2003A AMC 12 — Official Competition Problems (February 2024)
📅 2024 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
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
$AMC10+AMC12=123422$
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
9
第 9 题
坐标几何
A set S of points in the xy -plane is symmetric about the origin, both coordinate axes, and the line y=x . If (2,3) is in S , what is the smallest number of points in S ? (A) \ 1 (B) \ 2 (C) \ 4 (D) \ 8 (E) \ 16
💡 解题思路
If $(2,3)$ is in $S$ , then its reflection in the line $y = x$ , i.e. $(3,2)$ , is also in $S$ . Now by reflecting these points in both coordinate axes, we quickly deduce that every point of the form
10
第 10 题
分数与比例
Al, Bert, and Carl are the winners of a school drawing for a pile of Halloween candy, which they are to divide in a ratio of 3:2:1 , respectively. Due to some confusion they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes what he believes to be the correct share of candy, what fraction of the candy goes unclaimed? (A) \ \frac{1}{18} (B) \ \frac{1}{6} (C) \ \frac{2}{9} (D) \ \frac{5}{18} (E) \ \frac{5}{12}
💡 解题思路
Because the ratios are $3:2:1$ , Al, Bert, and Carl believe that they need to take $1/2$ , $1/3$ , and $1/6$ of the pile when they each arrive, respectively. After each person comes, $1/2$ , $2/3$ , a
11
第 11 题
几何·面积
A square and an equilateral triangle have the same perimeter. Let A be the area of the circle circumscribed about the square and B the area of the circle circumscribed around the triangle. Find A/B . (A) \ \frac{9}{16} (B) \ \frac{3}{4} (C) \ \frac{27}{32} (D) \ \frac{3√(6)}{8} (E) \ 1
💡 解题思路
Suppose that the common perimeter is $P$ . Then the side lengths of the square and the triangle are $\frac{P}{4}$ and $\frac{P}{3}$ , respectively.
12
第 12 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.
13
第 13 题
几何·面积
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 $D$ .
14
第 14 题
几何·面积
Points K, L, M, and N lie in the plane of the square ABCD such that AKB , BLC , CMD , and DNA are equilateral triangles. If ABCD has an area of 16, find the area of KLMN . [图] \textrm{(A)}\ 32 \textrm{(B)}\ 16+16√(3) \textrm{(C)}\ 48 \textrm{(D)}\ 32+16√(3) \textrm{(E)}\ 64
💡 解题思路
Since the area of square $\text{ABCD}$ is 16, the side length must be 4. Thus, the side length of triangle AKB is 4, and the height of $\text{AKB}$ , and thus $\text{DMC}$ , is $2\sqrt{3}$ .
15
第 15 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 point P is chosen at random in the interior of equilateral triangle ABC . What is the probability that \triangle ABP has a greater area than each of \triangle ACP and \triangle BCP ?
Square ABCD has sides of length 4 , and M is the midpoint of \overline{CD} . A circle with radius 2 and center M intersects a circle with radius 4 and center A at points P and D . What is the distance from P to \overline{AD} ? [图]
💡 解题思路
$APMD$ obviously forms a kite. Let the intersection of the diagonals be $E$ . $AE+EM=AM=2\sqrt{5}$ Let $AE=x$ . Then, $EM=2\sqrt{5}-x$ .
18
第 18 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
When a $5$ -digit number is divided by $100$ , the first $3$ digits become the quotient, $q$ , and the last $2$ digits become the remainder, $r$ .
19
第 19 题
坐标几何
A parabola with equation y=ax^2+bx+c is reflected about the x -axis. The parabola and its reflection are translated horizontally five units in opposite directions to become the graphs of y=f(x) and y=g(x) , respectively. Which of the following describes the graph of y=(f+g)(x)?
💡 解题思路
If we take the parabola $ax^2 + bx + c$ and reflect it over the x - axis, we have the parabola $-ax^2 - bx - c$ . Without loss of generality, let us say that the parabola is translated 5 units to the
20
第 20 题
统计
How many 15 -letter arrangements of 5 A's, 5 B's, and 5 C's have no A's in the first 5 letters, no B's in the next 5 letters, and no C's in the last 5 letters? \textrm{(A)}\ \sum_{k=0}^{5}\binom{5}{k}^{3} \textrm{(B)}\ 3^{5}· 2^{5} \textrm{(C)}\ 2^{15} \textrm{(D)}\ \frac{15!}{(5!)^{3}} \textrm{(E)}\ 3^{15} https://youtu.be/3MiGotKnC_U?t=2323 ~ ThePuzzlr https://youtu.be/0W3VmFp55cM?t=3737 ~ Sohil Rathi https://youtu.be/GmUWIXXf_uk?t=260 ~ Sohil Rathi
💡 解题思路
The answer is $\boxed{\textrm{(A)}}$ .
21
第 21 题
坐标几何
The graph of the polynomial \[P(x) = x^5 + ax^4 + bx^3 + cx^2 + dx + e\] has five distinct x -intercepts, one of which is at (0,0) . Which of the following coefficients cannot be zero? (A)\ a (B)\ b (C)\ c (D)\ d (E)\ e
💡 解题思路
Let the roots be $r_1=0, r_2, r_3, r_4, r_5$ . According to Viète's formulae , we have $d=r_1r_2r_3r_4 + r_1r_2r_3r_5 + r_1r_2r_4r_5 + r_1r_3r_4r_5 + r_2r_3r_4r_5$ . The first four terms contain $r_1=
22
第 22 题
坐标几何
Objects A and B move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object A starts at (0,0) and each of its steps is either right or up, both equally likely. Object B starts at (5,7) and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet? (A) \ 0.10 (B) \ 0.15 (C) \ 0.20 (D) \ 0.25 (E) \ 0.30
💡 解题思路
If $A$ and $B$ meet, their paths connect $(0,0)$ and $(5,7).$ There are $\binom{12}{5}=792$ such paths. Since the path is $12$ units long, they must meet after each travels $6$ units, so the probabili
23
第 23 题
几何·面积
How many perfect squares are divisors of the product 1! · 2! · 3! · \hdots · 9! ?
💡 解题思路
We want to find the number of perfect square factors in the product of all the factorials of numbers from $1 - 9$ . We can write this out and take out the factorials, and then find a prime factorizati
24
第 24 题
综合
If a≥ b > 1, what is the largest possible value of \log_{a}(a/b) + \log_{b}(b/a)?(A)\ -2 (B)\ 0 (C)\ 2 (D)\ 3 (E)\ 4
💡 解题思路
Using logarithmic rules, we see that
25
第 25 题
统计
Let f(x)= √(ax^2+bx) . For how many real values of a is there at least one positive value of b for which the domain of f and the range of f are the same set ? (A) \ 0 (B) \ 1 (C) \ 2 (D) \ 3 (E) \ \mathrm{infinitely \ many }
💡 解题思路
The function $f(x) = \sqrt{x(ax+b)}$ has a codomain of all non-negative numbers, or $0 \le f(x)$ . Since the domain and the range of $f$ are the same, it follows that the domain of $f$ also satisfies