2019A 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
几何·面积
The area of a pizza with radius 4 is N percent larger than the area of a pizza with radius 3 inches. What is the integer closest to N ?
💡 解题思路
The area of the larger pizza is $16\pi$ , while the area of the smaller pizza is $9\pi$ . Therefore, the larger pizza is $\frac{7\pi}{9\pi} \cdot 100\%$ bigger than the smaller pizza. $\frac{7\pi}{9\p
2
第 2 题
分数与比例
Suppose a is 150\% of b . What percent of a is 3b ?
💡 解题思路
Since $a=1.5b$ , that means $b=\frac{a}{1.5}$ . We multiply by $3$ to get a $3b$ term, yielding $3b=2a$ , and $2a$ is $\boxed{\textbf{(D) }200\%}$ of $a$ .
3
第 3 题
综合
A box contains 28 red balls, 20 green balls, 19 yellow balls, 13 blue balls, 11 white balls, and 9 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 15 balls of a single color will be drawn?
💡 解题思路
We try to find the worst case scenario where we can find the maximum number of balls that can be drawn while getting $<15$ of each color by applying the pigeonhole principle and through this we get a
4
第 4 题
规律与数列
What is the greatest number of consecutive integers whose sum is 45?
💡 解题思路
We might at first think that the answer would be $9$ , because $1+2+3 \dots +n = 45$ when $n = 9$ . But note that the problem says that they can be integers, not necessarily positive. Observe also tha
5
第 5 题
几何·面积
Two lines with slopes \dfrac{1}{2} and 2 intersect at (2,2) . What is the area of the triangle enclosed by these two lines and the line x+y=10 ?
💡 解题思路
Let's first work out the slope-intercept form of all three lines: $(x,y)=(2,2)$ and $y=\frac{x}{2} + b$ implies $2=\frac{2}{2} +b=> 2=1+b$ so $b=1$ , while $y=2x + c$ implies $2= 2 \cdot 2+c=> 2=4+c$
6
第 6 题
几何·面积
The figure below shows line \ell with a regular, infinite, recurring pattern of squares and line segments. [图] How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
💡 解题思路
Statement $1$ is true. A $180^{\circ}$ rotation about the point half way between an up-facing square and a down-facing square will yield the same figure.
7
第 7 题
统计
Melanie computes the mean \mu , the median M , and the modes of the 365 values that are the dates in the months of 2019 . Thus her data consist of 121s , 122s , . . . , 1228s , 1129s , 1130s , and 731s . Let d be the median of the modes. Which of the following statements is true?
💡 解题思路
First of all, $d$ obviously has to be smaller than $M$ , since when calculating $M$ , we must take into account the $29$ s, $30$ s, and $31$ s. So we can eliminate choices $B$ and $C$ . Since there ar
8
第 8 题
规律与数列
For a set of four distinct lines in a plane, there are exactly N distinct points that lie on two or more of the lines. What is the sum of all possible values of N ?
💡 解题思路
It is possible to obtain $0$ , $1$ , $3$ , $4$ , $5$ , and $6$ points of intersection, as demonstrated in the following figures:
9
第 9 题
数论
A sequence of numbers is defined recursively by a_1 = 1 , a_2 = \frac{3}{7} , and \[a_n=\frac{a_{n-2} · a_{n-1}}{2a_{n-2} - a_{n-1}}\] for all n ≥ 3 Then a_{2019} can be written as \frac{p}{q} , where p and q are relatively prime positive integers. What is p+q ?
💡 解题思路
Using the recursive formula, we find $a_3=\frac{3}{11}$ , $a_4=\frac{3}{15}$ , and so on. It appears that $a_n=\frac{3}{4n-1}$ , for all $n$ . Setting $n=2019$ , we find $a_{2019}=\frac{3}{8075}$ , so
10
第 10 题
几何·面积
The figure below shows 13 circles of radius 1 within a larger circle. All the intersections occur at points of tangency. What is the area of the region, shaded in the figure, inside the larger circle but outside all the circles of radius 1 ? [图]
For some positive integer k , the repeating base- k representation of the (base-ten) fraction \frac{7}{51} is 0.\overline{23}_k = 0.232323..._k . What is k ?
💡 解题思路
We can expand the fraction $0.\overline{23}_k$ as follows: $0.\overline{23}_k = 2\cdot k^{-1} + 3 \cdot k^{-2} + 2 \cdot k^{-3} + 3 \cdot k^{-4} + \cdots$
12
第 12 题
整数运算
Positive real numbers x ≠ 1 and y ≠ 1 satisfy \log_2{x} = \log_y{16} and xy = 64 . What is (\log_2{\tfrac{x}{y}})^2 ?
💡 解题思路
Let $\log_2{x} = \log_y{16}=k$ , so that $2^k=x$ and $y^k=16 \implies y=2^{\frac{4}{k}}$ . Then we have $(2^k)(2^{\frac{4}{k}})=2^{k+\frac{4}{k}}=2^6$ .
13
第 13 题
计数
How many ways are there to paint each of the integers 2, 3, ·s , 9 either red, green, or blue so that each number has a different color from each of its proper divisors? [图] (Diagram by Technodoggo)
💡 解题思路
The $5$ and $7$ can be painted with no restrictions because the set of integers does not contain a multiple or proper factor of $5$ or $7$ . There are 3 ways to paint each, giving us $\underline{9}$ w
14
第 14 题
综合
For a certain complex number c , the polynomial \[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\] has exactly 4 distinct roots. What is |c| ?
💡 解题思路
The polynomial can be factored further broken down into
15
第 15 题
规律与数列
Positive real numbers a and b have the property that \[√(\log{a)} + √(\log{b)} + \log √(a) + \log √(b) = 100\] and all four terms on the left are positive integers, where log denotes the base 10 logarithm. What is ab ?
💡 解题思路
Since $\sqrt{\log{a}}$ is a positive integer, we get $\log a = x^2$ for some integer $x$ ; since $\log \sqrt{a} = \tfrac 12 \log a$ is a positive integer, we get $x=2m$ . Thus $a=10^{4m^2}$ ; similarl
16
第 16 题
几何·面积
The numbers 1,2,\dots,9 are randomly placed into the 9 squares of a 3 × 3 grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
💡 解题思路
Note that odd sums can only be formed by $(e,e,o)$ or $(o,o,o),$ so we focus on placing the evens: we need to have each even be with another even in each row/column. Because there are only $5$ odd num
17
第 17 题
规律与数列
Let s_k denote the sum of the \textit{k} th powers of the roots of the polynomial x^3-5x^2+8x-13 . In particular, s_0=3 , s_1=5 , and s_2=9 . Let a , b , and c be real numbers such that s_{k+1} = a s_k + b s_{k-1} + c s_{k-2} for k = 2 , 3 , .... What is a+b+c ?
💡 解题思路
Applying Newton's Sums , we have \[s_{k+1}+(-5)s_k+(8)s_{k-1}+(-13)s_{k-2}=0,\] so \[s_{k+1}=5s_k-8s_{k-1}+13s_{k-2},\] we get the answer as $5+(-8)+13=10$ .
18
第 18 题
几何·面积
A sphere with center O has radius 6 . A triangle with sides of length 15, 15, and 24 is situated in space so that each of its sides is tangent to the sphere. What is the distance between O and the plane determined by the triangle? 3D: [图] Plane through triangle: [图]
💡 解题思路
The triangle is placed on the sphere so that its three sides are tangent to the sphere. The cross-section of the sphere created by the plane of the triangle is also the incircle of the triangle. To fi
19
第 19 题
几何·面积
In \triangle ABC with integer side lengths, \cos A = \frac{11}{16} , \cos B = \frac{7}{8} , and \cos C = -\frac{1}{4} . What is the least possible perimeter for \triangle ABC ?
💡 解题思路
Notice that by the Law of Sines, $a:b:c = \sin{A}:\sin{B}:\sin{C}$ , so let's flip all the cosines using $\sin^{2}{x} + \cos^{2}{x} = 1$ ( $\sin{x}$ is positive for $0^{\circ} < x < 180^{\circ}$ , so
20
第 20 题
概率
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval [0,1] . Two random numbers x and y are chosen independently in this manner. What is the probability that |x-y| > \tfrac{1}{2} ?
💡 解题思路
There are several cases depending on what the first coin flip is when determining $x$ and what the first coin flip is when determining $y$ .
21
第 21 题
综合
Let \[z=\frac{1+i}{√(2)}.\] What is \[(z^{1^2}+z^{2^2}+z^{3^2}+\dots+z^{{12}^2}) · (\frac{1}{z^{1^2}}+\frac{1}{z^{2^2}}+\frac{1}{z^{3^2}}+\dots+\frac{1}{z^{{12}^2}})?\]
💡 解题思路
Note that $z = \mathrm{cis }(45^{\circ})$ .
22
第 22 题
几何·面积
Circles \omega and \gamma , both centered at O , have radii 20 and 17 , respectively. Equilateral triangle ABC , whose interior lies in the interior of \omega but in the exterior of \gamma , has vertex A on \omega , and the line containing side \overline{BC} is tangent to \gamma . Segments \overline{AO} and \overline{BC} intersect at P , and \dfrac{BP}{CP} = 3 . Then AB can be written in the form \dfrac{m}{√(n)} - \dfrac{p}{√(q)} for positive integers m , n , p , q with gcd(m,n) = gcd(p,q) = 1 . What is m+n+p+q ? \phantom{}
Define binary operations \diamondsuit and \heartsuit by \[a \diamondsuit b = a^{\log_{7}(b)} and a \heartsuit b = a^{\frac{1}{\log_{7}(b)}}\] for all real numbers a and b for which these expressions are defined. The sequence (a_n) is defined recursively by a_3 = 3 \heartsuit 2 and \[a_n = (n \heartsuit (n-1)) \diamondsuit a_{n-1}\] for all integers n ≥ 4 . To the nearest integer, what is \log_{7}(a_{2019}) ?
💡 解题思路
First note that by log properties $a\diamondsuit b = 7^{(\log_7a)(\log_7b)}$ and $a \heartsuit b = 7^{\frac{\log_7a}{\log_7b}} = 7^{\log_ba}$ .
24
第 24 题
整数运算
For how many integers n between 1 and 50 , inclusive, is \[\frac{(n^2-1)!}{(n!)^n}\] an integer? (Recall that 0! = 1 .)
💡 解题思路
The main insight is that
25
第 25 题
几何·面积
Let \triangle A_0B_0C_0 be a triangle whose angle measures are exactly 59.999^\circ , 60^\circ , and 60.001^\circ . For each positive integer n , define A_n to be the foot of the altitude from A_{n-1} to line B_{n-1}C_{n-1} . Likewise, define B_n to be the foot of the altitude from B_{n-1} to line A_{n-1}C_{n-1} , and C_n to be the foot of the altitude from C_{n-1} to line A_{n-1}B_{n-1} . What is the least positive integer n for which \triangle A_nB_nC_n is obtuse?
💡 解题思路
For all nonnegative integers $n$ , let $\angle C_nA_nB_n=x_n$ , $\angle A_nB_nC_n=y_n$ , and $\angle B_nC_nA_n=z_n$ .