AMC数学竞赛真题2017年10B 15-16

AMC10数学竞赛是美国高中数学竞赛中的一项，是针对高中一年级及初中三年级学生的数学测试，该竞赛开始于2000年，分A赛和B赛，于每年的2月初和2月中举行，学生可任选参加一项即可。不管是对高校申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC10数学竞赛真题以及官方解答吧：

Problem 15

Rectangle  has  and . Point  is the foot of the perpendicular from  to diagonal . What is the area of ?

Solution

First, note that  because  is a right triangle. In addition, we have , so . Using similar triangles within , we get that  and .

Let  be the foot of the perpendicular from  to . Since  and  are parallel,  is similar to . Therefore, we have . Since , . Note that  is an altitude of  from , which has length . Therefore, the area of  is 

Solution 2

Alternatively, we can use coordinates. Denote  as the origin. We find the equation for  as , and  as . Solving for  yields . Our final answer then becomes 

Solution 3

We note that the area of  must equal area of  because they share the base and the height of both is the altitude of congruent triangles. Therefore, we find the area of  to be 

Problem 16

How many of the base-ten numerals for the positive integers less than or equal to  contain the digit ?

Solution

We can use complementary counting. There are  positive integers in total to consider, and there are  one-digit integers, two digit integers without a zero,  three digit integers without a zero, and  four-digit integers starting with a 1 without a zero. Therefore, the answer is .

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