2006 AMC amc10a 真题 答案 详解

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英文真题

2006 AMC 10A Problems Problem 1 Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? (A) 31 (B) 32 (C) 33 (D) 34 (E) 35 Problem 2 Define x ⊗ y = x^3 - y. What is h ⊗ (h ⊗ h)? (A) -h (B) 0 (C) h (D) 2h (E) h^3 Problem 3 The ratio of Mary's age to Alice's age is 3:5. Alice is 30 years old. How many years old is Mary? (A) 15 (B) 18 (C) 20 (D) 24 (E) 50 Problem 4 A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? (A) 17 (B) 19 (C) 21 (D) 22 (E) 23 Problem 5 Doug and Dave shared a pizza with 8 equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half of the pizza. The cost of a plain pizza was 8 dollars, and there was an additional cost of 2 dollars for putting anchovies on one half. Dave ate all of the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each then paid for what he had eaten. How many more dollars did Dave pay than Doug? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 6 What non-zero real value for x satisfies (7x)^14 = (14x)^7? (A) 1/7 (B) 2/7 (C) 1 (D) 7 (E) 14 Problem 7 The 8 × 18 rectangle ABCD is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is y? (A) 6 (B) 7 (C) 8 (D) 9 (E) 10 Problem 8 A parabola with equation y = x^2 + bx + c passes through the points (2,3) and (4,3). What is c? (A) 2 (B) 5 (C) 7 (D) 10 (E) 11

真题文字详解(英文)

Problem 1 The following problem is from both the 2006 AMC 12A #1 and 2006 AMC 10A #1, so both problems redirect to this page. Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas?
(A) 31 (B) 32 (C) 33 (D) 34 (E) 35

Solution The 5 sandwiches cost $5 \cdot 3 = 15$ dollars. The 8 sodas cost $8 \cdot 2 = 16$ dollars. In total, the purchase costs $15 + 16 =$ (A) 31 dollars.

See also Problem 2 The following problem is from both the 2006 AMC 12A #2 and 2006 AMC 10A #2, so both problems redirect to this page. Define $x \otimes y = x^3 - y$. What is $h \otimes (h \otimes h)$?
(A) −h (B) 0 (C) h (D) 2h (E) h³

Solution 1 By the definition of $\otimes$, we have $h \otimes h = h^3 - h$. Then, $h \otimes (h \otimes h) = h \otimes (h^3 - h) = h^3 - (h^3 - h) = $ (C) h.

See also Problem 3 The following problem is from both the 2006 AMC 12A #3 and 2006 AMC 10A #3, so both problems redirect to this page. The ratio of Mary's age to Alice's age is $3:5$. Alice is 30 years old. How old is Mary?
(A) 15 (B) 18 (C) 20 (D) 24 (E) 50

Solution 1 Let $m$ be Mary's age. Then $\frac{m}{30} = \frac{3}{5}$ Solving for m, we obtain $m = $ (B) 18. Solution 2 We can see this is a combined ratio of $8, (5+3)$. We can equalize by doing $30 \div 5 = 6$, and $6 \cdot 3 = $ (B) 18. With the common ratio of 8 and difference ratio of 6, we see $6 \cdot 8 = 30 + 18$. Therefore, we can see our answer is correct.

See also Problem 4 The following problem is from both the 2006 AMC 12A #4 and 2006 AMC 10A #4, so both problems redirect to this page. A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?
(A) 17 (B) 19 (C) 21 (D) 22 (E) 23

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2006 AMC 12A #1 and 2006 AMC 10A #1, so both problems redirect to this page. Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? Joe's快餐店的三明治每份$3美元，汽水每瓶$2美元。购买5份三明治和8瓶汽水需要多少钱？ (A) 31 (B) 32 (C) 33 (D) 34 (E) 35 Solution The 5 sandwiches cost 5 · 3 = 15 dollars. The 8 sodas cost 8 · 2 = 16 dollars. In total, the purchase costs 15 + 16 = (A) 31 dollars. 5份三明治花费5·3=15美元。8瓶汽水花费8·2=16美元。总花费为15+16=(A) 31美元。 See also Problem 2 The following problem is from both the 2006 AMC 12A #2 and 2006 AMC 10A #2, so both problems redirect to this page. Define x ⊗ y = x^3 - y. What is h ⊗ (h ⊗ h)? 定义x⊗y=x^3-y.h⊗(h⊗h)是什么？ (A) -h (B) 0 (C) h (D) 2h (E) h^3 Solution 1 By the definition of ⊗, we have h ⊗ h = h^3 - h. 根据对⊗的定义，我们有h⊗h=h^3-h。 Then, h ⊗ (h ⊗ h) = h ⊗ (h^3 - h) = h^3 - (h^3 - h) = (C) h. See also Problem 3 The following problem is from both the 2006 AMC 12A #3 and 2006 AMC 10A #3, so both problems redirect to this page. The ratio of Mary's age to Alice's age is 3 : 5. Alice is 30 years old. How old is Mary? Mary的年龄与Alice的年龄之比是3:5。Alice岁。Mary多大？ (A) 15 (B) 18 (C) 20 (D) 24 (E) 50 Solution 1