2003 AMC 10A — Official Competition Problems (January 2003)
📅 2003 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
计数
What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd counting numbers? (A) \ 0 (B) \ 1 (C) \ 2 (D) \ 2003 (E) \ 4006
💡 解题思路
The first $2003$ even counting numbers are $2,4,6,...,4006$ .
2
第 2 题
应用题
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost 4 per pair and each T-shirt costs 5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is 2366, how many members are in the League? (A) \ 77 (B) \ 91 (C) \ 143 (D) \ 182 (E) \ 286$
💡 解题思路
Since T-shirts cost $5$ dollars more than a pair of socks, T-shirts cost $5+4=9$ dollars.
3
第 3 题
分数与比例
A solid box is 15 cm by 10 cm by 8 cm. A new solid is formed by removing a cube 3 cm on a side from each corner of this box. What percent of the original volume is removed? (A) \ 4.5\% (B) \ 9\% (C) \ 12\% (D) \ 18\% (E) \ 24\%
💡 解题思路
The volume of the original box is $15\cdot10\cdot8=1200.$
4
第 4 题
统计
It takes Anna 30 minutes to walk uphill 1 km from her home to school, but it takes her only 10 minutes to walk from school to her home along the same route. What is her average speed, in km/hr, for the round trip? (A) \ 3 (B) \ 3.125 (C) \ 3.5 (D) \ 4 (E) \ 4.5
💡 解题思路
Since she walked $1$ km to school and $1$ km back home, her total distance is $1+1=2$ km.
5
第 5 题
综合
Let d and e denote the solutions of 2x^{2}+3x-5=0 . What is the value of (d-1)(e-1) ? (A) \ -\frac{5}{2} (B) \ 0 (C) \ 3 (D) \ 5 (E) \ 6
💡 解题思路
Using factoring:
6
第 6 题
方程
Define x \heartsuit y to be |x-y| for all real numbers x and y . Which of the following statements is not true? (A) \ x \heartsuit y = y \heartsuit x for all x and y(B) \ 2(x \heartsuit y) = (2x) \heartsuit (2y) for all x and y(C) \ x \heartsuit 0 = x for all x(D) \ x \heartsuit x = 0 for all x(E) \ x \heartsuit y > 0 if x ≠ y
💡 解题思路
We start by looking at the answers. Examining statement C, we notice:
7
第 7 题
几何·面积
How many non- congruent triangles with perimeter 7 have integer side lengths? (A) \ 1 (B) \ 2 (C) \ 3 (D) \ 4 (E) \ 5
💡 解题思路
By the triangle inequality , no side may have a length greater than the semiperimeter, which is $\frac{1}{2}\cdot7=3.5$ .
8
第 8 题
数论
What is the probability that a randomly drawn positive factor of 60 is less than 7 ? (A) \ \frac{1}{10} (B) \ \frac{1}{6} (C) \ \frac{1}{4} (D) \ \frac{1}{3} (E) \ \frac{1}{2}
💡 解题思路
For any positive integer $n$ which is not a perfect square, exactly half of its positive factors will be less than $\sqrt{n}$ , since each such factor can be paired with one that is larger than $\sqrt
The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge -to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? (A) \ 2 (B) \ 3 (C) \ 4 (D) \ 5 (E) \ 6
💡 解题思路
Let the squares be labeled $A$ , $B$ , $C$ , and $D$ .
11
第 11 题
规律与数列
The sum of the two 5-digit numbers AMC10 and AMC12 is 123422 . What is A+M+C ? (A) \ 10 (B) \ 11 (C) \ 12 (D) \ 13 (E) \ 14
💡 解题思路
$AMC10+AMC12=123422$
12
第 12 题
几何·面积
A point (x,y) is randomly picked from inside the rectangle with vertices (0,0) , (4,0) , (4,1) , and (0,1) . What is the probability that x<y ? (A) \ \frac{1}{8} (B) \ \frac{1}{4} (C) \ \frac{3}{8} (D) \ \frac{1}{2} (E) \ \frac{3}{4}
💡 解题思路
The rectangle has a width of $4$ and a height of $1$ .
13
第 13 题
规律与数列
The sum of three numbers is 20 . The first is four times the sum of the other two. The second is seven times the third. What is the product of all three? (A) \ 28 (B) \ 40 (C) \ 100 (D) \ 400 (E) \ 800
💡 解题思路
Let the numbers be $x$ , $y$ , and $z$ in that order. The given tells us that
14
第 14 题
数论
Let n be the largest integer that is the product of exactly 3 distinct prime numbers d , e , and 10d+e , where d and e are single digits. What is the sum of the digits of n ? (A) \ 12 (B) \ 15 (C) \ 18 (D) \ 21 (E) \ 24
💡 解题思路
Since we want $n$ to be as large as possible, we would like $d$ in $10d+e$ to be as large as possible. So, $d=7,$ the greatest single-digit prime. Then, $e$ cannot be $5$ because $10(7)+5 = 75,$ which
15
第 15 题
数论
What is the probability that an integer in the set \{1,2,3,...,100\} is divisible by 2 and not divisible by 3 ? (A) \ \frac{1}{6} (B) \ \frac{33}{100} (C) \ \frac{17}{50} (D) \ \frac{1}{2} (E) \ \frac{18}{25}
💡 解题思路
There are $100$ integers in the set.
16
第 16 题
数字运算
What is the units digit of 13^{2003} ? (A) \ 1 (B) \ 3 (C) \ 7 (D) \ 8 (E) \ 9
💡 解题思路
$13^{2003}\equiv 3^{2003}\pmod{10}$
17
第 17 题
几何·面积
The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? (A) \ \frac{3√(2)}{π} (B) \ \frac{3√(3)}{π} (C) \ √(3) (D) \ \frac{6}{π} (E) \ √(3)π
💡 解题思路
Let $s$ be the length of a side of the equilateral triangle and let $r$ be the radius of the circle.
18
第 18 题
方程
What is the sum of the reciprocals of the roots of the equation \frac{2003}{2004}x+1+\frac{1}{x}=0 ? (A) \ -\frac{2004}{2003} (B) \ -1 (C) \ \frac{2003}{2004} (D) \ 1 (E) \ \frac{2004}{2003}
💡 解题思路
Multiplying both sides by $x$ :
19
第 19 题
几何·面积
A semicircle of diameter 1 sits at the top of a semicircle of diameter 2 , as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a lune . Determine the area of this lune. [图] (A) \ \frac{1}{6}π-\frac{√(3)}{4} (B) \ \frac{√(3)}{4}-\frac{1}{12}π (C) \ \frac{√(3)}{4}-\frac{1}{24}π (D) \ \frac{√(3)}{4}+\frac{1}{24}π (E) \ \frac{√(3)}{4}+\frac{1}{12}π
A base-10 three digit number n is selected at random. Which of the following is closest to the probability that the base-9 representation and the base-11 representation of n are both three-digit numerals? (A) \ 0.3 (B) \ 0.4 (C) \ 0.5 (D) \ 0.6 (E) \ 0.7
💡 解题思路
To be a three digit number in base-10:
21
第 21 题
综合
Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected? (A) \ 22 (B) \ 25 (C) \ 27 (D) \ 28 (E) \ 729 https://youtu.be/3MiGotKnC_U?t=2554
💡 解题思路
Let the ordered triplet $(x,y,z)$ represent the assortment of $x$ chocolate chip cookies, $y$ oatmeal cookies, and $z$ peanut butter cookies.
22
第 22 题
几何·面积
In rectangle ABCD , we have AB=8 , BC=9 , H is on BC with BH=6 , E is on AD with DE=4 , line EC intersects line AH at G , and F is on line AD with GF \perp AF . Find the length of GF . [图] (A) \ 16 (B) \ 20 (C) \ 24 (D) \ 28 (E) \ 30
💡 解题思路
$\angle GHC = \angle AHB$ (Vertical angles are equal).
23
第 23 题
几何·面积
A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have 3 rows of small congruent equilateral triangles, with 5 small triangles in the base row. How many toothpicks would be needed to construct a large equilateral triangle if the base row of the triangle consists of 2003 small equilateral triangles? [图] (A) \ 1,004,004 (B) \ 1,005,006 (C) \ 1,507,509 (D) \ 3,015,018 (E) \ 6,021,018
💡 解题思路
There are $1+3+5+...+2003=1002^{2}=1004004$ small equilateral triangles.
24
第 24 题
规律与数列
Sally has five red cards numbered 1 through 5 and four blue cards numbered 3 through 6 . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? (A) \ 8 (B) \ 9 (C) \ 10 (D) \ 11 (E) \ 12
💡 解题思路
Let $R_i$ and $B_j$ designate the red card numbered $i$ and the blue card numbered $j$ , respectively.
25
第 25 题
数论
Let n be a 5 -digit number, and let q and r be the quotient and the remainder, respectively, when n is divided by 100 . For how many values of n is q+r divisible by 11 ? (A) \ 8180 (B) \ 8181 (C) \ 8182 (D) \ 9000 (E) \ 9090
💡 解题思路
$11|(q+r)$ implies that $11|(99q+q+r)$ and therefore $11|(100q+r)$ , so $11|n$ . Then, $n$ can range from $10010$ to $99990$ for a total of $\boxed{8181\Rightarrow \mathrm{(B)}}$ numbers.