2018A 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
分数与比例
A large urn contains 100 balls, of which 36 \% are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be 72 \% ? (No red balls are to be removed.)
💡 解题思路
There are $36$ red balls; for these red balls to comprise $72 \%$ of the urn, there must be only $14$ blue balls. Since there are currently $64$ blue balls, this means we must remove $\boxed{\textbf{(
2
第 2 题
应用题
While exploring a cave, Carl comes across a collection of 5 -pound rocks worth \14 each, 4 -pound rocks worth \11 each, and 1 -pound rocks worth \2 each. There are at least 20 of each size. He can carry at most 18$ pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
💡 解题思路
The value of $5$ -pound rocks is $\$14\div5=\$2.80$ per pound, and the value of $4$ -pound rocks is $\$11\div4=\$2.75$ per pound. Clearly, Carl should not carry more than three $1$ -pound rocks. Other
3
第 3 题
计数
How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6 -period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.)
💡 解题思路
We must place the classes into the periods such that no two classes are in the same period or in consecutive periods.
4
第 4 题
行程问题
Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let d be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of d ?
💡 解题思路
For each of the false statements, we identify its corresponding true statement. Note that:
5
第 5 题
规律与数列
What is the sum of all possible values of k for which the polynomials x^2 - 3x + 2 and x^2 - 5x + k have a root in common?
💡 解题思路
We factor $x^2-3x+2$ into $(x-1)(x-2)$ . Thus, either $1$ or $2$ is a root of $x^2-5x+k$ . If $1$ is a root, then $1^2-5\cdot1+k=0$ , so $k=4$ . If $2$ is a root, then $2^2-5\cdot2+k=0$ , so $k=6$ . T
6
第 6 题
统计
For positive integers m and n such that m+10<n+1 , both the mean and the median of the set \{m, m+4, m+10, n+1, n+2, 2n\} are equal to n . What is m+n ?
💡 解题思路
The mean and median are \[\frac{3m+4n+17}{6}=\frac{m+n+11}{2}=n,\] so $3m+17=2n$ and $m+11=n$ . Solving this gives $\left(m,n\right)=\left(5,16\right)$ for $m+n=\boxed{\textbf{(B)}~21}$ . (trumpeter)
7
第 7 题
整数运算
For how many (not necessarily positive) integer values of n is the value of 4000· (\tfrac{2}{5})^n an integer?
💡 解题思路
Note that \[4000\cdot \left(\frac{2}{5}\right)^n=\left(2^5\cdot5^3\right)\cdot \left(2\cdot5^{-1}\right)^n=2^{5+n}\cdot5^{3-n}.\] Since this expression is an integer, we need:
8
第 8 题
几何·面积
All of the triangles in the diagram below are similar to isosceles triangle ABC , in which AB=AC . Each of the 7 smallest triangles has area 1, and \triangle ABC has area 40 . What is the area of trapezoid DBCE ? [图]
💡 解题思路
Let $x$ be the area of $ADE$ . Note that $x$ is comprised of the $7$ small isosceles triangles and a triangle similar to $ADE$ with side length ratio $3:4$ (so an area ratio of $9:16$ ). Thus, we have
9
第 9 题
综合
Which of the following describes the largest subset of values of y within the closed interval [0,π] for which \[\sin(x+y)≤ \sin(x)+\sin(y)\] for every x between 0 and π , inclusive?
💡 解题思路
On the interval $[0, \pi]$ sine is nonnegative; thus $\sin(x + y) = \sin x \cos y + \sin y \cos x \le \sin x + \sin y$ for all $x, y \in [0, \pi]$ and equality only occurs when $\cos x = \cos y = 1$ ,
10
第 10 题
综合
💡 解题思路
We can solve this by graphing the equations. The second equation looks challenging to graph, but start by graphing it in the first quadrant only (which is easy since the inner absolute value signs can
Let S be a set of 6 integers taken from \{1,2,\dots,12\} with the property that if a and b are elements of S with a<b , then b is not a multiple of a . What is the least possible value of an element in S ?
💡 解题思路
We start with $2$ because $1$ is not an answer choice. We would have to include every odd number except $1$ to fill out the set, but then $3$ and $9$ would violate the rule, so that won't work.
13
第 13 题
整数运算
How many nonnegative integers can be written in the form \[a_7·3^7+a_6·3^6+a_5·3^5+a_4·3^4+a_3·3^3+a_2·3^2+a_1·3^1+a_0·3^0,\] where a_i\in \{-1,0,1\} for 0\le i \le 7 ?
💡 解题思路
This looks like balanced ternary, in which all the integers with absolute values less than $\frac{3^n}{2}$ are represented in $n$ digits. There are 8 digits. Plugging in 8 into the formula for the bal
14
第 14 题
数论
The solution to the equation \log_{3x} 4 = \log_{2x} 8 , where x is a positive real number other than \frac{1}{3} or \frac{1}{2} , can be written as \frac {p}{q} where p and q are relatively prime positive integers. What is p + q ?
💡 解题思路
We apply the Change of Base Formula, then rearrange: \begin{align*} \frac{\log_2{4}}{\log_2{(3x)}}&=\frac{\log_2{8}}{\log_2{(2x)}} \\ \frac{2}{\log_2{(3x)}}&=\frac{3}{\log_2{(2x)}} \\ 3\log_2{(3x)}&=2
15
第 15 题
几何·面积
A scanning code consists of a 7 × 7 grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of 49 squares. A scanning code is called \textit{symmetric} if its look does not change when the entire square is rotated by a multiple of 90 ^{\circ} counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes?
💡 解题思路
Draw a $7 \times 7$ square.
16
第 16 题
综合
Which of the following describes the set of values of a for which the curves x^2+y^2=a^2 and y=x^2-a in the real xy -plane intersect at exactly 3 points?
💡 解题思路
Substituting $y=x^2-a$ into $x^2+y^2=a^2$ , we get \[x^2+(x^2-a)^2=a^2 \implies x^2+x^4-2ax^2=0 \implies x^2(x^2-(2a-1))=0\] Since this is a quartic, there are $4$ total roots (counting multiplicity).
17
第 17 题
几何·面积
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square S so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from S to the hypotenuse is 2 units. What fraction of the field is planted? [图]
💡 解题思路
Note that the hypotenuse of the field is $5,$ and the area of the field is $6.$ Let $x$ be the side-length of square $S.$
18
第 18 题
几何·面积
Triangle ABC with AB=50 and AC=10 has area 120 . Let D be the midpoint of \overline{AB} , and let E be the midpoint of \overline{AC} . The angle bisector of \angle BAC intersects \overline{DE} and \overline{BC} at F and G , respectively. What is the area of quadrilateral FDBG ?
💡 解题思路
Let $BC = a$ , $BG = x$ , $GC = y$ , and the length of the perpendicular from $BC$ through $A$ be $h$ . By angle bisector theorem, we have that \[\frac{50}{x} = \frac{10}{y},\] where $y = -x+a$ . Ther
19
第 19 题
数论
Let A be the set of positive integers that have no prime factors other than 2 , 3 , or 5 . The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + ·s\] of the reciprocals of the elements of A can be expressed as \frac{m}{n} , where m and n are relatively prime positive integers. What is m+n ?
💡 解题思路
Note that the fractions of the form $\frac{1}{2^a3^b5^c},$ where $a,b,$ and $c$ are nonnegative integers, span all terms of the infinite sum.
20
第 20 题
几何·面积
Triangle ABC is an isosceles right triangle with AB=AC=3 . Let M be the midpoint of hypotenuse \overline{BC} . Points I and E lie on sides \overline{AC} and \overline{AB} , respectively, so that AI>AE and AIME is a cyclic quadrilateral. Given that triangle EMI has area 2 , the length CI can be written as \frac{a-√(b)}{c} , where a , b , and c are positive integers and b is not divisible by the square of any prime. What is the value of a+b+c ?
💡 解题思路
Observe that $\triangle{EMI}$ is isosceles right ( $M$ is the midpoint of diameter arc $EI$ since $m\angle MEI = m\angle MAI = 45^\circ$ ), so $MI=2,MC=\frac{3}{\sqrt{2}}$ . With $\angle{MCI}=45^\circ
21
第 21 题
综合
Which of the following polynomials has the greatest real root?
💡 解题思路
Denote the polynomials in the answer choices by $A(x),B(x),C(x),D(x),$ and $E(x),$ respectively.
22
第 22 题
几何·面积
The solutions to the equations z^2=4+4√(15)i and z^2=2+2\sqrt 3i, where i=√(-1), form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form p\sqrt q-r\sqrt s, where p,q,r, and s are positive integers and neither q nor s is divisible by the square of any prime number. What is p+q+r+s?
💡 解题思路
We solve each equation separately:
23
第 23 题
几何·面积
In \triangle PAT,\angle P=36^{\circ},\angle A=56^{\circ}, and PA=10. Points U and G lie on sides \overline{TP} and \overline{TA}, respectively, so that PU=AG=1. Let M and N be the midpoints of segments \overline{PA} and \overline{UG}, respectively. What is the degree measure of the acute angle formed by lines MN and PA? [图] ~MRENTHUSIASM
💡 解题思路
Let $P$ be the origin, and $PA$ lie on the $x$ -axis.
24
第 24 题
概率
Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 and 1, and Bob announces that he will choose his number uniformly at random from all the numbers between \tfrac{1}{2} and \tfrac{2}{3}. Armed with this information, what number should Carol choose to maximize her chance of winning?
💡 解题思路
The expected value of Alice's number is $\frac12\left(0+1\right)=\frac12,$ and the expected value of Bob's number is $\frac12\left(\frac12+\frac23\right)=\frac{7}{12}.$ To maximize her chance of winni
25
第 25 题
数字运算
For a positive integer n and nonzero digits a , b , and c , let A_n be the n -digit integer each of whose digits is equal to a ; let B_n be the n -digit integer each of whose digits is equal to b , and let C_n be the 2n -digit (not n -digit) integer each of whose digits is equal to c . What is the greatest possible value of a + b + c for which there are at least two values of n such that C_n - B_n = A_n^2 ?
💡 解题思路
By geometric series, we have \begin{alignat*}{8} A_n&=a\bigl(\phantom{ }\underbrace{111\cdots1}_{n\text{ digits}}\phantom{ }\bigr)&&=a\left(1+10+10^2+\cdots+10^{n-1}\right)&&=a\cdot\frac{10^n-1}{9}, \