2014A 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
综合
What is 10·(\tfrac{1}{2}+\tfrac{1}{5}+\tfrac{1}{10})^{-1}?
💡 解题思路
We have \[10\cdot\left(\frac{1}{2}+\frac{1}{5}+\frac{1}{10}\right)^{-1}\] Making the denominators equal gives \[\implies 10\cdot\left(\frac{5}{10}+\frac{2}{10}+\frac{1}{10}\right)^{-1}\] \[\implies 10
2
第 2 题
应用题
At the theater children get in for half price. The price for 5 adult tickets and 4 child tickets is \24.50 . How much would 8 adult tickets and 6$ child tickets cost?
💡 解题思路
Suppose $x$ is the price of an adult ticket. The price of a child ticket would be $\frac{x}{2}$ .
3
第 3 题
统计
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible?
💡 解题思路
Let's use casework on the yellow house. The yellow house $(\text{Y})$ is either the $3^\text{rd}$ house or the last house.
4
第 4 题
行程问题
Suppose that a cows give b gallons of milk in c days. At this rate, how many gallons of milk will d cows give in e days?
💡 解题思路
We need to multiply $b$ by $\frac{d}{a}$ for the new cows and $\frac{e}{c}$ for the new time, so the answer is $b\cdot \frac{d}{a}\cdot \frac{e}{c}=\frac{bde}{ac}$ , or $\boxed{\textbf{(A) } \frac{bde
5
第 5 题
统计
On an algebra quiz, 10\% of the students scored 70 points, 35\% scored 80 points, 30\% scored 90 points, and the rest scored 100 points. What is the difference between the mean and median score of the students' scores on this quiz?
💡 解题思路
WLOG, let there be $20$ students (the least whole number possible) who took the test. We have $2$ students score $70$ points, $7$ students score $80$ points, $6$ students score $90$ points and $5$ stu
6
第 6 题
规律与数列
The difference between a two-digit number and the number obtained by reversing its digits is 5 times the sum of the digits of either number. What is the sum of the two digit number and its reverse?
💡 解题思路
Let the two digits be $a$ and $b$ . Then, $5a + 5b = 10a + b - 10b - a = 9a - 9b$ , or $2a = 7b$ . This yields $a = 7$ and $b = 2$ because $a, b < 10$ . Then, $72 + 27 = \boxed{\textbf{(D) }99}.$
7
第 7 题
规律与数列
The first three terms of a geometric progression are \sqrt 3 , \sqrt[3]3 , and \sqrt[6]3 . What is the fourth term?
💡 解题思路
The terms are $\sqrt 3$ , $\sqrt[3]3$ , and $\sqrt[6]3$ , which are equivalent to $3^{\frac{3}{6}}$ , $3^{\frac{2}{6}}$ , and $3^{\frac{1}{6}}$ . So the next term will be $3^{\frac{0}{6}}=1$ , so the
8
第 8 题
应用题
A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: 10\% off the listed price if the listed price is at least \textdollar50 Coupon 2: \textdollar 20 off the listed price if the listed price is at least \textdollar100 Coupon 3: 18\% off the amount by which the listed price exceeds \textdollar100 For which of the following listed prices will coupon 1 offer a greater price reduction than either coupon 2 or coupon 3 ?
💡 解题思路
Let the listed price be $x$ . Since all the answer choices are above $\textdollar100$ , we can assume $x > 100$ . Thus the discounts after the coupons are used will be as follows:
9
第 9 题
统计
Five positive consecutive integers starting with a have average b . What is the average of 5 consecutive integers that start with b ?
💡 解题思路
Let $a=1$ . Our list is $\{1,2,3,4,5\}$ with an average of $15\div 5=3$ . Our next set starting with $3$ is $\{3,4,5,6,7\}$ . Our average is $25\div 5=5$ .
10
第 10 题
几何·面积
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length 1 . The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles?
💡 解题思路
Reflect each of the triangles over its respective side. Then since the areas of the triangles total to the area of the equilateral triangle, it can be seen that the triangles fill up the equilateral o
11
第 11 题
行程问题
David drives from his home to the airport to catch a flight. He drives 35 miles in the first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?
💡 解题思路
Note that he drives at $50$ miles per hour after the first hour and continues doing so until he arrives.
12
第 12 题
几何·面积
Two circles intersect at points A and B . The minor arcs AB measure 30^\circ on one circle and 60^\circ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle?
💡 解题思路
Let the radius of the larger and smaller circles be $x$ and $y$ , respectively. Also, let their centers be $O_1$ and $O_2$ , respectively. Then the ratio we need to find is \[\dfrac{\pi x^2}{\pi y^2}
13
第 13 题
计数
A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than 2 friends per room. In how many ways can the innkeeper assign the guests to the rooms?
💡 解题思路
We can discern three cases.
14
第 14 题
整数运算
Let a<b<c be three integers such that a,b,c is an arithmetic progression and a,c,b is a geometric progression. What is the smallest possible value of c ?
💡 解题思路
We have $b-a=c-b$ , so $a=2b-c$ . Since $a,c,b$ is geometric, $c^2=ab=(2b-c)b \Rightarrow 2b^2-bc-c^2=(2b+c)(b-c)=0$ . Since $a<b<c$ , we can't have $b=c$ and thus $c=-2b$ . Then our arithmetic progre
15
第 15 题
规律与数列
A five-digit palindrome is a positive integer with respective digits abcba , where a is non-zero. Let S be the sum of all five-digit palindromes. What is the sum of the digits of S ?
💡 解题思路
For each digit $a=1,2,\ldots,9$ there are $10\cdot10$ (ways of choosing $b$ and $c$ ) palindromes. So the $a$ s contribute $(1+2+\cdots+9)(100)(10^4+1)$ to the sum. For each digit $b=0,1,2,\ldots,9$ t
16
第 16 题
数论
The product (8)(888\dots8) , where the second factor has k digits, is an integer whose digits have a sum of 1000 . What is k ?
💡 解题思路
We can list the first few numbers in the form $8 \cdot (8....8)$
17
第 17 题
行程问题
A 4× 4× h rectangular box contains a sphere of radius 2 and eight smaller spheres of radius 1 . The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is h ?
💡 解题思路
Let one of the corners be $(0, 0, 0)$ . We can orient the box such that the center of the small sphere closest to the corner is $(1,1,1)$ , and the center of the large sphere is $(2, 2, h/2)$ .
18
第 18 题
数论
The domain of the function f(x)=\log_{\frac12}(\log_4(\log_{\frac14}(\log_{16}(\log_{\frac1{16}}x)))) is an interval of length \tfrac mn , where m and n are relatively prime positive integers. What is m+n ?
💡 解题思路
For all real numbers $a,b,$ and $c$ such that $b>0$ and $b\neq1,$ note that:
19
第 19 题
分数与比例
There are exactly N distinct rational numbers k such that |k|<200 and \[5x^2+kx+12=0\] has at least one integer solution for x . What is N ?
💡 解题思路
Factor the quadratic into \[\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0\] where $-n$ is our integer solution. Then, \[k = \frac{12}{n} + 5n,\] which takes rational values between $-200$ and $
20
第 20 题
几何·面积
In \triangle BAC , \angle BAC=40^\circ , AB=10 , and AC=6 . Points D and E lie on \overline{AB} and \overline{AC} respectively. What is the minimum possible value of BE+DE+CD ?
💡 解题思路
Let $C_1$ be the reflection of $C$ across $\overline{AB}$ , and let $C_2$ be the reflection of $C_1$ across $\overline{AC}$ . Then it is well-known that the quantity $BE+DE+CD$ is minimized when it is
21
第 21 题
规律与数列
For every real number x , let \lfloor x\rfloor denote the greatest integer not exceeding x , and let \[f(x)=\lfloor x\rfloor(2014^{x-\lfloor x\rfloor}-1).\] The set of all numbers x such that 1≤ x<2014 and f(x)≤ 1 is a union of disjoint intervals. What is the sum of the lengths of those intervals?
💡 解题思路
Let $\lfloor x\rfloor=k$ for some integer $1\leq k\leq 2013$ . Then we can rewrite $f(x)$ as $k(2014^{x-k}-1)$ . In order for this to be less than or equal to $1$ , we need $2014^{x-k}-1\leq\dfrac1k\i
22
第 22 题
整数运算
The number 5^{867} is between 2^{2013} and 2^{2014} . How many pairs of integers (m,n) are there such that 1≤ m≤ 2012 and \[5^n<2^m<2^{m+2}<5^{n+1}?\]
💡 解题思路
Between any two consecutive powers of $5$ there are either $2$ or $3$ powers of $2$ (because $2^2<5^1<2^3$ ). Consider the intervals $(5^0,5^1),(5^1,5^2),\dots (5^{866},5^{867})$ . We want the number
23
第 23 题
分数与比例
The fraction \[\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},\] where n is the length of the period of the repeating decimal expansion. What is the sum b_0+b_1+·s+b_{n-1} ?
💡 解题思路
the fraction $\dfrac{1}{99}$ can be written as \[\sum^{\infty}_{n=1}\dfrac{1}{10^{2n}}\] . similarly the fraction $\dfrac{1}{99^2}$ can be written as $\sum^{\infty}_{m=1}\dfrac{1}{10^{2m}}\sum^{\infty
24
第 24 题
综合
Let f_0(x)=x+|x-100|-|x+100| , and for n≥ 1 , let f_n(x)=|f_{n-1}(x)|-1 . For how many values of x is f_{100}(x)=0 ?
💡 解题思路
1. Draw the graph of $f_0(x)$ by dividing the domain into three parts. [asy] unitsize(0); int w = 250; int h = 125; xaxis(-w,w,Ticks(100.0),Arrows); yaxis(-h,h,Ticks(100.0),Arrows); draw((-100,-h)--(-
25
第 25 题
坐标几何
The parabola P has focus (0,0) and goes through the points (4,3) and (-4,-3) . For how many points (x,y)\in P with integer coordinates is it true that |4x+3y|≤ 1000 ?
💡 解题思路
The parabola is symmetric through $y=- \frac{4}{3}x$ , and the common distance is $5$ , so the directrix is the line through $(1,7)$ and $(-7,1)$ , which is the line \[3x-4y = -25.\] Using the point-l