2005 AMC 8 — Official Competition Problems (November 2019)
📅 2011 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
Connie multiplies a number by 2 and gets 60 as her answer. However, she should have divided the number by 2 to get the correct answer. What is the correct answer?
💡 解题思路
If $x$ is the number, then $2x=60$ and $x=30$ . Dividing the number by $2$ yields $\dfrac{30}{2} = \boxed{\textbf{(B)}\ 15}$ .
2
第 2 题
应用题
Karl bought five folders from Pay-A-Lot at a cost of \textdollar 2.50 each. Pay-A-Lot had a 20%-off sale the following day. How much could Karl have saved on the purchase by waiting a day?
💡 解题思路
Karl paid $5 \cdot 2.50 = \textdollar 12.50$ . $20 \%$ of this cost that he saved is $12.50 \cdot .2 = \boxed{\textbf{(C)}\ \textdollar 2.50}$ .
3
第 3 题
几何·面积
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal \overline{BD} of square ABCD ? [图]
💡 解题思路
Rotating square $ABCD$ counterclockwise $45^\circ$ so that the line of symmetry $BD$ is a vertical line makes it easier to see that $\boxed{\textbf{(D)}\ 4}$ squares need to be colored to match its co
4
第 4 题
几何·面积
A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are 6.1 cm, 8.2 cm and 9.7 cm. What is the area of the square in square centimeters?
💡 解题思路
The perimeter of the triangle is $6.1+8.2+9.7=24$ cm. A square's perimeter is four times its sidelength, since all its sidelengths are equal. If the square's perimeter is $24$ , the sidelength is $24/
5
第 5 题
综合
Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda?
💡 解题思路
Start by buying the largest packs first. After three $24$ -packs, $90-3(24)=18$ cans are left. After one $12$ -pack, $18-12=6$ cans are left. Then buy one more $6$ -pack. The total number of packs is
6
第 6 题
数字运算
Suppose d is a digit. For how many values of d is 2.00d5 > 2.005 ?
💡 解题思路
We see that $2.0055$ works but $2.0045$ does not. The digit $d$ can be numbers $5$ through $9$ , which is $\boxed{\textbf{(C)}\ 5}$ values.
7
第 7 题
行程问题
Bill walks \tfrac12 mile south, then \tfrac34 mile east, and finally \tfrac12 mile south. How many miles is he, in a direct line, from his starting point?
💡 解题思路
We have a right-angle triangle with sides 3/4 and 1. We can divide the simple pythagorean theorem identity 3,4,5 by 4 to get 3/4, 1, 5/4. This shows us the answer is $\boxed{\textbf{B}\ 1\tfrac{1}{4}}
8
第 8 题
整数运算
Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
💡 解题思路
Assume WLOG that $m$ and $n$ are both $1$ . Plugging into each of the choices, we get $4, 2, 6, 16,$ and $3$ . The only odd integer is $\boxed{\textbf{(E)}\ 3mn}$ .
9
第 9 题
几何·角度
In quadrilateral ABCD , sides \overline{AB} and \overline{BC} both have length 10, sides \overline{CD} and \overline{DA} both have length 17, and the measure of angle ADC is 60^\circ . What is the length of diagonal \overline{AC} ? [图]
💡 解题思路
Because $\overline{AD} = \overline{CD}$ , $\triangle ADC$ is an isosceles triangle with $\angle DAC = \angle DCA$ . Angles in a triangle add up to $180^\circ$ , and since $\angle ADC=60^\circ$ , the o
10
第 10 题
行程问题
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. How many minutes did it take Joe to get from home to school?
💡 解题思路
We can use the equation $d=rt$ where $d$ is the distance, $r$ is the rate, and $t$ is the time. The distances he ran and walked are equal, so $r_rt_r=r_wt_w$ , where $r_r$ is the rate at which he ran,
11
第 11 题
计数
The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its 90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up 90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?
💡 解题思路
The price Jack rings up is $\textdollar{(90.00)(1.06)(0.80)}$ . The price Jill rings up is $\textdollar{(90.00)(0.80)(1.06)}$ . By the commutative property of multiplication, these quantities are the
12
第 12 题
综合
Big Al, the ape, ate 100 bananas from May 1 through May 5. Each day he ate six more bananas than on the previous day. How many bananas did Big Al eat on May 5?
💡 解题思路
There are $5$ days from May 1 to May 5. The number of bananas he eats each day is an arithmetic sequence. He eats $n$ bananas on May 5, and $n-4(6)=n-24$ bananas on May 1. The sum of this arithmetic s
13
第 13 题
几何·面积
The area of polygon ABCDEF is 52 with AB=8 , BC=9 and FA=5 . What is DE+EF ? [图]
💡 解题思路
Notice that $AF + DE = BC$ , so $DE=4$ . Let $O$ be the intersection of the extensions of $AF$ and $DC$ , which makes rectangle $ABCO$ . The area of the polygon is the area of $FEDO$ subtracted from t
14
第 14 题
综合
The Little Twelve Basketball Conference has two divisions, with six teams in each division. Each team plays each of the other teams in its own division twice and every team in the other division once. How many conference games are scheduled?
💡 解题思路
Within each division, there are $\binom {6}{2} = 15$ pairings, and each of these games happens twice. The same goes for the other division so that there are $4(15)=60$ games within their own divisions
15
第 15 题
几何·面积
How many different isosceles triangles have integer side lengths and perimeter 23?
💡 解题思路
Let $b$ be the base of the isosceles triangles, and let $a$ be the lengths of the other legs. From this, $2a+b=23$ and $b=23-2a$ . From triangle inequality, $2a>b$ , then plug in the value from the pr
16
第 16 题
行程问题
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
💡 解题思路
The Martian can pull out $12$ socks, $4$ of each color, without having $5$ of the same kind yet. However, the next one he pulls out must be the fifth of one of the colors so he must remove $\boxed{\te
17
第 17 题
坐标几何
The results of a cross-country team's training run are graphed below. Which student has the greatest average speed? [图]
💡 解题思路
Average speed is distance over time, or the slope of the line through the point and the origin. $\boxed{\textbf{(E)}\ \text{Evelyn}}$ has the steepest line, and runs the greatest distance for the shor
18
第 18 题
数论
How many three-digit numbers are divisible by 13?
💡 解题思路
Let $k$ be any positive integer so that $13k$ is a multiple of $13$ . For the smallest three-digit number, $13k>100$ and $k>\frac{100}{13} \approx 7.7$ . For the greatest three-digit number, $13k<999$
19
第 19 题
几何·面积
What is the perimeter of trapezoid ABCD ? [图]
💡 解题思路
Draw altitudes from $B$ and $C$ to base $AD$ to create a rectangle and two right triangles. The side opposite $BC$ is equal to $50$ . The bases of the right triangles can be found using Pythagorean or
20
第 20 题
几何·面积
Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
💡 解题思路
Alice moves $5k$ steps and Bob moves $9k$ steps, where $k$ is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, $14k$ , is a multiple of $12$ . Since this n
21
第 21 题
几何·面积
How many distinct triangles can be drawn using three of the dots below as vertices? [图]
💡 解题思路
The number of ways to choose three points to make a triangle is $\binom 63 = 20$ . However, two* of these are a straight line so we subtract $2$ to get $\boxed{\textbf{(C)}\ 18}$ .
22
第 22 题
应用题
A company sells detergent in three different sized boxes: small (S), medium (M) and large (L). The medium size costs 50% more than the small size and contains 20% less detergent than the large size. The large size contains twice as much detergent as the small size and costs 30% more than the medium size. Rank the three sizes from best to worst buy. (A) SML (B) LMS (C) MSL (D) LSM (E) MLS
💡 解题思路
WLOG (Without Loss of Generality), suppose the small size costs $\textdollar 1$ and has $5$ oz. The medium size then costs $\textdollar 1.50$ and has $8$ oz. The large size costs $\textdollar 1.95$ an
23
第 23 题
几何·面积
Isosceles right triangle ABC encloses a semicircle of area 2π . The circle has its center O on hypotenuse \overline{AB} and is tangent to sides \overline{AC} and \overline{BC} . What is the area of triangle ABC ? [图] First, we notice half a square so first let's create a square. Once we have a square, we will have a full circle. This circle has a diameter of 4 which will be the side of the square. The area would be 4· 4 = 16. Divide 16 by 2 to get the original shape and you get [8]
💡 解题思路
We can figure out the radius of the semicircle from the question states that the area of the semicircle is $2\pi$ and we can multiply it by 2 to complete the circle to get $4\pi$ which we can find the
24
第 24 题
逻辑推理
A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"?
💡 解题思路
We can start at $200$ and work our way down to $1$ . We want to press the button that multiplies by $2$ the most, but since we are going down instead of up, we divide by $2$ instead. If we come across
25
第 25 题
几何·面积
A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle? [图]
💡 解题思路
Let the region within the circle and square be $a$ . In other words, it is the area inside the circle $\textbf{and}$ the square. Let $r$ be the radius. We know that the area of the circle minus $a$ is