}{2} \textbf {(B) } \frac{5}{3} \textbf {(C) } \frac{7}{4} \textbf {(D) } 2 \textbf {(E) } \frac{13}{4}
💡 解题思路
Let $m$ be the cost of a muffin and $b$ be the cost of a banana. From the given information, \[2m+16b=2(4m+3b)=8m+6b\Rightarrow 10b=6m\Rightarrow m=\frac{10}{6}b=\boxed{\frac{5}{3}\rightarrow \text{(B
5
第 5 题
几何·面积
Doug constructs a square window using 8 equal-size panes of glass, as shown. The ratio of the height to width for each pane is 5 : 2 , and the borders around and between the panes are 2 inches wide. In inches, what is the side length of the square window? [图]
💡 解题思路
We note that the total length must be the same as the total height, as it is given in the problem. Calling the width of each small rectangle $2x$ , and the height $5x$ , we can see that the length is
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第 6 题
应用题
Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at \frac{1}{3} off the regular price. What is the greatest number of balloons Orvin could buy? \textbf {(A) } 33 \textbf {(B) } 34 \textbf {(C) } 36 \textbf {(D) } 38 \textbf {(E) } 39
💡 解题思路
Since he pays $\dfrac{2}{3}$ the price for every second balloon, the price for two balloons is $\dfrac{5}{3}$ . Thus, if he had enough money to buy $30$ balloons before, he now has enough to buy $30 \
7
第 7 题
综合
Suppose A>B>0 and A is x % greater than B . What is x ? \textbf {(A) } 100(\frac{A-B}{B}) \textbf {(B) } 100(\frac{A+B}{B}) \textbf {(C) } 100(\frac{A+B}{A}) \textbf {(D) } 100(\frac{A-B}{A}) \textbf {(E) } 100(\frac{A}{B})
💡 解题思路
We have that A is $x\%$ greater than B, so $A=\frac{100+x}{100}(B)$ . We solve for $x$ . We get
8
第 8 题
行程问题
A truck travels \dfrac{b}{6} feet every t seconds. There are 3 feet in a yard. How many yards does the truck travel in 3 minutes? \textbf {(A) } \frac{b}{1080t} \textbf {(B) } \frac{30t}{b} \textbf {(C) } \frac{30b}{t} \textbf {(D) } \frac{10t}{b} \textbf {(E) } \frac{10b}{t}
💡 解题思路
Converting feet to yards and minutes to second, we see that the truck travels $\dfrac{b}{18}$ yards every $t$ seconds for $180$ seconds. We see that he does $\dfrac{180}{t}$ cycles of $\dfrac{b}{18}$
9
第 9 题
综合
For real numbers w and z , \[\cfrac{\frac{1}{w} + \frac{1}{z}}{\frac{1}{w} - \frac{1}{z}} = 2014.\] What is \frac{w+z}{w-z} ?
💡 解题思路
Multiply the numerator and denominator of the LHS (left hand side) by $wz$ to get $\frac{z+w}{z-w}=2014$ . Then since $z+w=w+z$ and $w-z=-(z-w)$ , $\frac{w+z}{w-z}=-\frac{z+w}{z-w}=-2014$ , or choice
10
第 10 题
数字运算
In the addition shown below A , B , C , and D are distinct digits. How many different values are possible for D ? \[\begin{array}[t]{r} ABBCB ; + \ BCADA ; \hline DBDDD \end{array}\] \textbf {(A) } 2 \textbf {(B) } 4 \textbf {(C) } 7 \textbf {(D) } 8 \textbf {(E) } 9
💡 解题思路
Note from the addition of the last digits that $A+B=D\text{ or }A+B=D+10$ . From the addition of the frontmost digits, $A+B$ cannot have a carry, since the answer is still a five-digit number. Also $A
11
第 11 题
计数
For the consumer, a single discount of n\% is more advantageous than any of the following discounts: (1) two successive 15\% discounts (2) three successive 10\% discounts (3) a 25\% discount followed by a 5\% discount What is the smallest possible positive integer value of n ?
💡 解题思路
Let the original price be $x$ . Then, for option $1$ , the discounted price is $(1-.15)(1-.15)x = .7225x$ . For option $2$ , the discounted price is $(1-.1)(1-.1)(1-.1)x = .729x$ . Finally, for option
12
第 12 题
综合
The largest divisor of 2,014,000,000 is itself. What is its fifth-largest divisor? \textbf {(A) } 125, 875, 000 \textbf {(B) } 201, 400, 000 \textbf {(C) } 251, 750, 000 \textbf {(D) } 402, 800, 000 \textbf {(E) } 503, 500, 000
💡 解题思路
Note that $2,014,000,000$ is divisible by $1,\ 2,\ 4,\ 5,\ 8$ . Then, the fifth largest factor would come from divisibility by $8$ , or $251,750,000$ , or $\boxed{\textbf{(C)}}$ .
13
第 13 题
几何·面积
Six regular hexagons surround a regular hexagon of side length 1 as shown. What is the area of \triangle{ABC} ? [图] \textbf {(A) } 2√(3) \textbf {(B) } 3√(3) \textbf {(C) } 1+3√(2) \textbf {(D) } 2+2√(3) \textbf {(E) } 3+2√(3)
💡 解题思路
We note that the $6$ triangular sections in $\triangle{ABC}$ can be put together to form a hexagon congruent to each of the seven other hexagons. By the formula for the area of the hexagon, we get the
14
第 14 题
时间问题
Danica drove her new car on a trip for a whole number of hours, averaging 55 miles per hour. At the beginning of the trip, abc miles was displayed on the odometer, where abc is a 3 -digit number with a\ge1 and a+b+c\le7 . At the end of the trip, the odometer showed cba miles. What is a^2+b^2+c^2 ? \textbf {(A) } 26 \textbf {(B) } 27 \textbf {(C) } 36 \textbf {(D) } 37 \textbf {(E) } 41
💡 解题思路
Let $h$ be the number of hours Danica drove. Note that $abc$ can be expressed as $100\cdot a+10\cdot b+c$ . From the given information, we have $100a+10b+c+55h=100c+10b+a$ . This can be simplified int
15
第 15 题
几何·面积
In rectangle ABCD , DC = 2 · CB and points E and F lie on \overline{AB} so that \overline{ED} and \overline{FD} trisect \angle ADC as shown. What is the ratio of the area of \triangle DEF to the area of rectangle ABCD ? [图]
💡 解题思路
Let the length of $AD$ be $x$ , so that the length of $AB$ is $2x$ and $\text{[}ABCD\text{]}=2x^2$ .
16
第 16 题
概率
Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the same value? \textbf {(A) } \frac{1}{36} \textbf {(B) } \frac{7}{72} \textbf {(C) } \frac{1}{9} \textbf {(D) } \frac{5}{36} \textbf {(E) } \frac{1}{6}
💡 解题思路
We split this problem into $2$ cases.
17
第 17 题
数论
What is the greatest power of 2 that is a factor of 10^{1002} - 4^{501} ?
💡 解题思路
We begin by factoring the $2^{1002}$ out. This leaves us with $5^{1002} - 1$ .
18
第 18 题
统计
A list of 11 positive integers has a mean of 10 , a median of 9 , and a unique mode of 8 . What is the largest possible value of an integer in the list? \textbf {(A) } 24 \textbf {(B) } 30 \textbf {(C) } 31 \textbf {(D) } 33 \textbf {(E) } 35
💡 解题思路
We start off with the fact that the median is $9$ , so we must have $a, b, c, d, e, 9, f, g, h, i, j$ , listed in ascending order. Note that the integers do not have to be distinct.
19
第 19 题
几何·面积
Two concentric circles have radii 1 and 2 . Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
💡 解题思路
Let the center of the two circles be $O$ . Now pick an arbitrary point $A$ on the boundary of the circle with radius $2$ . We want to find the range of possible places for the second point, $A'$ , suc
20
第 20 题
整数运算
For how many integers x is the number x^4-51x^2+50 negative? \textbf {(A) } 8 \textbf {(B) } 10 \textbf {(C) } 12 \textbf {(D) } 14 \textbf {(E) } 16
💡 解题思路
First, note that $50+1=51$ , so we factor the polynomial as $(x^2-50)(x^2-1)$ .
21
第 21 题
几何·角度
Trapezoid ABCD has parallel sides \overline{AB} of length 33 and \overline {CD} of length 21 . The other two sides are of lengths 10 and 14 . The angles A and B are acute. What is the length of the shorter diagonal of ABCD ?
💡 解题思路
[asy] size(7cm); pair A,B,C,D,CC,DD; A = (-2,7); B = (14,7); C = (10,0); D = (0,0); CC = (10,7); DD = (0,7); draw(A--B--C--D--cycle); //label("33",(A+B)/2,N); label("21",(C+D)/2,S); label("10",(A+D)/2
22
第 22 题
几何·面积
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles? (A) \dfrac{1+\sqrt2}4 (B) \dfrac{\sqrt5-1}2 (C) \dfrac{\sqrt3+1}4 (D) \dfrac{2\sqrt3}5 (E) \dfrac{\sqrt5}3 [图]
💡 解题思路
We connect the centers of the circle and one of the semicircles, then draw the perpendicular from the center of the middle circle to that side, as shown.
23
第 23 题
分数与比例
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? (A) \dfrac32 (B) \dfrac{1+\sqrt5}2 (C) \sqrt3 (D) 2 (E) \dfrac{3+\sqrt5}2 [图]
💡 解题思路
First, we draw the vertical cross-section passing through the middle of the frustum. Let the top base have a diameter of $2$ and the bottom base has a diameter of $2r$ .
24
第 24 题
几何·面积
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is \textit{bad} if it is not true that for every n from 1 to 15 one can find a subset of the numbers that appear consecutively on the circle that sum to n . Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? \textbf {(A) } 1 \textbf {(B) } 2 \textbf {(C) } 3 \textbf {(D) } 4 \textbf {(E) } 5 .
💡 解题思路
We see that there are $5!$ total ways to arrange the numbers. However, we can always rotate these numbers so that, for example, the number $1$ is always at the top of the circle. Thus, there are only
25
第 25 题
概率
In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad N , 0<N<10 , it will jump to pad N-1 with probability \frac{N}{10} and to pad N+1 with probability 1-\frac{N}{10} . Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?