2009A 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
行程问题
Kim's flight took off from Newark at 10:34 AM and landed in Miami at 1:18 PM. Both cities are in the same time zone. If her flight took h hours and m minutes, with 0 < m < 60 , what is h + m ?
💡 解题思路
There is $1$ hour and $60-34 = 26$ minutes between 10:34 AM and noon; and there is $1$ hour and $18$ minutes between noon and 1:18 PM. Hence the flight took $2$ hours and $26 + 18 = 44$ minutes,and 2+
2
第 2 题
综合
Which of the following is equal to 1 + \frac {1}{1 + \frac {1}{1 + 1}} ?
💡 解题思路
We compute:
3
第 3 题
综合
What number is one third of the way from \frac14 to \frac34 ?
💡 解题思路
We can rewrite the two given fractions as $\frac 3{12}$ and $\frac 9{12}$ . (We multiplied all numerators and denominators by $3$ .)
4
第 4 题
概率
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
💡 解题思路
Pre-Note: This solution is kinda just guessing, idk you decide.
5
第 5 题
立体几何
One dimension of a cube is increased by 1 , another is decreased by 1 , and the third is left unchanged. The volume of the new rectangular solid is 5 less than that of the cube. What was the volume of the cube?
💡 解题思路
Let the original cube have edge length $a$ . Then its volume is $a^3$ . The new box has dimensions $a-1$ , $a$ , and $a+1$ , hence its volume is $(a-1)a(a+1) = a^3-a$ . The difference between the two
6
第 6 题
整数运算
Suppose that P = 2^m and Q = 3^n . Which of the following is equal to 12^{mn} for every pair of integers (m,n) ?
💡 解题思路
We have $12^{mn} = (2\cdot 2\cdot 3)^{mn} = 2^{2mn} \cdot 3^{mn} = (2^m)^{2n} \cdot (3^n)^m = \boxed{\bold{E)} P^{2n} Q^m}$ .
7
第 7 题
规律与数列
The first three terms of an arithmetic sequence are 2x - 3 , 5x - 11 , and 3x + 1 respectively. The n th term of the sequence is 2009 . What is n ?
💡 解题思路
As this is an arithmetic sequence, the difference must be constant: $(5x-11) - (2x-3) = (3x+1) - (5x-11)$ . This solves to $x=4$ . The first three terms then are $5$ , $9$ , and $13$ . In general, the
8
第 8 题
几何·面积
Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
💡 解题思路
The area of the outer square is $4$ times that of the inner square. Therefore the side of the outer square is $\sqrt 4 = 2$ times that of the inner square.
9
第 9 题
函数
Suppose that f(x+3)=3x^2 + 7x + 4 and f(x)=ax^2 + bx + c . What is a+b+c ?
💡 解题思路
As $f(x)=ax^2 + bx + c$ , we have $f(1)=a\cdot 1^2 + b\cdot 1 + c = a+b+c$ .
10
第 10 题
整数运算
In quadrilateral ABCD , AB = 5 , BC = 17 , CD = 5 , DA = 9 , and BD is an integer. What is BD ?
💡 解题思路
By the triangle inequality we have $BD BC$ , hence $BD > BC - CD = 17 - 5 = 12$ .
11
第 11 题
几何·面积
The figures F_1 , F_2 , F_3 , and F_4 shown are the first in a sequence of figures. For n\ge3 , F_n is constructed from F_{n - 1} by surrounding it with a square and placing one more diamond on each side of the new square than F_{n - 1} had on each side of its outside square. For example, figure F_3 has 13 diamonds. How many diamonds are there in figure F_{20} ?
💡 解题思路
Split $F_n$ into $4$ congruent triangles by its diagonals (like in the pictures in the problem). This shows that the number of diamonds it contains is equal to $4$ times the $(n-2)$ th triangular numb
12
第 12 题
规律与数列
How many positive integers less than 1000 are 6 times the sum of their digits?
💡 解题思路
The sum of the digits is at most $9+9+9=27$ . Therefore the number is at most $6\cdot 27 = 162$ . Out of the numbers $1$ to $162$ the one with the largest sum of digits is $99$ , and the sum is $9+9=1
13
第 13 题
几何·角度
A ship sails 10 miles in a straight line from A to B , turns through an angle between 45^{\circ} and 60^{\circ} , and then sails another 20 miles to C . Let AC be measured in miles. Which of the following intervals contains AC^2 ? [图]
💡 解题思路
To answer the question we are asked, it is enough to compute $AC^2$ for two different angles, preferably for both extremes ( $45$ and $60$ degrees). You can use the law of cosines to do so.
14
第 14 题
几何·面积
A triangle has vertices (0,0) , (1,1) , and (6m,0) , and the line y = mx divides the triangle into two triangles of equal area. What is the sum of all possible values of m ?
💡 解题思路
Let's label the three points as $A=(0,0)$ , $B=(1,1)$ , and $C=(6m,0)$ .
15
第 15 题
综合
For what value of n is i + 2i^2 + 3i^3 + ·s + ni^n = 48 + 49i ? Note: here i = \sqrt { - 1} .
💡 解题思路
We know that $i^x$ cycles every $4$ powers so we group the sum in $4$ s. \[i+2i^2+3i^3+4i^4=2-2i\] \[5i^5+6i^6+7i^7+8i^8=2-2i\]
16
第 16 题
几何·面积
A circle with center C is tangent to the positive x and y -axes and externally tangent to the circle centered at (3,0) with radius 1 . What is the sum of all possible radii of the circle with center C ?
💡 解题思路
Let $r$ be the radius of our circle. For it to be tangent to the positive $x$ and $y$ axes, we must have $C=(r,r)$ . For the circle to be externally tangent to the circle centered at $(3,0)$ with radi
17
第 17 题
规律与数列
Let a + ar_1 + ar_1^2 + ar_1^3 + ·s and a + ar_2 + ar_2^2 + ar_2^3 + ·s be two different infinite geometric series of positive numbers with the same first term. The sum of the first series is r_1 , and the sum of the second series is r_2 . What is r_1 + r_2 ?
💡 解题思路
Using the formula for the sum of a geometric series we get that the sums of the given two sequences are $\frac a{1-r_1}$ and $\frac a{1-r_2}$ .
18
第 18 题
数论
For k > 0 , let I_k = 10\ldots 064 , where there are k zeros between the 1 and the 6 . Let N(k) be the number of factors of 2 in the prime factorization of I_k . What is the maximum value of N(k) ?
💡 解题思路
The number $I_k$ can be written as $10^{k+2} + 64 = 5^{k+2}\cdot 2^{k+2} + 2^6$ .
19
第 19 题
几何·面积
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were A and B , respectively. Each polygon had a side length of 2 . Which of the following is true?
💡 解题思路
In any regular polygon with side length $2$ , consider the isosceles triangle formed by the center of the polygon $S$ and two consecutive vertices $X$ and $Y$ . We are given that $XY=2$ . Obviously $S
20
第 20 题
几何·面积
Convex quadrilateral ABCD has AB = 9 and CD = 12 . Diagonals AC and BD intersect at E , AC = 14 , and \triangle AED and \triangle BEC have equal areas. What is AE ?
💡 解题思路
Let $[ABC]$ denote the area of triangle $ABC$ . $[AED] = [BEC]$ , so $[ABD] = [AED] + [AEB] = [BEC] + [AEB] = [ABC]$ . Since triangles $ABD$ and $ABC$ share a base, they also have the same height and
21
第 21 题
综合
Let p(x) = x^3 + ax^2 + bx + c , where a , b , and c are complex numbers. Suppose that What is the number of nonreal zeros of x^{12} + ax^8 + bx^4 + c ?
💡 解题思路
From the three zeroes, we have $p(x) = (x - (2009 + 9002\pi i))(x - 2009)(x - 9002)$ .
22
第 22 题
几何·面积
A regular octahedron has side length 1 . A plane parallel to two of its opposite faces cuts the octahedron into the two congruent solids. The polygon formed by the intersection of the plane and the octahedron has area \frac {a\sqrt {b}}{c} , where a , b , and c are positive integers, a and c are relatively prime, and b is not divisible by the square of any prime. What is a + b + c ?
💡 解题思路
Firstly, note that the intersection of the plane must be a hexagon. Consider the net of the octahedron. Notice that the hexagon becomes a line on the net. Also, notice that, given the parallel to the
23
第 23 题
几何·面积
Functions f and g are quadratic , g(x) = - f(100 - x) , and the graph of g contains the vertex of the graph of f . The four x -intercepts on the two graphs have x -coordinates x_1 , x_2 , x_3 , and x_4 , in increasing order, and x_3 - x_2 = 150 . Then x_4 - x_1 = m + n\sqrt p , where m , n , and p are positive integers, and p is not divisible by the square of any prime. What is m + n + p ?
💡 解题思路
The two quadratics are $180^{\circ}$ rotations of each other about $(50,0)$ . Since we are only dealing with differences of roots, we can translate them to be symmetric about $(0,0)$ . Now $x_3 = - x_
24
第 24 题
行程问题
The tower function of twos is defined recursively as follows: T(1) = 2 and T(n + 1) = 2^{T(n)} for n\ge1 . Let A = (T(2009))^{T(2009)} and B = (T(2009))^A . What is the largest integer k for which \[\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k times}\] is defined?
💡 解题思路
Testing the first two (or three) positive integers instead of 2009, $k$ seems to always be 4 more. Put E and go on to tackle #25 :) ~ dolphin7
25
第 25 题
规律与数列
The first two terms of a sequence are a_1 = 1 and a_2 = \frac {1}{\sqrt3} . For n\ge1 , What is |a_{2009}| ?
💡 解题思路
Consider another sequence $\{\theta_1, \theta_2, \theta_3...\}$ such that $a_n = \tan{\theta_n}$ , and $0 \leq \theta_n < 180$ .