2011B 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 \frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?
💡 解题思路
Add up the numbers in each fraction to get $\frac{12}{9} - \frac{9}{12}$ , which equals $\frac{4}{3} - \frac{3}{4}$ . Doing the subtraction yields $\boxed{\frac{7}{12}\ \textbf{(C)}}$
2
第 2 题
统计
Josanna's test scores to date are 90, 80, 70, 60, and 85. Her goal is to raise her test average at least 3 points with her next test. What is the minimum test score she would need to accomplish this goal?
💡 解题思路
Take the average of her current test scores, which is \[\frac{90+80+70+60+85}{5} = \frac{385}{5} = 77\]
3
第 3 题
应用题
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid A dollars and Bernardo had paid B dollars, where A < B. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
💡 解题思路
The total amount of money that was spent during the trip was \[A + B\] So each person should pay \[\frac{A+B}{2}\] if they were to share the costs equally. Because LeRoy has already paid $A$ dollars o
4
第 4 题
数字运算
In multiplying two positive integers a and b , Ron reversed the digits of the two-digit number a . His erroneous product was 161. What is the correct value of the product of a and b ?
💡 解题思路
Taking the prime factorization of $161$ reveals that it is equal to $23*7.$ Therefore, the only ways to represent $161$ as a product of two positive integers is $161*1$ and $23*7.$ Because neither $16
5
第 5 题
数论
Let N be the second smallest positive integer that is divisible by every positive integer less than 7 . What is the sum of the digits of N ?
💡 解题思路
$N$ must be divisible by every positive integer less than $7$ , or $1, 2, 3, 4, 5,$ and $6$ . Each number that is divisible by each of these is a multiple of their least common multiple. $LCM(1,2,3,4,
6
第 6 题
几何·面积
Two tangents to a circle are drawn from a point A . The points of contact B and C divide the circle into arcs with lengths in the ratio 2 : 3 . What is the degree measure of \angle{BAC} ?
💡 解题思路
In order to solve this problem, use of the tangent-tangent intersection theorem (Angle of intersection between two tangents dividing a circle into arc length A and arc length B = 1/2 (Arc A° - Arc B°)
7
第 7 题
分数与比例
Let x and y be two-digit positive integers with mean 60 . What is the maximum value of the ratio \frac{x}{y} ?
💡 解题思路
If $x$ and $y$ have a mean of $60$ , then $\frac{x+y}{2}=60$ and $x+y=120$ . To maximize $\frac{x}{y}$ , we need to maximize $x$ and minimize $y$ . Since they are both two-digit positive integers, the
8
第 8 题
几何·面积
Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
💡 解题思路
To find Keiko's speed, all we need to find is the difference between the distance around the inside edge of the track and the distance around the outside edge of the track, and divide it by the differ
9
第 9 题
概率
Two real numbers are selected independently and at random from the interval [-20,10] . What is the probability that the product of those numbers is greater than zero?
💡 解题思路
For the product to be greater than zero, we must have either both numbers negative or both positive.
10
第 10 题
几何·面积
Rectangle ABCD has AB=6 and BC=3 . Point M is chosen on side AB so that \angle AMD=\angle CMD . What is the degree measure of \angle AMD ? \textrm{(A)}\ 15 \textrm{(B)}\ 30 \textrm{(C)}\ 45 \textrm{(D)}\ 60 \textrm{(E)}\ 75
💡 解题思路
Since $AB \parallel CD$ , $\angle AMD = \angle CDM$ , so $\angle AMD = \angle CMD = \angle CDM$ , so $\bigtriangleup CMD$ is isosceles, and hence $CM=CD=6$ . Therefore, $\triangle BMC$ is a 30-60-90 t
11
第 11 题
坐标几何
A frog located at (x,y) , with both x and y integers, makes successive jumps of length 5 and always lands on points with integer coordinates. Suppose that the frog starts at (0,0) and ends at (1,0) . What is the smallest possible number of jumps the frog makes?
💡 解题思路
Since the frog always jumps in length $5$ and lands on a lattice point, the sum of its coordinates must change either by $5$ (by jumping parallel to the x- or y-axis), or by $3$ or $4$ (3-4-5 right tr
12
第 12 题
几何·面积
A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is the probability that the dart lands within the center square? [图]
💡 解题思路
Let's assume that the side length of the octagon is $x$ . The area of the center square is just $x^2$ . The triangles are all $45-45-90$ triangles, with a side length ratio of $1:1:\sqrt{2}$ . The are
13
第 13 题
规律与数列
Brian writes down four integers w > x > y > z whose sum is 44 . The pairwise positive differences of these numbers are 1, 3, 4, 5, 6 and 9 . What is the sum of the possible values of w ?
💡 解题思路
Assume that $y-z=a, x-y=b, w-x=c.$ $w-z$ results in the greatest pairwise difference, and thus it is $9$ . This means $a+b+c=9$ . $a,b,c$ must be in the set ${1,3,4,5,6}$ . The only way for 3 numbers
14
第 14 题
几何·角度
A segment through the focus F of a parabola with vertex V is perpendicular to \overline{FV} and intersects the parabola in points A and B . What is \cos(\angle AVB) ?
💡 解题思路
Name the directrix of the parabola $l$ . Define $d(X,k)$ to be the distance between a point $X$ and a line $k$ .
15
第 15 题
数论
How many positive two-digit integers are factors of 2^{24}-1 ? ~ pi_is_3.14
💡 解题思路
Repeating difference of squares :
16
第 16 题
几何·面积
Rhombus ABCD has side length 2 and \angle B = 120 °. Region R consists of all points inside the rhombus that are closer to vertex B than any of the other three vertices. What is the area of R ?
💡 解题思路
Suppose that $P$ is a point in the rhombus $ABCD$ and let $\ell_{BC}$ be the perpendicular bisector of $\overline{BC}$ . Then $PB < PC$ if and only if $P$ is on the same side of $\ell_{BC}$ as $B$ . T
17
第 17 题
规律与数列
Let f(x) = 10^{10x}, g(x) = \log_{10}(\frac{x}{10}), h_1(x) = g(f(x)) , and h_n(x) = h_1(h_{n-1}(x)) for integers n ≥ 2 . What is the sum of the digits of h_{2011}(1) ?
A pyramid has a square base with side of length 1 and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube?
💡 解题思路
We can use the Pythagorean Theorem to split one of the triangular faces into two 30-60-90 triangles with side lengths $\frac{1}{2}, 1$ and $\frac{\sqrt{3}}{2}$ .
19
第 19 题
坐标几何
A lattice point in an xy -coordinate system is any point (x, y) where both x and y are integers. The graph of y = mx + 2 passes through no lattice point with 0 < x ≤ 100 for all m such that \frac{1}{2} < m < a . What is the maximum possible value of a ?
💡 解题思路
It is very easy to see that the $+2$ in the graph does not impact whether it passes through the lattice.
20
第 20 题
几何·面积
Triangle ABC has AB = 13, BC = 14 , and AC = 15 . The points D, E , and F are the midpoints of \overline{AB}, \overline{BC} , and \overline{AC} respectively. Let X ≠ E be the intersection of the circumcircles of \triangle BDE and \triangle CEF . What is XA + XB + XC ?
💡 解题思路
Let us also consider the circumcircle of $\triangle ADF$ .
21
第 21 题
统计
The arithmetic mean of two distinct positive integers x and y is a two-digit integer. The geometric mean of x and y is obtained by reversing the digits of the arithmetic mean. What is |x - y| ?
💡 解题思路
Answer: (D)
22
第 22 题
几何·面积
Let T_1 be a triangle with side lengths 2011 , 2012 , and 2013 . For n ≥ 1 , if T_n = \triangle ABC and D, E , and F are the points of tangency of the incircle of \triangle ABC to the sides AB , BC , and AC , respectively, then T_{n+1} is a triangle with side lengths AD, BE , and CF , if it exists. What is the perimeter of the last triangle in the sequence (T_n) ?
💡 解题思路
Answer: (D)
23
第 23 题
坐标几何
A bug travels in the coordinate plane, moving only along the lines that are parallel to the x -axis or y -axis. Let A = (-3, 2) and B = (3, -2) . Consider all possible paths of the bug from A to B of length at most 20 . How many points with integer coordinates lie on at least one of these paths?
💡 解题思路
We declare a point $(x, y)$ to make up for the extra steps that the bug has to move. If the point $(x, y)$ satisfies the property that $|x - 3| + |y + 2| + |x + 3| + |y - 2| \le 20$ , then it is in th
24
第 24 题
几何·面积
Let P(z) = z^8 + (4√(3) + 6)z^4 - (4√(3) + 7) . What is the minimum perimeter among all the 8 -sided polygons in the complex plane whose vertices are precisely the zeros of P(z) ?
💡 解题思路
Answer: (B)
25
第 25 题
概率
For every m and k integers with k odd, denote by [\frac{m}{k}] the integer closest to \frac{m}{k} . For every odd integer k , let P(k) be the probability that \[[\frac{n}{k}] + [\frac{100 - n}{k}] = [\frac{100}{k}]\] for an integer n randomly chosen from the interval 1 ≤ n ≤ 99! . What is the minimum possible value of P(k) over the odd integers k in the interval 1 ≤ k ≤ 99 ?
