Math Kangaroo袋鼠数学竞赛真题解答（二）

上周我们给了一组袋鼠竞赛的题目（见《袋鼠数学竞赛题选（一）》），今天我们给出第二组题目，仍然是每个年级组各一题，并附上解答。

1-2年级组

Place each of the numbers 1, 2, 3, 4, 5 into one box so that everything is correct. Which number goes into the box marked with question mark?

把数字1，2，3，4，5分别填入下图中的方框内，使得其中的算式都成立。标注问号的方框内应该填哪个数字？

3-4年级组

Josip has 4 toys - a car, a doll, a ball and a ship. He wants to put them on a line on a shelf. The ship has to be next to the car and the doll has to be next to the car. In how many ways can he arrange them so that all the conditions are fulfilled?

Josip有四个玩具：一辆车，一个洋娃娃，一个球和一只船。他想把这些玩具在架子上排成一排。船要跟车紧挨着，洋娃娃也要跟车紧挨着。他有多少种不同的方式排列这些玩具？

5-6年级组

On Jump Street, there are 9 houses in a row. At least one person lives in each house. Any two neighbouring houses together are inhabited by at most six people. What is the largest number of people that could be living on Jump Street?

有9栋房子依次座落在Jump街上。每栋房子里至少住了一个人，而且任意两栋相邻的房子里住的人数不超过6人。Jump街上最多住了多少人？

7-8年级组

A bush has 10 branches. Each branch has either 5 leaves only or 2 leaves and 1 flower. Which of the following could be the total number of leaves the bush has?

一丛灌木有10个分枝。每个分枝上要么有5片叶子，要么有2片叶子和1朵花。这丛灌木的叶子数量可能是多少？

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下面是第二组题目的参考答案和解析。

1-2年级组

Place each of the numbers 1, 2, 3, 4, 5 into one box so that everything is correct. Which number goes into the box marked with question mark?

把数字1，2，3，4，5分别填入下图中的方框内，使得其中的算式都成立。标注问号的方框内应该填哪个数字？

【答案：E】标注了问号的方框里的数是最上方的两个数的和，同时也是它右方和最下方的两个数之和。所以方框里的数应该是所有数里面最大的。

3-4年级组

Josip has 4 toys - a car, a doll, a ball and a ship. He wants to put them on a line on a shelf. The ship has to be next to the car and the doll has to be next to the car. In how many ways can he arrange them so that all the conditions are fulfilled?

Josip有四个玩具：一辆车，一个洋娃娃，一个球和一只船。他想把这些玩具在架子上排成一排。船要跟车紧挨着，洋娃娃也要跟车紧挨着。他有多少种不同的方式排列这些玩具？

【答案：B】船和洋娃娃都要紧挨着车，所以这三种玩具共有两种排列方式。排好这三种玩具后，球可以排在最前面或最后面，所以有两种选择。根据乘法原则得答案。

5-6年级组

On Jump Street, there are 9 houses in a row. At least one person lives in each house. Any two neighbouring houses together are inhabited by at most six people. What is the largest number of people that could be living on Jump Street?

有9栋房子依次座落在Jump街上。每栋房子里至少住了一个人，而且任意两栋相邻的房子里住的人数不超过6人。Jump街上最多住了多少人？

【答案：D】考虑能使人最多的方案。第1，2栋房子一共住6人，第3，4栋也一共住6人，第5，6栋共住6人，第7，8栋也住6人。第9栋房子住5人。这个住房方案是可以实现的，只需第奇数栋房子住5人，第偶数栋房子住1人。

7-8年级组

A bush has 10 branches. Each branch has either 5 leaves only or 2 leaves and 1 flower. Which of the following could be the total number of leaves the bush has?

一丛灌木有10个分枝。每个分枝上要么有5片叶子，要么有2片叶子和1朵花。这丛灌木的叶子数量可能是多少？

【答案：E】称有5片叶子的分枝为第一类分枝，有2片叶子和1朵花的分枝为第二类分枝。如果10个分枝都是第一类分枝，则共有50片叶子。把一个第一类分枝换成第二类分枝，叶子就减少3片。所以，50和最终的叶子数量之差必定是3的倍数，或者说，最终的叶子数量和50关于3同余。50除以3的余数为2，而选项A~D的数除以3的余数都不是2。