考题1-2 2015 AMC 10A

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 1

What is the value of 

Solution

.

Problem 2

A box contains a collection of triangular and square tiles. There are  tiles in the box, containing  edges total. How many square tiles are there in the box?

Solution

Let  be the amount of triangular tiles and  be the amount of square tiles.

Triangles have  edges and squares have  edges, so we have a system of equations.

We have  tiles total, so .

We have  edges total, so .

Solving gives,  and , so the answer is .

Alternate Solution

If all of the tiles were triangles, there would be  edges. This is not enough, so there need to be some squares. Trading a triangle for a square results in one additional edge each time, so we must trade out  triangles for squares. Answer: 

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