2014第五届两岸四地华杯总决赛小学高年级(含解析)
第五届两岸四地华罗庚金杯少年数学精英邀请赛
笔试一试卷（小学高年级组）
（2014年7月23日，60分钟）
一、填空题Ⅰ（每题8分，共32分）
1．算式()()
()2
135791012141618
56789
?????????
????
的计算结果是．
2．今年是2014年，2014不是完全平方数，但可以将它的各位数字改变顺序，使得到的新四位数是完全平方数．例如2
102432
=．已知用数字2、0、1、4各一个还能组成另一个四位完全平方数，那么这个新的四位完全平方数是．
3．在不同的历史时期，“斤”和“两”之间的进制不同．成语“半斤八两”就是由16进制而来的．为方便计算，我们认为古代16两是1斤，每斤为现代的600克；现代的10两是1斤，每斤为现代的500克．现有一批药品，有一部分按古制称，另一部分按现制称，统计发现，“斤”数和是5，“两”数和是68．那么，这批药品共有克．
4．两个完全相同的正三角形可以拼成一个菱形，如果正三角形的边长为10，则这个菱形内部最大的正方形面积为．
二、填空题Ⅱ（每题10分，共40分）
5．
2
2
1
3
3
6
5
6
7
6
6
A=+
+
+
+
+
，与100
A?最接近的整数是．
6．如图所示，由75个小方格组成了155
?的图案，图中一些小方格已经被涂上了阴影．现在要继续把一些空白的小方格涂上阴影，保证任意22
?的方格中阴影小方格的数量都多于一半，那么最少需要再把个空白小方格涂上阴影．
7．自然数A 除本身以外最大的约数是d ，自然数2A +除本身以外最大的约数是2d +，那么A 的值是 ．
8．如图，三横、三竖、三斜共9条街道，编号为1~9的9个路口，A ，B ，C ，D ，E 五位警察在其中5个不同的路口站岗．如果两个警察在同一街道上，那么他们就能互相看到，他们各自说了如下的一句话： A ：
“我能看到另4位警察．” B ：
“我能看到另4位警察中的3位．” C ：“我能看到另4位警察中的2位．” C ：“我只能看到B ．”
E ：
“我谁也看不到．” 已知他们恰有1人说谎，且说谎的警察所在路口的编号是五位警察中最小的，那么，A ，
B ，
C ，
D ，
E 所在路口编号依次组成的五位数是 ．
9
8
7
6
5432
1
三、填空题Ⅲ（每题12分，共48分）
9．四个小伙伴想办“亲情套餐”，即缴纳一定金钱后，四人之间发短信免费，但是尴尬的是，当地只有10元的“三人间免费”的A 套餐和5元的“两人间免费”的B 套餐．四人想了一下，觉得可以开几个这种套餐，使得小伙伴们可以通过由中间人转发器到相互之间都可以免费发短信．那么在花钱最少的情况下，他们有 种开通套餐的方式．
10．如图，正八边形中连出3条对角线围成一个三角形（图中阴影部分），如果该正八边形的边长为60，那么阴影部分的面积是 ．
11．A B两地相距291千米，甲、乙两人同时从A地出发匀速前往B地，与此同时丙从B地出发匀速前往A地．当乙走了p千米后与丙相遇时，甲走了q千米；又过了一段时间，当甲、丙相遇时，乙共走了r千米．如果p、q、r均是质数，那么，p、q、r的和是．
12．第12题作答要求：
（1）请在答题卡第12题的万位及千位，填涂上你认为本试卷中一道最佳试题的题号，答题范围为01~11．
（2）请在答题卡第12题的百位，填涂上你认为本试卷整体的难度级别，最简单为“1”，最难为“9”，总计九个级别，答题范围为1~9．
（3）请在答题卡第12题的十位及个位，填涂上你认为本试卷中一道最难试题的题号，答题范围为01~11．
所有答题范围内的作答均可得分，所有的评定都将视为本人对本试卷的有效评定，不作答或者超出作答范围不得分．
第五届两岸四地华罗庚金杯少年数学精英邀请赛
笔试一试卷（小学高年级组）
参考答案
部分解析
一、填空题Ⅰ（每题8分，共32分） 1．算式
()()()
2
13579101214161856789?????????????的计算结果是________．
【考点】速算巧算 【难度】☆☆☆ 【答案】2 【解析】原式135792526272829
5678956789??????????????=
?????????
52372229256789
25678956789
?????????????=
=?????????．
2．今年是2014年，2014不是完全平方数，但可以将它的各位数字改变顺序，使得到的新四位数是完全平方数．例如2102432=．已知用数字2、0、1、4各一个还能组成另一个四位完全平方数，那么这个新的四位完全平方数是________． 【考点】数论 【难度】☆☆☆ 【答案】2401 【解析】2492401=．
3．在不同的历史时期，“斤”和“两”之间的进制不同．成语“半斤八两”就是由16进制而来的．为方便计算，我们认为古代16两是1斤，每斤为现代的600克；现代的10两是1斤，每斤为现代的500克．现有一批药品，有一部分按古制称，另一部分按现制称，统计发现，“斤”数和是5，“两”数和是68．那么，这批药品共有________克． 【考点】应用题 【难度】☆☆☆ 【答案】2800
【解析】假设全是按照古制称量，5斤是：16580
?=（两），
实际“两”数和是68，所以按现制称量的“斤”数是：(8068)(1610)2
-÷-=（斤），所以，这批药品共有：600350022800
?+?=（克）．
4．两个完全相同的正三角形可以拼成一个菱形，如果正三角形的边长为10，则这个菱形内部最大的正方形面积为________．
【考点】几何
【难度】☆☆☆
【答案】50
【解析】当正方形如图摆放时，面积最大．面积最大为：1010250
?÷=．
二、填空题Ⅱ（每题10分，共40分）
5．
2
2
2
2
1
3
3
6
5
6
7
6
6
A=+
+
+
+
+
，与100
A?最接近的整数是________．
【考点】速算巧算【难度】☆☆☆【答案】314
【解析】设原式
2
2
2
2
2
1
3
3
6
5
6
7
6
9
6
x
=+
+
+
+
+
，
2
11
66
6
x=+>
+
，
22
1111
6627
66
x=+<+<
+
，
当6x =时，
原式2
22
222
22
22
2
1113=3+=3+33366+6+975556+6+6+
332776+6+
96+
26
=++21=3+=3.1429966+989， 当27x =时， 原式22
1133 3.14082927366843
2
5669
=+
=+=+
+
+
+
，
所以与算式计算结果100倍最接近的整数是314．
6．如图所示，由75个小方格组成了155?的图案，图中一些小方格已经被涂上了阴影．现在要继续把一些空白的小方格涂上阴影，保证任意22?的方格中阴影小方格的数量都多于一半，那么最少需要再把________个空白小方格涂上阴影．
【考点】操作与策略 【难度】☆☆☆ 【答案】17
【解析】不存在右图四种情况的相连空白格（即任意两个空白格不存在公共顶点）将空白格
分成如图不相连的7块．
其中D 、G 两块为51?，每块至少需要涂其中2个格子；同理，C 、F 两块，每块至少需要涂其中1个格子．
而对于A 块，可分为1块51?和1块21?，所以至少需要涂其中213+=个格子．B 块与A 块完全相同．
而对于E 块，可分为1块51?和1块22?，22?中不能有任何2块空白，所以22?中至少需要涂上三个格子，这样E 块至少需要涂其中235+=个格子．
综上所述，A 、B 、C 、D 、E 、F 、G 这7块依次分别至少需要涂上3、3、1、2、5、1、2个格子，那么一共至少涂上331251217++++++=(个)格子． 右图，给出了一种构造（实际上只有E 、F 这2块各有2种选择，其它阴影格子都是惟一选择）．
7．自然数A 除本身以外最大的约数是d ，自然数2A +除本身以外最大的约数是2d +，那么A 的值是________． 【考点】数论 【难度】☆☆☆ 【答案】7
【解析】如果A 是偶数，2A d =
，2
22
A d ++=，矛盾．所以A 是奇数，A 和2A +互质，d 和2d +互质．若A 不是质数，设A md =，m 是A 最小的质因数，则：22d md ++ ∴2222d md m m +++-，∴222d m +-，
∴21d m +-与m 是A 最小的质因数矛盾，∴A 是质数，∴1d =，23d +=， ∵2d +是2A +除本身以外的最大约数， ∴29A +=，∴7A =．
8．如图，三横、三竖、三斜共9条街道，编号为1~9的9个路口，A ，B ，C ，D ，E 五位警察在其中5个不同的路口站岗．如果两个警察在同一街道上，那么他们就能互相看到，他们各自说了如下的一句话： A ：
“我能看到另4位警察．” B ：
“我能看到另4位警察中的3位．” C ：“我能看到另4位警察中的2位．”
C ：“我只能看到B ．”
E ：
“我谁也看不到．” 已知他们恰有1人说谎，且说谎的警察所在路口的编号是五位警察中最小的，那么，A ，
B ，
C ，
D ，
E 所在路口编号依次组成的五位数是________．
9
8
7
6
5432
1
【考点】数论 【难度】☆☆☆☆ 【答案】25496
【解析】A 说的话与D 、E 矛盾，所以A 说的是错的，B 、C 、D 、E 说的是对的．
如果有人站在1号位置，他可以看到2、4、8、3、6号，共5个位置，所以E 不能站在1号位置，否则2、4、8、3、6号位置均不能站人，则位置不够． 同理，3号能看到6个位置，4号能看到6个位置，5号能看到5个位置，7号能看到6个位置，8号能看到6个位置，这些位置E 都不能站．所以E 只能站在2、6、9．
如果E 站在2号位置，1、4、5、7不能站人，剩下的3、6、8、9这4个位置中没人能同时看到另外3个位置，与B 可以看到3个人矛盾；
如果E 站在9号位置，3、5、7、8不能站人，剩下的1、2、4、9这4个位置中能同时看到另外3个位置的只有1，那么B 在1，而1是最小编号，A 所在位置的编号一定比1大，这与题意中“说慌的警察所在路口的编号是五位警察中最小的”矛盾；
所以E 在6号位置，1、3、7、8不能站人，剩下的2、4、5、9中能同时看到另外3个位置的只有5，那么B 在5；说慌的A 所在路口的编号是五位警察中最小的，所以A 在2；4能看到在2的A 与在5的B ，据“D 只能看到B ”，所以D 不在4，
D 在9；剩下的C 在4．
所以，答案为25496．
三、填空题Ⅲ（每题12分，共48分）
9．四个小伙伴想办“亲情套餐”，即缴纳一定金钱后，四人之间发短信免费，但是尴尬的是，当地只有10元的“三人间免费”的A 套餐和5元的“两人间免费”的B 套餐．四人想了一下，觉得可以开几个这种套餐，使得小伙伴们可以通过由中间人转发器到相互之间都可以免费发短信．那么在花钱最少的情况下，他们有________种开通套餐的方式． 【考点】计数 【难度】☆☆☆ 【答案】28
【解析】4人如果想相互之间免费发短信，最少花15元．可以办1个三人套餐1个两人套餐，或3个两人套餐．
第一种情况，办1个三人套餐和1个两人套餐，从4个人选出3个人办套餐共有4
种方法，3人中选出1人跟没办3人套餐的人办两人套餐，所以，共有4312
?=种方法；
第二种情况，办3个两人套餐，A和B、B和C、C和D，共有4321212
???÷=种方法；
第三种情况，办3个两人套餐，A和B、C、D办，共有4种方法．
综上所述，共有1212428
++=种方法．
10．如图，正八边形中连出3条对角线围成一个三角形（图中阴影部分），如果该正八边形的边长为60，那么阴影部分的面积是________．
【考点】几何
【难度】☆☆☆☆
【答案】900
【解析】以AB为底，做OAB
△的高OH．据对称性，H是中点，
∴OH是BEF
OH=．
△的中位线∴30
根据正八边形的对称性，ABCD是菱形∴60
==
AB BC
∴ABC
△的面积为60302900
?÷=．
11．A B两地相距291千米，甲、乙两人同时从A地出发匀速前往B地，与此同时丙从B地出发匀速前往A地．当乙走了p千米后与丙相遇时，甲走了q千米；又过了一段时间，当甲、丙相遇时，乙共走了r千米．如果p、q、r均是质数，那么，p、q、r的和是________．
【考点】行程问题
【难度】☆☆☆
【答案】221
【解析】当乙走了p 千米时，甲和丙共走了291p q -+千米；当乙走了r 千米时，甲和丙共
走了291千米；∴
291291
p r
p q =-+化简，得291(291)p r p q =-+
∵p r <，p 、r 均是质数， ∴(,)1p r =，∴291r ，3r =或97．
3r =时，据质数p r <，∴2p =，代入291(291)p r p q =-+得194=2912+q -，矛
盾！
∴97r =代入291(291)p r p q =-+，得4291p q =+
∵4293p >，73p >，又据97p r <=且p 的尾数不能是9（否则q 尾数是5）， ∴83p =，代入后可得，41q =．
综上述，p ，q ，r 的和是834197221++=．
12．第12题作答要求：
（1）请在答题卡第12题的万位及千位，填涂上你认为本试卷中一道最佳试题的题号，答题范围为01~11．
（2）请在答题卡第12题的百位，填涂上你认为本试卷整体的难度级别，最简单为“1”，最难为“9”，总计九个级别，答题范围为1~9．
（3）请在答题卡第12题的十位及个位，填涂上你认为本试卷中一道最难试题的题号，答题范围为01~11．
所有答题范围内的作答均可得分，所有的评定都将视为本人对本试卷的有效评定，不作答或者超出作答范围不得分． 【考点】送分题 【难度】 【答案】随便填 【解析】送分题
2017年英语专四真题及答案-(1)
2017年英语专业四级真题及答案 III. 语言知识： 11. ___B_____ combination of techniques authors use, all stories---from the briefest anecdotes to the longest novels ----have a plot. A. Regarding B. Whatever. C. In so far as D. No matter 12. She followed the receptionist down a luxurious corridor to a closed door, ____B______ the women gave a quick knock before opening it.. A. wherein B. on which C. but when D. then 13. Ms Ennab is one of the first Palestinian ______C____ with seven years’ racing experience. A. woman drivers B. women driver C. women drivers D. woman driver 14. “I wondered if I could have a word with you.”The past tense in the sentence refers to a __B___. A. past event for exact time reference B. present event for tentativeness C. present event for uncertainty D. past event for politeness 15. “If I were you, I wouldn’t wait to propose to her.” The subjunctive mood in the sentence is used to ____D______. A. alleviate hostility B. express unfavorable feelings C. indicate uncertainty D. make a suggestion 16. “It’s a shame that the city official should have gone back on his word.” The modal auxiliary SHOULD express __B_____. A obligation B disappointment C future in the past D. tentativeness 17. Timothy Ray Brown, the first man cured of HIV, initially opted against the stem cell transplantation that _____D______ history. A. could have later made B. should have made later C. might make later D. would later make 18. Some Martian rock structures look strikingly like structures on Earth that are known ___C___by microbes. A. having been created B. being created C. to have been created D. to be created 19. At that moment, with the crowd watching me, I was not afraid in the ordinary sense, as ______ if I ____A_______alone. A. would have been …had been B. should be … had been C. could be … were D. might have been… were 20. You must fire __C____ incompetent assistant of yours A. the B. an C. that D. whichever 21. Some narratives seem more like plays, heavy with dialogue by which writers allow their
2014年湖北省高考英语完形填空试题及其详细解析
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