2006 AMC 10A — Official Competition Problems (January 2006)
📅 2006 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
Sandwiches at Joe's Fast Food cost \3 each and sodas cost \2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
💡 解题思路
The $5$ sandwiches cost $5\cdot 3=15$ dollars. The $8$ sodas cost $8\cdot 2=16$ dollars. In total, the purchase costs $15+16=\boxed{\textbf{(A) }31}$ dollars.
2
第 2 题
行程问题
Define x\otimes y=x^3-y . What is h\otimes (h\otimes h) ?
💡 解题思路
By the definition of $\otimes$ , we have $h\otimes h=h^{3}-h$ .
3
第 3 题
分数与比例
The ratio of Mary's age to Alice's age is 3:5 . Alice is 30 years old. How old is Mary?
💡 解题思路
Let $m$ be Mary's age. Then $\frac{m}{30}=\frac{3}{5}$ . Solving for $m$ , we obtain $m=\boxed{\textbf{(B) }18}.$
4
第 4 题
规律与数列
A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
💡 解题思路
From the greedy algorithm , we have $9$ in the hours section and $59$ in the minutes section. $9+5+9=\boxed{\textbf{(E) }23}$
5
第 5 题
数论
Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was 8 dollars, and there was an additional cost of 2 dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
💡 解题思路
Dave and Doug paid $8+2=10$ dollars in total. Doug paid for three slices of plain pizza, which cost $\frac{3}{8}\cdot 8=3$ . Dave paid $10-3=7$ dollars. Dave paid $7-3=\boxed{\textbf{(D) }4}$ more dol
6
第 6 题
综合
What non-zero real value for x satisfies (7x)^{14}=(14x)^7 ?
💡 解题思路
Taking the seventh root of both sides, we get $(7x)^2=14x$ .
7
第 7 题
几何·面积
The 8×18 rectangle ABCD is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is y ? [图] (A)\ 6 (B)\ 7 (C)\ 8 (D)\ 9 (E)\ 10
💡 解题思路
Since the two hexagons are going to be repositioned to form a square without overlap, the area will remain the same. The rectangle's area is $18\cdot8=144$ . This means the square will have four sides
8
第 8 题
方程
A parabola with equation y=x^2+bx+c passes through the points (2,3) and (4,3) . What is c ?
💡 解题思路
Substitute the points $(2,3)$ and $(4,3)$ into the given equation for $(x,y)$ .
9
第 9 题
规律与数列
How many sets of two or more consecutive positive integers have a sum of 15 ?
💡 解题思路
Notice that if the consecutive positive integers have a sum of $15$ , then their average (which could be a fraction) must be a divisor of $15$ . If the number of integers in the list is odd, then the
10
第 10 题
整数运算
For how many real values of x is √(120-\sqrt{x)} an integer?
💡 解题思路
For $\sqrt{120-\sqrt{x}}$ to be an integer, $120-\sqrt{x}$ must be a perfect square.
11
第 11 题
坐标几何
Which of the following describes the graph of the equation (x+y)^2=x^2+y^2 ?
💡 解题思路
Expanding the left side, we have
12
第 12 题
几何·面积
Rolly wishes to secure his dog with an 8-foot rope to a square shed that is 16 feet on each side. His preliminary drawings are shown. [图] Which of these arrangements give the dog the greater area to roam, and by how many square feet?
A player pays \textdollar 5 to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)
💡 解题思路
The probability of rolling an even number on the first turn is $\frac{1}{2}$ and the probability of rolling the same number on the next turn is $\frac{1}{6}$ . The probability of winning is $\frac{1}{
14
第 14 题
行程问题
A number of linked rings, each 1 cm thick, are hanging on a peg. The top ring has an outside diameter of 20 cm. The outside diameter of each of the outer rings is 1 cm less than that of the ring above it. The bottom ring has an outside diameter of 3 cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring? [图]
💡 解题思路
The inside diameters of the rings are the positive integers from $1$ to $18$ . The total distance needed is the sum of these values plus $2$ for the top of the first ring and the bottom of the last ri
15
第 15 题
计数
Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?
💡 解题思路
Since $d = rt$ , we note that Odell runs one lap in $\frac{2 \cdot 50\pi}{250} = \frac{2\pi}{5}$ minutes, while Kershaw also runs one lap in $\frac{2 \cdot 60\pi}{300} = \frac{2\pi}{5}$ minutes. They
16
第 16 题
几何·面积
A circle of radius 1 is tangent to a circle of radius 2 . The sides of \triangle ABC are tangent to the circles as shown, and the sides \overline{AB} and \overline{AC} are congruent. What is the area of \triangle ABC ? [图]
💡 解题思路
Let the centers of the smaller and larger circles be $O_1$ and $O_2$ , respectively. Let their tangent points to $\triangle ABC$ be $D$ and $E$ , respectively. We can then draw the following diagram:
17
第 17 题
几何·面积
In rectangle ADEH , points B and C trisect \overline{AD} , and points G and F trisect \overline{HE} . In addition, AH=AC=2 , and AD=3 . What is the area of quadrilateral WXYZ shown in the figure? [图]
💡 解题思路
By symmetry , $WXYZ$ is a square.
18
第 18 题
数字运算
A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?
💡 解题思路
There are $10\cdot10\cdot10\cdot10 = 10^4$ ways to choose 4 digits.
19
第 19 题
几何·面积
How many non- similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression ?
💡 解题思路
The sum of the angles of a triangle is $180$ degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it $\fra
20
第 20 题
数论
Six distinct positive integers are randomly chosen between 1 and 2006 , inclusive. What is the probability that some pair of these integers has a difference that is a multiple of 5 ?
💡 解题思路
For two numbers to have a difference that is a multiple of $5$ , the numbers must be congruent $\bmod{5}$ (their remainders after division by $5$ must be the same).
21
第 21 题
数字运算
How many four-digit positive integers have at least one digit that is a 2 or a 3 ? https://youtu.be/0W3VmFp55cM?t=3291 ~ pi_is_3.14
💡 解题思路
Since we are asked for the number of positive $4$ -digit integers with at least $2$ or $3$ in it, we can find this by finding the total number of $4$ -digit integers and subtracting off those which do
22
第 22 题
方程
Two farmers agree that pigs are worth 300 dollars and that goats are worth 210 dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a 390 dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
💡 解题思路
The problem can be restated as an equation of the form $300p + 210g = x$ , where $p$ is the number of pigs, $g$ is the number of goats, and $x$ is the positive debt. The problem asks us to find the lo
23
第 23 题
几何·面积
Circles with centers A and B have radius 3 and 8, respectively. A common internal tangent intersects the circles at C and D , respectively. Lines AB and CD intersect at E , and AE=5 . What is CD ? [图]
Centers of adjacent faces of a unit cube are joined to form a regular octahedron . What is the volume of this octahedron?
💡 解题思路
We can break the octahedron into two square pyramids by cutting it along a plane perpendicular to one of its internal diagonals. [asy] import three; real r = 1/2; triple A = (-0.5,1.5,0); size(400); c
25
第 25 题
概率
A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
💡 解题思路
Call this cube $ABCDEFGH$ , with $A$ being the starting point.