2014B AMC 12 — Official Competition Problems (February 2024)
📅 2024 A 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?
💡 解题思路
She has $p$ pennies and $n$ nickels, where $n + p = 13$ . If she had $n+1$ nickels then $n+1 = p$ , so $2n+ 1 = 13$ and $n=6$ . So she has 6 nickels and 7 pennies, which clearly have a value of $\boxe
2
第 2 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?
💡 解题思路
If every balloon costs $n$ dollars, then Orvin has $30n$ dollars. For every balloon he buys for $n$ dollars, he can buy another for $\frac{2n}{3}$ dollars. This means it costs him $\frac{5n}{3}$ dolla
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?
💡 解题思路
If the first and last legs of his trip account for $\frac{1}{3}$ and $\frac{1}{5}$ of his trip, then the middle leg accounts for $1 - \frac{1}{3} - \frac{1}{5} = \frac{7}{15}$ ths of his trip. This is
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?
💡 解题思路
Let $m$ stand for the cost of a muffin, and let $b$ stand for the value of a banana. We need to find $\frac{m}{b}$ , the ratio of the price of the muffins to that of the bananas. We have \[2(4m + 3b)
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
Let the height of the panes equal $5x$ , and let the width of the panes equal $2x$ . Now notice that the total width of the borders equals $10$ , and the total height of the borders is $6$ . We have \
6
第 6 题
规律与数列
Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume \frac{3}{4} of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?
💡 解题思路
Let the size of Ed's drink equal $x$ ounces, and let the size of Ann's drink equal $\frac{3}{2}x$ ounces. After both consume $\frac{3}{4}$ of their drinks, Ed and Ann have $\frac{x}{4}$ and $\frac{3x}
7
第 7 题
整数运算
For how many positive integers n is \frac{n}{30-n} also a positive integer?
💡 解题思路
We know that $n \le 30$ or else $30-n$ will be negative, resulting in a negative fraction. We also know that $n \ge 15$ or else the fraction's denominator will exceed its numerator making the fraction
8
第 8 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 ? \[\begin{tabular}{cccccc}&A&B&B&C&B ; +&B&C&A&D&A ; \hline &D&B&D&D&D\end{tabular}\]
💡 解题思路
From the first column, we see $A+B < 10$ because it yields a single digit answer. From the fourth column, we see that $C+D$ equals $D$ and therefore $C = 0$ . We know that $A+B = D$ . Therefore, the n
9
第 9 题
几何·面积
Convex quadrilateral ABCD has AB=3 , BC=4 , CD=13 , AD=12 , and \angle ABC=90^{\circ} , as shown. What is the area of the quadrilateral? [图]
💡 解题思路
Note that by the pythagorean theorem, $AC=5$ . Also note that $\angle CAD$ is a right angle because $\triangle CAD$ is a right triangle. The area of the quadrilateral is the sum of the areas of $\tria
10
第 10 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 ≥{1} and a+b+c ≤{7} . At the end of the trip, the odometer showed cba miles. What is a^2+b^2+c^2? .
💡 解题思路
We know that the number of miles she drove is divisible by $5$ , so $a$ and $c$ must either be equal or differ by $5$ . We can quickly conclude that the former is impossible; otherwise, she would have
11
第 11 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?
💡 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.
12
第 12 题
几何·面积
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have?
💡 解题思路
Define $T$ to be the set of all integral triples $(a, b, c)$ such that $a \ge b \ge c$ , $b+c > a$ , and $a, b, c < 5$ . Now we enumerate the elements of $T$ :
13
第 13 题
几何·面积
Real numbers a and b are chosen with 1<a<b such that no triangle with positive area has side lengths 1 , a , and b or \frac{1}{b} , \frac{1}{a} , and 1 . What is the smallest possible value of b ?
💡 解题思路
Notice that $1>\frac{1}{a}>\frac{1}{b}$ . Using the triangle inequality, we find \[a+1 > b \implies a>b-1\] \[\frac{1}{a}+\frac{1}{b} > 1\] In order for us the find the lowest possible value for $b$ ,
14
第 14 题
几何·面积
A rectangular box has a total surface area of 94 square inches. The sum of the lengths of all its edges is 48 inches. What is the sum of the lengths in inches of all of its interior diagonals?
💡 解题思路
Let the side lengths of the rectangular box be $x, y$ and $z$ . From the information we get
15
第 15 题
数论
When p = \sum\limits_{k=1}^{6} k ln {k} , the number e^p is an integer. What is the largest power of 2 that is a factor of e^p ?
💡 解题思路
Let's write out the sum. Our sum is equal to \[1 \ln{1} + 2 \ln{2} + 3 \ln{3} + 4 \ln{4} + 5 \ln{5} + 6 \ln{6} =\] \[\ln{1^1} + \ln{2^2} + \ln{3^3} + \ln {4^4} + \ln{5^5} + \ln {6^6} =\] \[\ln{(1^1\ti
16
第 16 题
综合
Let P be a cubic polynomial with P(0) = k , P(1) = 2k , and P(-1) = 3k . What is P(2) + P(-2) ?
💡 解题思路
Let $P(x) = Ax^3+Bx^2 + Cx+D$ . Plugging in $0$ for $x$ , we find $D=k$ , and plugging in $1$ and $-1$ for $x$ , we obtain the following equations: \[A+B+C+k=2k\] \[-A+B-C+k=3k\] Adding these two equa
17
第 17 题
坐标几何
Let P be the parabola with equation y=x^2 and let Q = (20, 14) . There are real numbers r and s such that the line through Q with slope m does not intersect P if and only if r < m < s . What is r + s ?
💡 解题思路
Let $y = m(x - 20) + 14$ . Equating them:
18
第 18 题
几何·面积
The numbers 1, 2, 3, 4, 5 are to be arranged in a circle. An arrangement is 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 $4
19
第 19 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? [图] (A) \dfrac32 (B) \dfrac{1+\sqrt5}2 (C) \sqrt3 (D) 2 (E) \dfrac{3+\sqrt5}2
💡 解题思路
First, we draw the vertical cross-section passing through the middle of the frustum. let the top base equal 2 and the bottom base to be equal to 2r [asy] size(7cm); pair A,B,C,D; real r = (3+sqrt(5))/
20
第 20 题
整数运算
For how many positive integers x is \log_{10}(x-40) + \log_{10}(60-x) < 2 ?
💡 解题思路
The domain of the LHS implies that \[400$ . \[\] Begin from the left-hand side \[\log_{10}[(x-40)(60-x)]<2\]
21
第 21 题
几何·面积
In the figure, ABCD is a square of side length 1 . The rectangles JKHG and EBCF are congruent. What is BE ? [图]
💡 解题思路
Draw the altitude from $H$ to $AB$ and call the foot $L$ . Then $HL=1$ . Consider $HJ$ . It is the hypotenuse of both right triangles $\triangle HGJ$ and $\triangle HLJ$ , and we know $JG=HL=1$ , so w
22
第 22 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?
💡 解题思路
A long, but straightforward bash:
23
第 23 题
数论
The number 2017 is prime. Let S = \sum \limits_{k=0}^{62} \dbinom{2014}{k} . What is the remainder when S is divided by 2017?
💡 解题思路
Note that $2014\equiv -3 \mod2017$ . We have for $k\ge1$ \[\dbinom{2014}{k}\equiv \frac{(-3)(-4)(-5)....(-2-k)}{k!}\mod 2017\] \[\equiv (-1)^k\dbinom{k+2}{k} \mod 2017\] Therefore \[\sum \limits_{k=0}
24
第 24 题
几何·面积
Let ABCDE be a pentagon inscribed in a circle such that AB = CD = 3 , BC = DE = 10 , and AE= 14 . The sum of the lengths of all diagonals of ABCDE is equal to \frac{m}{n} , where m and n are relatively prime positive integers. What is m+n ?
💡 解题思路
Let $BE=a$ , $AD=b$ , and $AC=CE=BD=c$ . Let $F$ be on $AE$ such that $CF \perp AE$ . [asy] size(200); defaultpen(linewidth(0.4)+fontsize(10)); pen s = linewidth(0.8)+fontsize(8); pair O,A,B,C,D,E0,F;
25
第 25 题
规律与数列
Find the sum of all the positive solutions of 2\cos2x (\cos2x - \cos{( \frac{2014π^2}{x} ) } ) = \cos4x - 1
💡 解题思路
Rewrite $\cos{4x} - 1$ as $2\cos^2{2x} - 2$ . Now let $a = \cos{2x}$ , and let $b = \cos{\left( \frac{2014\pi^2}{x} \right) }$ . We have: \[2a(a - b) = 2a^2 - 2\]