2005 AMC 10B — Official Competition Problems (January 2005)
📅 2005 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
A scout troop buys 1000 candy bars at a price of five for 2 dollars. They sell all the candy bars at the price of two for 1 dollar. What was their profit, in dollars?
A positive number x has the property that x\% of x is 4 . What is x ?
💡 解题思路
Since $x\%$ means $0.01x$ , the statement " $x\% \text{ of } x \text{ is 4}$ " can be rewritten as " $0.01x \cdot x = 4$ ":
3
第 3 题
分数与比例
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?
💡 解题思路
After the first day, there is $1-\left(\dfrac{1}{3}\cdot1\right)=\frac{2}{3}$ gallons left. After the second day, there is a total of $\dfrac{2}{3}-\left(\dfrac{1}{3}\times\dfrac{2}{3}\right)=\dfrac{2
4
第 4 题
综合
For real numbers a and b , define a \diamond b = √(a^2 + b^2) . What is the value of (5 \diamond 12) \diamond ((-12) \diamond (-5)) ?
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
💡 解题思路
Let $m =$ Brianna's money. We have $\frac15 m = \frac13 (\mbox{CDs}) \Rightarrow \frac35 m = (\mbox{CDs})$ . Thus, the money left over is $m-\frac35m = \frac25m$ , so the answer is $\boxed{\textbf{(C)
6
第 6 题
综合
At the beginning of the school year, Lisa's goal was to earn an A on at least 80\% of her 50 quizzes for the year. She earned an A on 22 of the first 30 quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A ?
💡 解题思路
Lisa's goal was to get an $A$ on $80\% \cdot 50 = 40$ quizzes. She already has $A$ 's on $22$ quizzes, so she needs to get $A$ 's on $40-22=18$ more. There are $50-30=20$ quizzes left, so she can affo
7
第 7 题
几何·面积
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smallest circle to the area of the largest square?
💡 解题思路
Let the side of the largest square be $x$ . It follows that the diameter of the inscribed circle is also $x$ . Therefore, the square's diagonal inscribed in the circle is $x$ . The side length of the
8
第 8 题
几何·面积
An 8 -foot by 10 -foot bathroom floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded? [图]
💡 解题思路
There are $80$ tiles. Each tile has $[\mbox{square} - 4 \cdot (\mbox{quarter circle})]$ shaded. Thus:
9
第 9 题
概率
One fair die has faces 1, 1, 2, 2, 3, 3 and another has faces 4, 4, 5, 5, 6, 6. The dice are rolled and the numbers on the top faces are added. What is the probability that the sum will be odd?
💡 解题思路
In order to obtain an odd sum, exactly one out of the two dice must have an odd number. We can easily find the total probability using casework.
10
第 10 题
几何·面积
In \triangle ABC , we have AC=BC=7 and AB=2 . Suppose that D is a point on line AB such that B lies between A and D and CD=8 . What is BD ?
💡 解题思路
Draw height $CH$ (Perpendicular line from point C to line AD). We have that $BH=1$ . By the Pythagorean Theorem , $CH=\sqrt{48}$ . Since $CD=8$ , $HD=\sqrt{8^2-48}=\sqrt{16}=4$ , and $BD=HD-1$ , so $B
11
第 11 题
规律与数列
The first term of a sequence is 2005 . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the {2005}^{th} term of the sequence? (A) 29 (B) 55 (C) 85 (D) 133 (E) 250
💡 解题思路
Performing this operation several times yields the results of $133$ for the second term, $55$ for the third term, and $250$ for the fourth term. The sum of the cubes of the digits of $250$ equal $133$
12
第 12 题
数论
Twelve fair dice are rolled. What is the probability that the product of the numbers on the top faces is prime?
💡 解题思路
In order for the product of the numbers to be prime, $11$ of the dice have to be a $1$ , and the other die has to be a prime number. There are $3$ prime numbers ( $2$ , $3$ , and $5$ ), and there is o
13
第 13 题
数论
How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12 ?
💡 解题思路
To find the multiples of $3$ or $4$ but not $12$ , you need to find the number of multiples of $3$ and $4$ , and then subtract twice the number of multiples of $12$ , because you overcount and do not
14
第 14 题
几何·面积
Equilateral \triangle ABC has side length 2 , M is the midpoint of \overline{AC} , and C is the midpoint of \overline{BD} . What is the area of \triangle CDM ? [图]
💡 解题思路
The area of a triangle can be given by $\frac12 ab \sin C$ . $MC=1$ because it is the midpoint of a side, and $CD=2$ because it is the same length as $BC$ . Each angle of an equilateral triangle is $6
15
第 15 题
概率
An envelope contains eight bills: 2 ones, 2 fives, 2 tens, and 2 twenties. Two bills are drawn at random without replacement. What is the probability that their sum is 20$ or more?
💡 解题思路
The only way to get a total of \$ $20$ or more is if you pick a twenty and another bill, or if you pick both tens. There are a total of $\dbinom{8}{2}=\dfrac{8\times7}{2\times1}=28$ ways to choose $2$
16
第 16 题
方程
The quadratic equation x^2+mx+n has roots twice those of x^2+px+m , and none of m,n, and p is zero. What is the value of n/p ?
💡 解题思路
Let $x^2 + px + m = 0$ have roots $a$ and $b$ . Then
17
第 17 题
综合
Suppose that 4^a = 5 , 5^b = 6 , 6^c = 7 , and 7^d = 8 . What is a · b· c · d ?
All of David's telephone numbers have the form 555-abc-defg , where a , b , c , d , e , f , and g are distinct digits and in increasing order, and none is either 0 or 1 . How many different telephone numbers can David have?
💡 解题思路
The only digits available to use in the phone number are $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , and $9$ . There are only $7$ spots left among the $8$ numbers, so we need to find the number of ways
19
第 19 题
统计
On a certain math exam, 10\% of the students got 70 points, 25\% got 80 points, 20\% got 85 points, 15\% got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?
💡 解题思路
To begin, we see that the remaining $30\%$ of the students got $95$ points. Assume that there are $20$ students; we see that $2$ students got $70$ points, $5$ students got $80$ points, $4$ students go
20
第 20 题
统计
What is the average (mean) of all 5-digit numbers that can be formed by using each of the digits 1, 3, 5, 7, and 8 exactly once?
💡 解题思路
The average of all valid numbers is simply the expected value of a randomly chosen valid number. In other words, it is $E[\text{ten thousands digit}]\cdot 10^4+E[\text{thousands digit}]\cdot 10^3+\cdo
21
第 21 题
概率
Forty slips are placed into a hat, each bearing a number 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , or 10 , with each number entered on four slips. Four slips are drawn from the hat at random and without replacement. Let p be the probability that all four slips bear the same number. Let q be the probability that two of the slips bear a number a and the other two bear a number b ≠ a . What is the value of q/p ?
💡 解题思路
There are $10$ ways to determine which number to pick. There are $4!$ ways to then draw those four slips with that number, and $40 \cdot 39 \cdot 38 \cdot 37$ total ways to draw four slips. Thus $p =
22
第 22 题
数论
For how many positive integers n less than or equal to 24 is n! evenly divisible by 1 + 2 + ·s + n?
💡 解题思路
Since $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$ , the condition is equivalent to having an integer value for $\frac{n!} {\frac{n(n+1)}{2}}$ . This reduces, when $n\ge 1$ , to having an integer value for
23
第 23 题
几何·面积
In trapezoid ABCD we have \overline{AB} parallel to \overline{DC} , E as the midpoint of \overline{BC} , and F as the midpoint of \overline{DA} . The area of ABEF is twice the area of FECD . What is AB/DC ?
💡 解题思路
Since the heights of both trapezoids are equal, and the area of $ABEF$ is twice the area of $FECD$ ,
24
第 24 题
数字运算
Let x and y be two-digit integers such that y is obtained by reversing the digits of x . The integers x and y satisfy x^2 - y^2 = m^2 for some positive integer m . What is x + y + m ?
💡 解题思路
Let $x = 10a+b, y = 10b+a$ . The given conditions imply $x>y$ , which implies $a>b$ , and they also imply that both $a$ and $b$ are nonzero.
25
第 25 题
规律与数列
A subset B of the set of integers from 1 to 100 , inclusive, has the property that no two elements of B sum to 125 . What is the maximum possible number of elements in B ?
💡 解题思路
The question asks for the maximum possible number of elements. The integers from $1$ to $24$ can be included because you cannot make $125$ with integers from $1$ to $24$ without the other number being