2006 AMC 10B — Official Competition Problems (January 2006)
📅 2006 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
综合
What is (-1)^{1} + (-1)^{2} + ... + (-1)^{2006} ?
💡 解题思路
Since $-1$ raised to an odd integer is $-1$ and $-1$ raised to an even integer exponent is $1$ :
2
第 2 题
综合
For real numbers x and y , define x \spadesuit y = (x+y)(x-y) . What is 3 \spadesuit (4 \spadesuit 5) ? (A) \ -72 (B) \ -27 (C) \ -24 (D) \ 24 (E) \ 72
💡 解题思路
Since $x \spadesuit y = (x+y)(x-y)$ :
3
第 3 题
综合
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
💡 解题思路
Let $x$ be the number of points scored by the Cougars, and $y$ be the number of points scored by the Panthers. The problem is asking for the value of $y$ . \begin{align*} x+y &= 34 \\ x-y &= 14 \\ 2x
4
第 4 题
几何·面积
Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
💡 解题思路
The area painted red is equal to the area of the smaller circle and the area painted blue is equal to the area of the larger circle minus the area of the smaller circle.
5
第 5 题
几何·面积
A 2 × 3 rectangle and a 3 × 4 rectangle are contained within a square without overlapping at any point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square?
💡 解题思路
By placing the $2 \times 3$ rectangle adjacent to the $3 \times 4$ rectangle with the 3 side of the $2 \times 3$ rectangle next to the 4 side of the $3 \times 4$ rectangle, we get a figure that can be
6
第 6 题
几何·面积
A region is bounded by semicircular arcs constructed on the side of a square whose sides measure \frac{2}{π} , as shown. What is the perimeter of this region? [图]
💡 解题思路
Since the side of the square is the diameter of the semicircle, the radius of the semicircle is $\frac{1}{2}\cdot\frac{2}{\pi}=\frac{1}{\pi}$ .
7
第 7 题
综合
Which of the following is equivalent to √(\frac{x){1-\frac{x-1}{x}}} when x < 0 ? (A) \ -x (B) \ x (C) \ 1 (D) \ √(\frac{x){2}} (E) \ x√(-1)
A square of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle? [图]
💡 解题思路
Since the area of the square is $40$ , the length of a side is $\sqrt{40}=2\sqrt{10}$ . The distance between the center of the semicircle and one of the bottom vertices of the square is half the lengt
9
第 9 题
综合
Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?
💡 解题思路
The calorie to gram ratio of Francesca's lemonade is $\frac{25+386+0}{100+100+400}=\frac{411\textrm{ calories}}{600\textrm{ grams}}=\frac{137\textrm{ calories}}{200\textrm{ grams}}$
10
第 10 题
几何·面积
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15 . What is the greatest possible perimeter of the triangle?
💡 解题思路
Let $x$ be the length of the first side.
11
第 11 题
规律与数列
What is the tens digit in the sum 7!+8!+9!+...+2006!
💡 解题思路
Since $10!$ is divisible by $100$ , any factorial greater than $10!$ is also divisible by $100$ . The last two digits of all factorials greater than $10!$ are $00$ , so the last two digits of $10!+11!
12
第 12 题
方程
The lines x=\frac{1}{4}y+a and y=\frac{1}{4}x+b intersect at the point (1,2) . What is a+b ?
💡 解题思路
Since $(1,2)$ is a solution to both equations, plugging in $x=1$ and $y=2$ will give the values of $a$ and $b$ .
13
第 13 题
分数与比例
Joe and JoAnn each bought 12 ounces of coffee in a 16 ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
💡 解题思路
After drinking and adding cream, Joe's cup has $2$ ounces of cream.
14
第 14 题
方程
Let a and b be the roots of the equation x^2-mx+2=0 . Suppose that a+\frac1b and b+\frac1a are the roots of the equation x^2-px+q=0 . What is q ? https://youtu.be/3dfbWzOfJAI?t=457 ~ pi_is_3.14592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067
💡 解题思路
In a quadratic equation of the form $x^2 + bx + c = 0$ , the product of the roots is $c$ ( Vieta's Formulas ).
15
第 15 题
几何·面积
Rhombus ABCD is similar to rhombus BFDE . The area of rhombus ABCD is 24 and \angle BAD = 60^\circ . What is the area of rhombus BFDE ? [图]
💡 解题思路
Using the property that opposite angles are equal in a rhombus , $\angle DAB = \angle DCB = 60 ^\circ$ and $\angle ADC = \angle ABC = 120 ^\circ$ . It is easy to see that rhombus $ABCD$ is made up of
16
第 16 题
综合
Leap Day, February 29 , 2004 , occurred on a Sunday. On what day of the week will Leap Day, February 29 , 2020 , occur?
💡 解题思路
There are $365$ days in a year, plus $1$ extra day if there is a Leap Day, which occurs on years that are multiples of $4$ (with a few exceptions that don't affect this problem).
17
第 17 题
概率
Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same? https://youtu.be/5UojVH4Cqqs?t=1160 ~ pi_is_3.14
💡 解题思路
Since there are the same amount of total balls in Alice's bag as in Bob's bag, and there is an equal chance of each ball being selected, the color of the ball that Alice puts in Bob's bag doesn't matt
18
第 18 题
规律与数列
Let a_1 , a_2 , ... be a sequence for which a_1=2 , a_2=3 , and a_n=\frac{a_{n-1}}{a_{n-2}} for each positive integer n \ge 3 . What is a_{2006} ? (A) \ \frac{1}{2} (B) \ \frac{2}{3} (C) \ \frac{3}{2} (D) \ 2 (E) \ 3
💡 解题思路
Looking at the first few terms of the sequence:
19
第 19 题
几何·面积
A circle of radius 2 is centered at O . Square OABC has side length 1 . Sides AB and CB are extended past B to meet the circle at D and E , respectively. What is the area of the shaded region in the figure, which is bounded by BD , BE , and the minor arc connecting D and E ? [图] (A) \ \frac{π}{3}+1-√(3) (B) \ \frac{π}{2}(2-√(3)) (C) \ π(2-√(3)) (D) \ \frac{π}{6}+\frac{√(3)+1}{2} (E) \ \frac{π}{3}-1+√(3)
💡 解题思路
The shaded area is equivalent to the area of sector $DOE$ minus the area of triangle $DOE$ plus the area of triangle $DBE$ .
20
第 20 题
几何·面积
In rectangle ABCD , we have A=(6,-22) , B=(2006,178) , D=(8,y) , for some integer y . What is the area of rectangle ABCD ? (A) \ 4000 (B) \ 4040 (C) \ 4400 (D) \ 40,000 (E) \ 40,400
💡 解题思路
Let the slope of $AB$ be $m_1$ and the slope of $AD$ be $m_2$ .
21
第 21 题
分数与比例
For a particular peculiar pair of dice, the probabilities of rolling 1 , 2 , 3 , 4 , 5 , and 6 , on each die are in the ratio 1:2:3:4:5:6 . What is the probability of rolling a total of 7 on the two dice? (A) \ \frac{4}{63} (B) \ \frac{1}{8} (C) \ \frac{8}{63} (D) \ \frac{1}{6} (E) \ \frac{2}{7}
💡 解题思路
Let $x$ be the probability of rolling a $1$ . The probabilities of rolling a $2$ , $3$ , $4$ , $5$ , and $6$ are $2x$ , $3x$ , $4x$ , $5x$ , and $6x$ , respectively.
22
第 22 题
规律与数列
Elmo makes N sandwiches for a fundraiser. For each sandwich he uses B globs of peanut butter at 4\cent per glob and J blobs of jam at 5\cent per blob. The cost of the peanut butter and jam to make all the sandwiches is \ 2.53 . Assume that B , J , and N are positive integers with N>1 . What is the cost of the jam Elmo uses to make the sandwiches? (A) \ \ 1.05 (B) \ \ 1.25 (C) \ \ 1.45 (D) \ \ 1.65 (E) \ \ 1.85
💡 解题思路
The peanut butter and jam for each sandwich costs $4B\cent+5J\cent$ , so the peanut butter and jam for $N$ sandwiches costs $N(4B+5J)\cent$ .
23
第 23 题
几何·面积
A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? [图] (A) \ 15 (B) \ 17 (C) \ \frac{35}{2} (D) \ 18 (E) \ \frac{55}{3}
💡 解题思路
Label the points in the figure as shown below, and draw the segment $CF$ . This segment divides the quadrilateral into two triangles, let their areas be $x$ and $y$ .
24
第 24 题
几何·面积
Circles with centers O and P have radii 2 and 4 , respectively, and are externally tangent. Points A and B on the circle with center O and points C and D on the circle with center P are such that AD and BC are common external tangents to the circles. What is the area of the concave hexagon AOBCPD ? [图] (A) \ 18√(3) (B) \ 24√(2) (C) \ 36 (D) \ 24√(3) (E) \ 32√(2) https://youtu.be/cdjZ9Xd3Yt8 [Class:Geometry] ~ Education, the Study of Everything
💡 解题思路
When we see this problem, it practically screams similar triangles at us. Extend $OP$ to the left until it intersects lines $AD$ and $BC$ at point $E$ . Triangles $EBO$ and $ECP$ are similar, and by s
25
第 25 题
数论
Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? (A) \ 4 (B) \ 5 (C) \ 6 (D) \ 7 (E) \ 8
💡 解题思路
Let $S$ be the set of the ages of Mr. Jones' children (in other words $i \in S$ if Mr. Jones has a child who is $i$ years old). Then $|S| = 8$ and $9 \in S$ . Let $m$ be the positive integer seen on t