考题3-4 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 3

Ann made a -step staircase using  toothpicks as shown in the figure. How many toothpicks does she need to add to complete a -step staircase?

Solution

We can see that a -step staircase requires  toothpicks and a -step staircase requires  toothpicks. Thus, to go from a -step to -step staircase,  additional toothpicks are needed and to go from a -step to -step staircase,  additional toothpicks are needed. Applying this pattern, to go from a -step to -step staircase,  additional toothpicks are needed and to go from a -step to -step staircase,  additional toothpicks are needed. Our answer is 

Problem 4

Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?

Solution 1

Assign a variable to the number of eggs Mia has, say . Then, because we are given that Sofia has twice the number of eggs Mia has, Sofia has  eggs, and Pablo, having three times the number of eggs as Sofia, has  eggs.

For them to all have the same number of eggs, they must each have  eggs. This means Pablo must give  eggs to Mia and a  eggs to Sofia, so the answer is 

Solution 2

Let Mia have  egg. Sofia and Pablo would then have  and  eggs, respectively. If we split the eggs evenly, each would have  eggs. Since Sofia needs  more egg, Pablo must give her  of his  eggs, or 

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