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第十二届少文杯四年级试题及解析-成都

第十二届少文杯四年级试题及解析-成都
第十二届少文杯四年级试题及解析-成都

第十二届“少文杯”数学邀请赛

四年级试真题及卷详解

【2012·少文杯·计算综合·第1题】396+402+403+399+401

【解析】原式=400-4+400+2+400+3+400-1+400+1

=400×5-4+2+3-1+1

=2000+1

=2001

【2012·少文杯·计算综合·第2题】2012×2012÷2012+2012

【解析】原式=2012+2012

=4024

【2012·少文杯·计算综合·第3题】90÷7-13÷7

【解析】原式=(90-13)÷7

=77÷7

=11

【2012·少文杯·计算综合·第4题】25×44×125×16

【解析】原式=25×4×11×125×8×2

=100×11×1000×2

=2200000

【2012·少文杯·计算综合·第5题】3+6+9+12+……+63

【解析】项数=(63-3)÷3+1=21

和=(3+63)×21÷2=693

【2012·少文杯·应用题·第6题】爸爸和妈妈一个月的工资收入一共有6500元,已知爸爸比妈妈每个月多挣1500元,那么妈妈每个月的工资收入是元。

【解析】本题是一个典型的和差问题,由题意可知,爸爸比妈妈挣的多,所以由较小数=(和+差)÷2得,妈妈挣的工资是(6500-1500)÷2=2500元。

【2012·少文杯·数字游戏·第7题】观察下表,这里有四个数字被掩盖住了,那么这四个数字所组成的不重复的最大四位数是。

【解析】通过观察计算每一行每一列的和都是等于30,比如横行5+4+10+7+4=30,竖列

8+4+7+9+2=30。所以经计算得如下表格:

空白部分分别是1、5、6、8,这四个数组成的数最大是8651。

【2012·少文杯·数论·第8题】采购员为酒店采购了99个杯子,却发现发票上字迹模糊看

不清楚金额,只能看到3.5,采购员只记得金额一定是在340元到370元之间,请帮她算出来

用了元。

【解析】本题要利用整除的特征,买了993一定能从平均分成99份,35一定能被99整除,而能被99整除的数有个特点——将一个数两位为一段,段

段相加能被99整除那么这个数一定能被99整除,例如:2376两位为一段分成23和76,23+76=99,99显然能被99整除,所以2376一定能被99整除。那么在这个题目中,要使

35能被99整除,那么加35的和也能被99整除,经计算发现在340~370之间,34+65=99能被99整除。所以采购员用了346.5元。

【2012·少文杯·推理策略·第9题】现有容量一样的a、b、c、d四个酒杯,分别装有40毫升、50毫升、70毫升、100毫升的鸡尾酒。实习调酒师将最多那杯分别往其余三个酒杯里各倒

入10毫升酒,这样称为一次练习,那么当他练习了100次以后,a酒杯里装有毫升的酒。【解析】对于这样的问题,我们应该做几次练习,希望能找到周期和规律。

不妨想做几次练习看看:

①②③④⑤

a:40->50->60->70->80->50

b:50->60->70->80->50->60

c:70->80->50->60->70->80

d:100->70->80->50->60->70

我们通过上表可看出,四个酒杯中的鸡尾酒总数在50、60、70、80这四个数中循环。并且这里有个规律:第五次练习后与第一次调整后的情况完全一样。现在我们就知道了调整的周期是4。100÷4=25,这说明在第100次练习后的情况应该与第四次练习后的情况相同。所以a酒杯是80毫升。

【2012·少文杯·计数·第10题】如图所示,小蚂蚁要从进口A走到出口B,箭头所示均可以行走的道路,B1、B2、B3均为出口,那么再不走回头路的情况下,它有种走出去的路线。

【解析】本题可使用标数法解决,起点标1,每一步的方法数都是上一步所有的方法数的和。如下图:

通过此图可知,一共有2+4+2=8种走法。

【2012·少文杯·数字游戏·第10题】下面的文字代表不同的数字,观察竖式,“好”字用数字表示应为。

【解析】加减数字谜主要有以下几点:首位分析、末尾分析、进退位。看首位,“少”大于“年”;看末尾,一个末尾是“年”的数减“少”得到个位是3的数,可以是11-8、12-9,即“少”、“年”分别是(8,1)或者(9,2),这两种情况算出”好”的答案都是6。

【2012·少文杯·计数·第11题】数一数,下图中有个三角形.

【解析】不规则的图形可用标数法,再根据标号的个数分类枚举。

只有一个标号:①、③、⑤、⑥、⑦;

有两个标号:①④、③⑥、⑤⑥;

有三个标号:①②③、③⑥⑦;

有七个标号:①②③④⑤⑥⑦。

一共有11个三角形。

【2012·少文杯·推理策略·第12题】甲、乙、丙、丁与小文5人一起参加围棋赛,每两个人都要比赛1盘,比到现在为止.甲赛了4盘,乙赛了3盘,丙赛了2盘,丁赛了1盘,那么小文赛了盘。

解析】根据题意,将进行了比赛的两人进行连线得到如下点线图。通过这个点线图可知,小文赛了2盘。

【2012·少文杯·应用题·第13题】商店有大、中、小三种型号的玩具共18个,已知大型玩具的个数是小型玩具个数的8倍,那么,中型玩具有个。

【解析】根据题意,可以进行从小到大的枚举。如果小型玩具个数是1个,那么大型玩具是8个,中型玩具有18-1-8=9个;如果小型玩具是2个,那么大型玩具是2×8=16个那么中型玩具是18-2-16=0个。所以中型玩具是9个。

【2012·少文杯·应用题·第14题】听说大牛年龄是小牛年龄的4倍,再过10年大牛的年龄就比小牛年龄的2倍多4岁。那么,你能算出大牛和小牛现在各多少岁吗?

【解析】本题是一个与年龄有关的变倍问题。根据题意可设小牛今年是x岁,那么有

由条件“再过10大牛的年龄比小牛的年龄2倍多4岁”可得方程:

4x+10=2×(x+10)+4

4x+10=2x+20+4

2x=14

x=7

所以,小牛今年是7岁,大牛今年是7×4=28岁。

【2012·少文杯·应用题·第15题】度假村里新修了一个环形游泳池,长225米内环边上每隔5米就有一把太阳伞,每把伞之间都会有相同数量的沙滩椅;而长576米的外环边上每隔8米才有一把太阳伞,每把太阳伞之间都有3把沙滩椅;已知游泳池一共摆有306把沙滩椅,那么,内环边上每两把伞之间有几把沙滩椅?

【解析】本题是一个环形间隔问题(间隔数=节点数),内环的间隔数量是225÷5=45个;外环的间隔数量是576÷8=72个。外环总共有沙滩椅3×72=216把,所以内环有沙滩椅306-216=90把,所以内环每两把伞之间有90÷45=2把沙滩椅。

【2012·少文杯·应用题·第16题】28只地鼠准备一天内在一块地面挖很多坑,已知13只大地鼠一天里比15只小地鼠要多挖4个坑,一只大地鼠比一只小地鼠一天多挖6个坑,那么,请推算出地鼠们这一天里一共挖了多少个坑呢?

【解析】根据题意,如果大地鼠和小地鼠数量都是13只,那么大地鼠总共比小地鼠多挖13×6=78个,但是题目中13只大地鼠只比15只小地鼠多挖了4个,所以,还有78-4=74个坑是用2只小地鼠挖的,一只小地鼠挖坑74÷2=37个,一只大地鼠挖坑37+6=43个。则这一天总共挖了13×43+15×37=1114个。

【2012·少文杯·应用题·第17题】魔术师的花园里种有一棵果子树,11月1日的白天,魔术师发现树上只有100粒果子,他非常不满意,于是夜晚时他把一瓶神奇的药水浇灌给了果子树,药水可以使树上第二天白天新增加一粒果子,并且以后每天新结的果子都比前一天多1颗。11月某一日的夜晚,魔术师清点数上果实,发现一共已经结了271粒果子,那么,请你推算出,这个时候是11月多少号呢?

【解析】这是一道与等差数列有关的题目。根据题意11月1日100粒果子、11月2日变成

100+1=101粒,2日增加1粒;11月3日变成100+1+2=103粒,3日增加2粒;11月4日变成100+1+2+3=106粒,4日增加3粒……如此往复,1+2+3+……+n=271-100=171,(n+1)日增加n粒。经计算,当n=18时,1+2+3+4+……+n=171,所以推算出这个时间是11月19日。

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