2012 AMC 10A — Official Competition Problems (February 2012)
📅 2012 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
时间问题
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
💡 解题思路
Cagney can frost one in $20$ seconds, and Lacey can frost one in $30$ seconds. Working together, they can frost one in $\frac{20\cdot30}{20+30} = \frac{600}{50} = 12$ seconds. In $300$ seconds ( $5$ m
2
第 2 题
几何·面积
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
💡 解题思路
Cutting the square in half will bisect one pair of sides while the other side will remain unchanged. Thus, the new square is $\frac{8}{2}$ by $8$ , or $\boxed{\textbf{(E)}\ 4\ \text{by}\ 8}$ .
3
第 3 题
综合
A bug crawls along a number line, starting at -2 . It crawls to -6 , then turns around and crawls to 5 . How many units does the bug crawl altogether?
Let \angle ABC = 24^\circ and \angle ABD = 20^\circ . What is the smallest possible degree measure for \angle CBD ?
💡 解题思路
$\angle ABD$ and $\angle ABC$ share ray $AB$ . In order to minimize the value of $\angle CBD$ , $D$ should be located between $A$ and $C$ .
5
第 5 题
统计
Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year?
💡 解题思路
Half of the 100 adult cats are female, so there are $\frac{100}{2}$ = $50$ female cats. Half of those female adult cats have a litter of kittens, so there would be $\frac{50}{2}$ = $25$ litters. Since
6
第 6 题
规律与数列
The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers?
💡 解题思路
Let the two numbers equal $x$ and $y$ . From the information given in the problem, two equations can be written:
7
第 7 题
分数与比例
In a bag of marbles, \frac{3}{5} of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?
💡 解题思路
Assume that there are 5 total marbles in the bag. The actual number does not matter, since all we care about is the ratios, and the only operation performed on the marbles in the bag is doubling.
8
第 8 题
规律与数列
The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number?
💡 解题思路
Let the three numbers be equal to $a$ , $b$ , and $c$ . We can now write three equations:
9
第 9 题
概率
A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two of each 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7?
💡 解题思路
The total number of combinations when rolling two dice is $6*6 = 36$ .
10
第 10 题
几何·面积
Mary divides a circle into 12 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
💡 解题思路
Let $a_1$ be the first term of the arithmetic progression and $a_{12}$ be the last term of the arithmetic progression. From the formula of the sum of an arithmetic progression (or arithmetic series),
11
第 11 题
几何·面积
Externally tangent circles with centers at points A and B have radii of lengths 5 and 3 , respectively. A line externally tangent to both circles intersects ray AB at point C . What is BC ?
💡 解题思路
Let $D$ and $E$ be the points of tangency on circles $A$ and $B$ with line $CD$ . $AB=8$ . Also, let $BC=x$ . As $\angle ADC$ and $\angle BEC$ are right angles (a radius is perpendicular to a tangent
12
第 12 题
数论
A year is a leap year if and only if the year number is divisible by 400 (such as 2000) or is divisible by 4 but not 100 (such as 2012). The 200th anniversary of the birth of novelist Charles Dickens was celebrated on February 7, 2012, a Tuesday. On what day of the week was Dickens born?
💡 解题思路
In this solution we refer to moving to the left as decreasing the year or date number and moving to the right as increasing the year or date number. Every non-leap year we move to the right results in
13
第 13 题
统计
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
💡 解题思路
The iterative average of any 5 integers $a,b,c,d,e$ is defined as:
14
第 14 题
几何·面积
Chubby makes nonstandard checkerboards that have 31 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
💡 解题思路
There are 15 rows with 15 black tiles, and 16 rows with 16 black tiles, so the answer is $15^2+16^2 =225+256= \boxed{\textbf{(B)}\ 481}$
15
第 15 题
几何·面积
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of \triangle ABC ?
💡 解题思路
$AC$ intersects $BC$ at a right angle, (this can be proved by noticing that the slopes of the two lines are negative reciprocals of each other) so $\triangle ABC \sim \triangle BED$ . The hypotenuse o
16
第 16 题
行程问题
Three runners start running simultaneously from the same point on a 500-meter circular track. They each run clockwise around the course maintaining constant speeds of 4.4, 4.8, and 5.0 meters per second. The runners stop once they are all together again somewhere on the circular course. How many seconds do the runners run?
💡 解题思路
First consider the first two runners. The faster runner will lap the slower runner exactly once, or run 500 meters farther. Let $x$ be the time these runners run in seconds.
17
第 17 题
数论
Let a and b be relatively prime positive integers with a>b>0 and \dfrac{a^3-b^3}{(a-b)^3} = \dfrac{73}{3} . What is a-b ?
💡 解题思路
Since $a$ and $b$ are relatively prime, $a^3-b^3$ and $(a-b)^3$ are both integers as well. Then, for the given fraction to simplify to $\frac{73}{3}$ , the denominator $(a-b)^3$ must be a multiple of
18
第 18 题
几何·面积
The closed curve in the figure is made up of 9 congruent circular arcs each of length \frac{2π}{3} , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve? [图]
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break?
💡 解题思路
Let Paula work at a rate of $p$ , the two helpers work at a combined rate of $h$ , and the time it takes to eat lunch be $L$ , where $p$ and $h$ are in house/hours and L is in hours. Then the labor on
20
第 20 题
几何·面积
A 3 × 3 square is partitioned into 9 unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated 90 ^{\circ} clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability the grid is now entirely black?
💡 解题思路
First, look for invariants. The center, unaffected by rotation, must be black. So automatically, the chance is less than $\frac{1}{2}.$ Note that a $90^{\circ}$ rotation requires that black squares be
21
第 21 题
几何·面积
Let points A = (0 ,0 ,0) , B = (1, 0, 0) , C = (0, 2, 0) , and D = (0, 0, 3) . Points E , F , G , and H are midpoints of line segments \overline{BD}, \overline{AB}, \overline {AC}, and \overline{DC} respectively. What is the area of EFGH ?
💡 解题思路
Consider a tetrahedron with vertices at $A,B,C,D$ in the $xyz$ -space. The length of $EF$ is just one-half of $AD$ because it is the midsegment of $\triangle ABD.$ The same concept applies to the othe
22
第 22 题
规律与数列
The sum of the first m positive odd integers is 212 more than the sum of the first n positive even integers. What is the sum of all possible values of n ?
💡 解题思路
The sum of the first $m$ odd integers is given by $m^2$ . The sum of the first $n$ even integers is given by $n(n+1)$ .
23
第 23 题
计数
Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen? (A)\ 60 (B)\ 170 (C)\ 290 (D)\ 320 (E)\ 660
💡 解题思路
Note that if $n$ is the number of friends each person has, then $n$ can be any integer from $1$ to $4$ , inclusive, truly.
24
第 24 题
整数运算
Let a , b , and c be positive integers with a\geb\gec such that a^2-b^2-c^2+ab=2011 and a^2+3b^2+3c^2-3ab-2ac-2bc=-1997 . What is a ?
💡 解题思路
Add the two equations.
25
第 25 题
概率
Real numbers x , y , and z are chosen independently and at random from the interval [0,n] for some positive integer n . The probability that no two of x , y , and z are within 1 unit of each other is greater than \frac {1}{2} . What is the smallest possible value of n ?
💡 解题思路
Since $x,y,z$ are all reals located in $[0, n]$ , the number of choices for each one is continuous so we use geometric probability.