2011 A AMC 10A — Official Competition Problems (February 2011 A)
📅 2011 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
应用题
A cell phone plan costs 20 dollars each month, plus 5 cents per text message sent, plus 10 cents for each minute used over 30 hours. In January Michelle sent 100 text messages and talked for 30.5 hours. How much did she have to pay?
💡 解题思路
The base price of Michelle's cell phone plan is $20$ dollars. If she sent $100$ text messages and it costs $5$ cents per text, then she must have spent $500$ cents for texting, or $5$ dollars. She tal
2
第 2 题
工程问题
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
💡 解题思路
To find how many small bottles we need, we can simply divide $500$ by $35$ . This simplifies to $\frac{100}{7}=14 \frac{2}{7}.$ Since the answer must be an integer greater than $14$ , we have to round
3
第 3 题
统计
Suppose [ ab ] denotes the average of a and b , and { abc } denotes the average of a , b , and c . What is \{\{1 1 0\} [0 1] 0\}?
💡 解题思路
Average $1$ , $1$ , and $0$ to get $\frac23$ . Average $0$ , and $1$ , to get $\frac12$ . Average $\frac23$ , $\frac12$ , and $0$ . to get $\boxed{\textbf{(D)}\ \frac7{18}}$
4
第 4 题
规律与数列
Let X and Y be the following sums of arithmetic sequences: \begin{eqnarray*}X =& 10+12+14+·s+100, ; Y =& 12+14+16+·s+102.\end{eqnarray*} What is the value of Y - X?
💡 解题思路
We see that both sequences have equal numbers of terms, so reformat the sequence to look like:
5
第 5 题
统计
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12 , 15 , and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
💡 解题思路
Let us say that there are $f$ fifth graders. According to the given information, there must be $2f$ fourth graders and $4f$ third graders. The average time run by each student is equal to the total am
6
第 6 题
综合
Set A has 20 elements, and set B has 15 elements. What is the smallest possible number of elements in A \cup B ?
💡 解题思路
$A \cup B$ will be smallest if $B$ is completely contained in $A$ , in which case all the elements in $B$ would be counted for in $A$ . So the total would be the number of elements in $A$ , which is $
7
第 7 题
方程
Which of the following equations does NOT have a solution? (A)\:(x+7)^2=0(B)\:|-3x|+5=0(C)\:√(-x)-2=0(D)\:√(x)-8=0(E)\:|-3x|-4=0
💡 解题思路
$|-3x|+5=0$ has no solution because absolute values only output nonnegative numbers.
8
第 8 题
分数与比例
Last summer 30% of the birds living on Town Lake were geese, 25% were swans, 10% were herons, and 35% were ducks. What percent of the birds that were not swans were geese?
💡 解题思路
75% of the total birds were not swans. Out of that 75%, there was $30\% / 75\% = \boxed{40\%\text{\textbf{ (C)}}}$ of the birds that were not swans that were geese.
9
第 9 题
几何·面积
A rectangular region is bounded by the graphs of the equations y=a, y=-b, x=-c, and x=d , where a,b,c, and d are all positive numbers. Which of the following represents the area of this region?
💡 解题思路
We have a rectangle of side lengths $a-(-b)=a+b$ and $d-(-c)=c+d.$ Thus the area of this rectangle is $(a + b)(c + d) = \boxed{\textbf{(A)}\ ac + ad + bc + bd}$ .
10
第 10 题
统计
A majority of the 30 students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than 1 . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was \17.71$ . What was the cost of a pencil in cents?
💡 解题思路
The total cost of the pencils can be found by $(\text{students}\cdot\text{pencils purchased by each}\cdot\text{price of each pencil})$ .
11
第 11 题
几何·面积
Square EFGH has one vertex on each side of square ABCD . Point E is on AB with AE=7· EB . What is the ratio of the area of EFGH to the area of ABCD ? (A) \frac{49}{64} (B) \frac{25}{32} (C) \frac78 (D) \frac{5√(2)}{8} (E) \frac{√(14)}{4}
💡 解题思路
Let $8$ be the length of the sides of square $ABCD$ . Then, the length of one of the sides of square $EFGH$ is $\sqrt{7^2+1^2}=\sqrt{50}$ , and hence the ratio in the areas is $\frac{\sqrt{50}^2}{8^2}
12
第 12 题
综合
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was 61 points. How many free throws did they make?
💡 解题思路
For the points made from two-point shots and from three-point shots to be equal, the numbers of made shots are in a $3:2$ ratio. Therefore, assume they made $3x$ and $2x$ two- and three- point shots,
13
第 13 题
数字运算
How many even integers are there between 200 and 700 whose digits are all different and come from the set \{1,2,5,7,8,9\} ? (A) 12 (B) 20 (C) 72 (D) 120 (E) 200
💡 解题思路
We split up into cases of the hundreds digits being $2$ or $5$ . If the hundred digits is $2$ , then the units digits must be $8$ in order for the number to be even and then there are $4$ remaining ch
14
第 14 题
几何·面积
A pair of standard 6 -sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
💡 解题思路
For the circumference to be greater than the area, we must have $\pi d > \pi \left( \frac{d}{2} \right) ^2$ , or $d<4$ . Now since $d$ is determined by a sum of two dice, the only possibilities for $d
15
第 15 题
统计
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles? (A)\ 140 (B)\ 240 (C)\ 440 (D)\ 640 (E)\ 840
💡 解题思路
We know that $\frac{\text{total miles}}{\text{total gas}}=55$ . Let $x$ be the distance in miles the car traveled during the time it ran on gasoline, then the amount of gas used is $0.02x$ . The total
16
第 16 题
综合
Which of the following is equal to √(9-6\sqrt{2)}+√(9+6\sqrt{2)} ? (A) 3\sqrt2 (B) 2\sqrt6 (C) \frac{7\sqrt2}{2} (D) 3\sqrt3 (E) 6
💡 解题思路
We find the answer by squaring, then square rooting the expression.
17
第 17 题
规律与数列
In the eight term sequence A , B , C , D , E , F , G , H , the value of C is 5 and the sum of any three consecutive terms is 30 . What is A+H ?
💡 解题思路
Let $A=x$ . Then from $A+B+C=30$ , we find that $B=25-x$ . From $B+C+D=30$ , we then get that $D=x$ . Continuing this pattern, we find $E=25-x$ , $F=5$ , $G=x$ , and finally $H=25-x$ . So $A+H=x+25-x=
18
第 18 题
几何·面积
Circles A, B, and C each has radius 1 . Circles A and B share one point of tangency. Circle C has a point of tangency with the midpoint of \overline{AB}. What is the area inside circle C but outside circle A and circle B? [图]
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?
💡 解题思路
Let the population of the town in $1991$ be $p^2$ . Let the population in $2001$ be $q^2+9$ . It follows that $p^2+150=q^2+9$ . Rearrange this equation to get $141=q^2-p^2=(q-p)(q+p)$ . Since $q$ and
20
第 20 题
几何·面积
Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect?
💡 解题思路
Fix a point $A$ from which we draw a clockwise chord. In order for the clockwise chord from another point $B$ to intersect that of point $A$ , $A$ and $B$ must be no more than $r$ units apart. By draw
21
第 21 题
概率
Two counterfeit coins of equal weight are mixed with 8 identical genuine coins. The weight of each of the counterfeit coins is different from the weight of each of the genuine coins. A pair of coins is selected at random without replacement from the 10 coins. A second pair is selected at random without replacement from the remaining 8 coins. The combined weight of the first pair is equal to the combined weight of the second pair. What is the probability that all 4 selected coins are genuine?
💡 解题思路
If we pick $4$ indistinguishable real coins from the set of $8$ real coins, there are $\binom{8}{4}$ ways to pick the coins. If we then place the coins in four distinguishable slots on the scale, ther
22
第 22 题
计数
Each vertex of convex pentagon ABCDE is to be assigned a color. There are 6 colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
💡 解题思路
Let vertex $A$ be any vertex, then vertex $B$ be one of the diagonal vertices to $A$ , $C$ be one of the diagonal vertices to $B$ , and so on. We consider cases for this problem.
23
第 23 题
概率
Seven students count from 1 to 1000 as follows: Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000. Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers. Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers. Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers. Finally, George says the only number that no one else says. What number does George say?
💡 解题思路
First look at the numbers Alice says. $1, 3, 4, 6, 7, 9 \cdots$ skipping every number that is congruent to $2 \pmod 3$ . Thus, Barbara says those numbers EXCEPT every second - being $2 + 3^1 \equiv 5
24
第 24 题
立体几何
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra? (A) \frac{1}{12} (B) \frac{\sqrt2}{12} (C) \frac{\sqrt3}{12} (D) \frac{1}{6} (E) \frac{\sqrt2}{6}
💡 解题思路
A regular unit tetrahedron can be split into eight tetrahedra that have lengths of $\frac{1}{2}$ . The volume of a regular tetrahedron can be found using the formula for the area of a pyramid: $\frac{
25
第 25 题
几何·面积
Let R be a unit square region and n ≥ 4 an integer. A point X in the interior of R is called n-ray partitional if there are n rays emanating from X that divide R into n triangles of equal area. How many points are 100 -ray partitional but not 60 -ray partitional?
💡 解题思路
There must be four rays emanating from $X$ that intersect the four corners of the square region. Depending on the location of $X$ , the number of rays distributed among these four triangular sectors w