2009 AMC 10B — Official Competition Problems (February 2009)
📅 2009 B 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
Each morning of her five-day workweek, Jane bought either a 50 -cent muffin or a 75 -cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
💡 解题思路
If Jane bought one more bagel but one fewer muffin, then her total cost for the week would increase by $25$ cents.
2
第 2 题
综合
Which of the following is equal to \dfrac{\frac{1}{3}-\frac{1}{4}}{\frac{1}{2}-\frac{1}{3}} ? (A) \frac 14 (B) \frac 13 (C) \frac 12 (D) \frac 23 (E) \frac 34
💡 解题思路
Multiplying the numerator and the denominator by the same value does not change the value of the fraction. We can multiply both by $12$ , getting $\dfrac{4-3}{6-4} = \boxed{\dfrac 12}$ .
3
第 3 题
工程问题
Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? (A)\ 10 (B)\ 12 (C)\ 15 (D)\ 18 (E)\ 25
💡 解题思路
Losing three cans of paint corresponds to being able to paint five fewer rooms. So $\frac 35 \cdot 25 = \boxed{15}$ . The answer is $\mathrm{(C)}$ .
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第 4 题
几何·面积
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths 15 and 25 meters. What fraction of the yard is occupied by the flower beds? (A)\frac {1}{8} (B)\frac {1}{6} (C)\frac {1}{5} (D)\frac {1}{4} (E)\frac {1}{3}
💡 解题思路
Each triangle has leg length $\frac 12 \cdot (25 - 15) = 5$ meters and area $\frac 12 \cdot 5^2 = \frac {25}{2}$ square meters. Thus the flower beds have a total area of $25$ square meters. The entire
5
第 5 题
分数与比例
Twenty percent less than 60 is one-third more than what number? (A)\ 16 (B)\ 30 (C)\ 32 (D)\ 36 (E)\ 48
💡 解题思路
Twenty percent less than 60 is $\frac 45 \cdot 60 = 48$ . One-third more than a number n is $\frac 43n$ . Therefore $\frac 43n = 48$ and the number is $\boxed {36}$ . The answer is $\mathrm{(D)}$ .
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第 6 题
规律与数列
Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? (A)\ 10 (B)\ 12 (C)\ 16 (D)\ 18 (E)\ 24
💡 解题思路
The age of each person is a factor of $128 = 2^7$ . So the twins could be $2^0 = 1, 2^1 = 2, 2^2 = 4, 2^3 = 8$ years of age and, consequently Kiana could be $128$ , $32$ , $8$ , or $2$ years old, resp
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第 7 题
行程问题
By inserting parentheses, it is possible to give the expression \[2×3 + 4×5\] several values. How many different values can be obtained? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6
💡 解题思路
The three operations can be performed on any of $3! = 6$ orders. However, if the addition is performed either first or last, then multiplying in either order produces the same result. So at most four
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第 8 题
应用题
In a certain year the price of gasoline rose by 20\% during January, fell by 20\% during February, rose by 25\% during March, and fell by x\% during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is x(A)\ 12 (B)\ 17 (C)\ 20 (D)\ 25 (E)\ 35
💡 解题思路
Let $p$ be the price at the beginning of January. The price at the end of March was $(1.2)(0.8)(1.25)p = 1.2p.$ Because the price at the end of April was $p$ , the price decreased by $0.2p$ during Apr
9
第 9 题
几何·角度
Segment BD and AE intersect at C , as shown, AB=BC=CD=CE , and \angle A = \frac 52 \angle B . What is the degree measure of \angle D ? [图] (A) 52.5 (B) 55 (C) 57.7 (D) 60 (E) 62.5
A flagpole is originally 5 meters tall. A hurricane snaps the flagpole at a point x meters above the ground so that the upper part, still attached to the stump, touches the ground 1 meter away from the base. What is x ? (A) 2.0 (B) 2.1 (C) 2.2 (D) 2.3 (E) 2.4
💡 解题思路
The broken flagpole forms a right triangle with legs $1$ and $x$ , and hypotenuse $5-x$ . The Pythagorean theorem now states that $1^2 + x^2 = (5-x)^2$ , hence $10x = 24$ , and $x=\boxed{2.4}$ .
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第 11 题
数字运算
How many 7 -digit palindromes (numbers that read the same backward as forward) can be formed using the digits 2 , 2 , 3 , 3 , 5 , 5 , 5 ? (A) 6 (B) 12 (C) 24 (D) 36 (E) 48
💡 解题思路
A seven-digit palindrome is a number of the form $\overline{abcdcba}$ . Clearly, $d$ must be $5$ , as we have an odd number of fives. We are then left with $\{a,b,c\} = \{2,3,5\}$ . There are $3!$ per
12
第 12 题
几何·面积
Distinct points A , B , C , and D lie on a line, with AB=BC=CD=1 . Points E and F lie on a second line, parallel to the first, with EF=1 . A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7
💡 解题思路
Consider the classical formula for triangle area: $\frac 12 \cdot b \cdot h$ . Each of the triangles that we can make has exactly one side lying on one of the two parallel lines. If we pick this side
13
第 13 题
方程
As shown below, convex pentagon ABCDE has sides AB=3 , BC=4 , CD=6 , DE=3 , and EA=7 . The pentagon is originally positioned in the plane with vertex A at the origin and vertex B on the positive x -axis. The pentagon is then rolled clockwise to the right along the x -axis. Which side will touch the point x=2009 on the x -axis? [图] (A) \overline{AB} (B) \overline{BC} (C) \overline{CD} (D) \overline{DE} (E) \overline{EA}
💡 解题思路
The perimeter of the polygon is $3+4+6+3+7 = 23$ . Hence as we roll the polygon to the right, every $23$ units the side $\overline{AB}$ will be the bottom side.
14
第 14 题
综合
On Monday, Millie puts a quart of seeds, 25\% of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only 25\% of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
💡 解题思路
On Monday, day 1, the birds find $\frac 14$ quart of millet in the feeder. On Tuesday they find \[\frac 14 + \frac 34 \cdot \frac 14\] quarts of millet. On Wednesday, day 3, they find \[\frac 14 + \fr
15
第 15 题
规律与数列
When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of water the total weight is b kilograms. In terms of a and b , what is the total weight in kilograms when the bucket is full of water? (A)\ \frac23a + \frac13b (B)\ \frac32a - \frac12b (C)\ \frac32a + b (D)\ \frac32a + 2b (E)\ 3a - 2b
💡 解题思路
Let $x$ be the weight of the bucket and let $y$ be the weight of the water in a full bucket. Then we are given that $x + \frac 23y = a$ and $x + \frac 12y = b$ . Hence $\frac 16y = a-b$ , so $y = 6a-6
16
第 16 题
几何·面积
Points A and C lie on a circle centered at O , each of \overline{BA} and \overline{BC} are tangent to the circle, and \triangle ABC is equilateral. The circle intersects \overline{BO} at D . What is \frac{BD}{BO} ? (A) \frac {\sqrt2}{3} (B) \frac {1}{2} (C) \frac {\sqrt3}{3} (D) \frac {\sqrt2}{2} (E) \frac {\sqrt3}{2}
💡 解题思路
[asy] unitsize(1.5cm); defaultpen(0.8); pair B=(0,0), A=(3,0), C=3*dir(60), O=intersectionpoint( C -- (C+3*dir(-30)), A -- (A+3*dir(90)) ); pair D=intersectionpoint(B--O, circle(O,length(A-O))); draw(
17
第 17 题
几何·面积
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from (c,0) to (3,3) , divides the entire region into two regions of equal area. What is c ? [图]
💡 解题思路
For $c\geq 1.5$ the shaded area is at most $1.5$ , which is too little. Hence $c<1.5$ , and therefore the point $(2,1)$ is indeed inside the shaded part, as shown in the picture.
18
第 18 题
几何·面积
Rectangle ABCD has AB=8 and BC=6 . Point M is the midpoint of diagonal \overline{AC} , and E is on AB with \overline{ME}\perp\overline{AC} . What is the area of \triangle AME ? (A) \frac{65}{8} (B) \frac{25}{3} (C) 9 (D) \frac{75}{8} (E) \frac{85}{8}
💡 解题思路
Set $A$ to $(0,0)$ . Since $M$ is the midpoint of the diagonal, it would be $(4,3)$ . The diagonal $AC$ would be the line $y = \frac{3x}{4}$ . Since $ME$ is perpendicular to $AC$ , its line would be i
19
第 19 题
分数与比例
A particular 12 -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 1 , it mistakenly displays a 9 . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? (A)\ \frac 12 (B)\ \frac 58 (C)\ \frac 34 (D)\ \frac 56 (E)\ \frac {9}{10}
💡 解题思路
The clock will display the incorrect time for the entire hours of $1, 10, 11$ and $12$ . So the correct hour is displayed $\frac 23$ of the time. The minutes will not display correctly whenever either
20
第 20 题
几何·面积
Triangle ABC has a right angle at B , AB=1 , and BC=2 . The bisector of \angle BAC meets \overline{BC} at D . What is BD ? [图] (A) \frac {\sqrt3 - 1}{2} (B) \frac {\sqrt5 - 1}{2} (C) \frac {\sqrt5 + 1}{2} (D) \frac {\sqrt6 + \sqrt2}{2} (E) 2\sqrt 3 - 1
💡 解题思路
By the Pythagorean Theorem, $AC=\sqrt5$ . Then, from the Angle Bisector Theorem, we have:
21
第 21 题
数论
What is the remainder when 3^0 + 3^1 + 3^2 + ·s + 3^{2009} is divided by 8? (A)\ 0 (B)\ 1 (C)\ 2 (D)\ 4 (E)\ 6
💡 解题思路
The sum of any four consecutive powers of 3 is divisible by $3^0 + 3^1 + 3^2 +3^3 = 40$ and hence is divisible by 8. Therefore
22
第 22 题
几何·面积
A cubical cake with edge length 2 inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where M is the midpoint of a top edge. The piece whose top is triangle B contains c cubic inches of cake and s square inches of icing. What is c+s ? [图] (A) \frac{24}{5} (B) \frac{32}{5} (C) 8+\sqrt5 (D) 5+\frac{16\sqrt5}{5} (E) 10+5\sqrt5
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture? (A)\frac {1}{16} (B)\frac 18 (C)\frac {3}{16} (D)\frac 14 (E)\frac {5}{16}
💡 解题思路
After $10$ minutes $(600$ seconds $),$ Rachel will have completed $6$ laps and be $30$ seconds from completing her seventh lap. Because Rachel runs one-fourth of a lap in $22.5$ seconds, she will be i
24
第 24 题
几何·角度
The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with 9 trapezoids, let x be the angle measure in degrees of the larger interior angle of the trapezoid. What is x ? [图] (A) 100 (B) 102 (C) 104 (D) 106 (E) 108
💡 解题思路
Extend all the legs of the trapezoids. They will all intersect in the middle of the bottom side of the picture, forming the situation shown below.
25
第 25 题
概率
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? (A)\frac 18 (B)\frac {3}{16} (C)\frac 14 (D)\frac 38 (E)\frac 12
💡 解题思路
There are two possible stripe orientations for each of the six faces of the cube, so there are $2^6 = 64$ possible stripe combinations. There are three pairs of parallel faces so, if there is an encir