2002 AMC amc12 真题 答案 详解

英文真题

2002 AMC 12P Problems Problem 1 Which of the following numbers is a perfect square? (A) $4^45^56^6$ (B) $4^45^66^5$ (C) $4^55^46^6$ (D) $4^65^54^5$ (E) $4^65^56^4$ Problem 2 The function f is given by the table If $u_0 = 4$ and $u_{n+1} = f(u_n)$ for $n \geq 0$, find $u_{2002}$ (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 3 The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum of the three dimensions. (A) 36 (B) 38 (C) 42 (D) 44 (E) 92 Problem 4 Let a and b be distinct real numbers for which $$\frac{a}{b} + \frac{a + 10b}{b + 10a} = 2.$$ Find $\frac{a}{b}$ (A) 0.4 (B) 0.5 (C) 0.6 (D) 0.7 (E) 0.8 Problem 5 For how many positive integers m is $$\frac{2002}{m^2 - 2}$$ a positive integer? (A) one (B) two (C) three (D) four (E) more than four Problem 6 Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female? (A) $\frac{1}{3}$ (B) $\frac{4}{11}$ (C) $\frac{2}{5}$ (D) $\frac{4}{9}$ (E) $\frac{1}{2}$ Problem 7 How many three-digit numbers have at least one 2 and at least one 3? (A) 52 (B) 54 (C) 56 (D) 58 (E) 60

真题文字详解(英文)

Problem1
Which of the following numbers is a perfect square?
(A) $4^45^56^6$ (B) $4^45^66^5$ (C) $4^55^54^6$ (D) $4^65^56^5$ (E) $4^65^56^4$

Solution 1
For a positive integer to be a perfect square, all the primes in its prime factorization must have an even exponent. With a quick glance at the answer choices, we can eliminate options
(A) because $5^5$ is an odd power
(B) because $6^5 = 2^5 \cdot 3^5$ and $3^5$ is an odd power
(D) because $6^5 = 2^5 \cdot 3^5$ and $3^5$ is an odd power, and
(E) because $5^5$ is an odd power.

This leaves option (C), in which $4^5 = (2^2)^5 = 2^{10}$, and since $10$, $4$, and $6$ are all even, (C) is a perfect square. Thus, our answer is
(C) $4^55^54^6$.