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 3

Problem

Real numbers , , and  satisfy the inequalities , , and . Which of the following numbers is necessarily positive?

Solution

Notice that  must be positive because . Therefore the answer is .

The other choices:

 As  grows closer to ,  decreases and thus becomes less than .

  can be as small as possible (), so  grows close to  as  approaches .

 For all , , and thus it is always negative.

 The same logic as above, but when  this time.

Problem 4

Supposed that  and  are nonzero real numbers such that . What is the value of ?

Solutions

Solution 1

Rearranging, we find , or . Substituting, we can convert the second equation into .

Solution 2

Substituting each  and  with , we see that the given equation holds true, as . Thus, 

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