2010B 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
Makarla attended two meetings during her 9 -hour work day. The first meeting took 45 minutes and the second meeting took twice as long. What percent of her work day was spent attending meetings?
💡 解题思路
The total number of minutes in her $9$ -hour work day is $9 \times 60 = 540.$ The total amount of time spend in meetings in minutes is $45 + 45 \times 2 = 135.$ The answer is then $\frac{135}{540}$ $=
2
第 2 题
几何·面积
A big L is formed as shown. What is its area?
💡 解题思路
We find the area of the big rectangle to be $8 \times 5 = 40$ , and the area of the smaller rectangle to be $(8 - 2) \times (5 - 2) = 18$ . The answer is then $40 - 18 = 22$ $(A)$ .
3
第 3 题
应用题
A ticket to a school play cost x dollars, where x is a whole number. A group of 9th graders buys tickets costing a total of \48 , and a group of 10th graders buys tickets costing a total of \64 . How many values for x are possible?
💡 解题思路
We find the greatest common factor of $48$ and $64$ to be $16$ . The number of factors of $16$ is $5$ which is the answer $(E)$ .
4
第 4 题
整数运算
A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
💡 解题思路
$31 \equiv 3 \pmod {7}$ so the week cannot start with Saturday, Sunday, Tuesday or Wednesday as that would result in an unequal number of Mondays and Wednesdays. Therefore, Monday, Thursday, and Frida
5
第 5 题
概率
Lucky Larry's teacher asked him to substitute numbers for a , b , c , d , and e in the expression a-(b-(c-(d+e))) and evaluate the result. Larry ignored the parenthese but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for a , b , c , and d were 1 , 2 , 3 , and 4 , respectively. What number did Larry substitute for e ?
💡 解题思路
We simply plug in the numbers \[1 - 2 - 3 - 4 + e = 1 - (2 - (3 - (4 + e)))\] \[-8 + e = -2 - e\] \[2e = 6\] \[e = 3 \;\;(D)\]
6
第 6 题
综合
At the beginning of the school year, 50\% of all students in Mr. Well's class answered "Yes" to the question "Do you love math", and 50\% answered "No." At the end of the school year, 70\% answered "Yes" and 30\% answered "No." Altogether, x\% of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of x ?
💡 解题思路
The minimum possible value would be $70 - 50 = 20\%$ . The maximum possible value would be $30 + 50 = 80\%$ . The difference is $80 - 20 = \boxed{\textbf{(D) }60}$ .
7
第 7 题
行程问题
Shelby drives her scooter at a speed of 30 miles per hour if it is not raining, and 20 miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of 16 miles in 40 minutes. How many minutes did she drive in the rain?
💡 解题思路
Let $x$ be the time it is not raining, and $y$ be the time it is raining, in hours.
8
第 8 题
统计
Every high school in the city of Euclid sent a team of 3 students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed 37 th and 64 th , respectively. How many schools are in the city?
💡 解题思路
There are $x$ schools. This means that there are $3x$ people. Because no one's score was the same as another person's score, that means that there could only have been $1$ median score. This implies t
9
第 9 题
几何·面积
Let n be the smallest positive integer such that n is divisible by 20 , n^2 is a perfect cube, and n^3 is a perfect square. What is the number of digits of n ?
💡 解题思路
We know that $n^2 = k^3$ and $n^3 = m^2$ . Cubing and squaring the equalities respectively gives $n^6 = k^9 = m^4$ . Let $a = n^6$ . Now we know $a$ must be a perfect $36$ -th power because $lcm(9,4)
10
第 10 题
统计
The average of the numbers 1, 2, 3,·s, 98, 99, and x is 100x . What is x ?
💡 解题思路
We first sum the first $99$ numbers: $\frac{99(100)}{2}=99\cdot50$ . Then, we know that the sum of the series is $99\cdot50+x$ . There are $100$ terms, so we can divide this sum by $100$ and set it eq
11
第 11 题
数论
A palindrome between 1000 and 10,000 is chosen at random. What is the probability that it is divisible by 7 ?
💡 解题思路
View the palindrome as some number with form (decimal representation): $a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0$ . But because the number is a palindrome, $a_3 = a_0, a_2 = a_1$ . Recombi
12
第 12 题
综合
For what value of x does \[\log_{√(2)}√(x)+\log_{2}{x}+\log_{4}{x^2}+\log_{8}{x^3}+\log_{16}{x^4}=40?\]
In \triangle ABC , \cos(2A-B)+\sin(A+B)=2 and AB=4 . What is BC ?
💡 解题思路
We note that $-1$ $\le$ $\sin x$ $\le$ $1$ and $-1$ $\le$ $\cos x$ $\le$ $1$ . Therefore, there is no other way to satisfy this equation other than making both $\cos(2A-B)=1$ and $\sin(A+B)=1$ , since
14
第 14 题
规律与数列
Let a , b , c , d , and e be positive integers with a+b+c+d+e=2010 and let M be the largest of the sum a+b , b+c , c+d and d+e . What is the smallest possible value of M ?
💡 解题思路
We want to try make $a+b$ , $b+c$ , $c+d$ , and $d+e$ as close as possible so that $M$ , the maximum of these, is smallest.
15
第 15 题
整数运算
For how many ordered triples (x,y,z) of nonnegative integers less than 20 are there exactly two distinct elements in the set \{i^x, (1+i)^y, z\} , where i=√(-1) ?
💡 解题思路
We have either $i^{x}=(1+i)^{y}\neq z$ , $i^{x}=z\neq(1+i)^{y}$ , or $(1+i)^{y}=z\neq i^x$ .
16
第 16 题
数论
Positive integers a , b , and c are randomly and independently selected with replacement from the set \{1, 2, 3,\dots, 2010\} . What is the probability that abc + ab + a is divisible by 3 ?
💡 解题思路
We group this into groups of $3$ , because $3|2010$ . This means that every residue class mod 3 has an equal probability.
17
第 17 题
统计
The entries in a 3 × 3 array include all the digits from 1 through 9 , arranged so that the entries in every row and column are in increasing order. How many such arrays are there?
💡 解题思路
Observe that all tables must have 1s and 9s in the corners, 8s and 2s next to those corner squares, and 4-6 in the middle square. Also note that for each table, there exists a valid table diagonally s
18
第 18 题
概率
A frog makes 3 jumps, each exactly 1 meter long. The directions of the jumps are chosen independently at random. What is the probability that the frog's final position is no more than 1 meter from its starting position?
💡 解题思路
We will let the moves be complex numbers $a$ , $b$ , and $c$ , each of magnitude one. The starts on the origin. It is relatively easy to show that exactly one element in the set \[\{|a + b + c|, |a +
19
第 19 题
规律与数列
A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than 100 points. What was the total number of points scored by the two teams in the first half?
💡 解题思路
Let $a,ar,ar^{2},ar^{3}$ be the quarterly scores for the Raiders. We know $r > 1$ because the sequence is said to be increasing. We also know that each of $a, ar, ar^2, ar^3$ is an integer. We start b
20
第 20 题
规律与数列
A geometric sequence (a_n) has a_1=\sin x , a_2=\cos x , and a_3= \tan x for some real number x . For what value of n does a_n=1+\cos x ?
💡 解题思路
By the defintion of a geometric sequence, we have $\cos^2x=\sin x \tan x$ . Since $\tan x=\frac{\sin x}{\cos x}$ , we can rewrite this as $\cos^3x=\sin^2x$ .
21
第 21 题
整数运算
Let a > 0 , and let P(x) be a polynomial with integer coefficients such that What is the smallest possible value of a ?
💡 解题思路
We observe that because $P(1) = P(3) = P(5) = P(7) = a$ , if we define a new polynomial $R(x)$ such that $R(x) = P(x) - a$ , $R(x)$ has roots when $P(x) = a$ ; namely, when $x=1,3,5,7$ .
22
第 22 题
整数运算
Let ABCD be a cyclic quadrilateral. The side lengths of ABCD are distinct integers less than 15 such that BC· CD=AB· DA . What is the largest possible value of BD ?
💡 解题思路
Let $AB = a$ , $BC = b$ , $CD = c$ , and $AD = d$ . We see that by the Law of Cosines on $\triangle ABD$ and $\triangle CBD$ , we have:
23
第 23 题
方程
Monic quadratic polynomial P(x) and Q(x) have the property that P(Q(x)) has zeros at x=-23, -21, -17, and -15 , and Q(P(x)) has zeros at x=-59,-57,-51 and -49 . What is the sum of the minimum values of P(x) and Q(x) ?
💡 解题思路
$P(x) = (x - a)^2 - b, Q(x) = (x - c)^2 - d$ . Notice that $P(x)$ has roots $a\pm \sqrt {b}$ , so that the roots of $P(Q(x))$ are the roots of $Q(x) = a + \sqrt {b}, a - \sqrt {b}$ . For each individu
24
第 24 题
规律与数列
The set of real numbers x for which \[\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1\] is the union of intervals of the form a<x\le b . What is the sum of the lengths of these intervals?
💡 解题思路
Because the right side of the inequality is a horizontal line, the left side can be translated horizontally by any value and the intervals will remain the same. For simplicity of calculation, we will
25
第 25 题
数论
For every integer n\ge2 , let pow(n) be the largest power of the largest prime that divides n . For example pow(144)=pow(2^4·3^2)=3^2 . What is the largest integer m such that 2010^m divides
💡 解题思路
Because 67 is the largest prime factor of 2010, it means that in the prime factorization of $\prod_{n=2}^{5300}\text{pow}(n)$ , there'll be $p_1 ^{e_1} \cdot p_2 ^{e_2} \cdot .... 67^x ...$ where $x$