2004年波士顿首次奥林匹克数学竞赛-小学五年级，你会做吗？？(ZT)

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First Greater Boston Math Olympiad, May 23 rd , 2004 Grade 5 Problems
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First Greater Boston Math Olympiad 5 th Grade
Your name: _________________________________________________________
Try to solve as many problems as you can, in any order you choose. There are six
problems, and a correct solution of each of them wins you the number of points shown in
parentheses. Show your work. If your answer is wrong but your method is correct, you
will get partial credit. If necessary, use back sides of pages or attach additional sheets,
putting your name on them.
Good luck!
***PLEASE DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO***

1. (6) Solve: INK + INK + INK + INK + INK + INK = PEN
(INK and PEN are 3-digit numbers, and different letters stand for different digits).
Answer:

2. Money in Wonderland comes in $5 and $7 bills.
(a) (4) What is the smallest amount of money you need to have in order to buy
a slice of pizza which costs $1 and get back your change in full?
(The pizza man has plenty of $5 and $7 bills.) For example, having $7 won't do, since
the pizza man can only give you $5 back.
Answer:
Explanation:
(b) (8) Vending machines in Wonderland accept only exact payments
(do not give back change). List all positive integer numbers which CANNOT be used
as prices in such vending machines. (That is, find the sums of money that cannot be paid
by exact change.)
Answer:
Explanation:


3. Two people play a game. They put 3 piles of matches on the table:
the first one contains 1 match, the second one 3 matches, and the third one 4 matches.
Then they take turns making moves. In a move, a player may take any nonzero number
of matches FROM ONE PILE. The player who takes the last match from the table
loses the game.
(a) (5) The player who makes the first move can win the game.
What is the winning first move?
Answer:
(b) (6) How can he win? (Describe his strategy.)
Answer:


4. (a) (4) How many times in a half-day ( = 12 hours) the hour and the minute hand
of a clock form the right angle with each other?
Answer:
Explanation:
(b) (8) How many times in a half-day the seconds hand of a clock falls on the line
bisecting the angle between the hour and the minute hands?
Answer:
Explanation:


5. (a) (4) Put 5 points on the plane so that each 3 of them are vertices of an isosceles triangle (i.e., a triangle
with two equal sides), and no three points lie on the same line.
(b) (8) Do the same with 6 points.
Answers:

6. (7) The number A2...2B has 2004 digits (all digits standing between A and B are 2).
This number is divisible by 72. Find the digits A and B. (Hint: use the fact that 72 = 8 × 9.
Find B first and then A.)
Answer:
Explanation:

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中文：

1，（6分）求解：INK + INK + INK + INK + INK + INK = PEN
（INK和PEN是三位数，不同的字母代表不同的数字）。
答案：

2，好奇岛上的纸币分5元和7元两种。
（a）（4分）要买一片1元钱的比萨饼，你最少需要付多少钱，才能全部拿回要找回的零钱？
（卖皮萨的人有足够的5元和7元钞票）比如，只有7元钱不行，因为你只能拿回5元钱。
答案：
解释：
（b）（8分）好奇岛上的自动售货机只接受准确数目的钱（不找零钱）。请列出所有不能在售货机上用作标价的正整数。
（也就是，找出哪些数目的钱不能被找零。）


3，两个人玩火柴游戏。在桌上放三堆火柴：第一堆1根，第二堆3根，第三堆4根。然后轮流移动火柴，
每个人每次只能从一堆火柴中拿任意数目的火柴（不能不拿），最后拿的人输。
（a）（5分）第一个拿火柴的人会赢。他第一次需要哪几根呢？
答案：
（b）（6分）他怎么拿才能赢（请说明策略方法）？
答案：

4，（a）（4分）半天（12个小时）当中，钟表的时针和分针有多少次互相形成直角？
答案：
解释：
（b）（8分）半天当中，钟表的秒针有多少次落在时针和分针形成的夹角的等分线上？
答案：
解释：

5，（a）（4分）把5个点放在平面上，使每3个点成为一个等腰三角形的顶点， 并且不能有三个点位于同一条直线上。
（b）（8分）用6个点同样做一下。
答案：

6，（7分）A2...2B是一个2004位的整数（A和B之间的数字均为2）。这个数可被72整除。请找出数字A和B。
（提示：利用72=8x9， 先找出B再找出A。）
答案：
解释：