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问题 解答题

在正2004边形A1A2…A2004各顶点上随意填上1，2，…501中的一个数．试证明：一定存在四个顶点满足如下条件：

（1）这四个顶点构成的四边形为矩形；

（2）此四边形相对两顶点所填数之和相等．

答案

证明：（1）由题意知，顶点A1与Ai=1002为一组关于中心对称的点，其中i=1，2，…1002．

则2004个顶点可分为1002组，

顺次连接每两组的顶点，均可得到一个四边形，

由于对角线互相平分且相等，

所以，得到的四边形是矩形．

（2）由题意，设在顶点A1上所填的数为a1，则

2≤a1+ai=1002≤501×2，

即2到1002共有1001个不同的数，

又1002组有1002个数，由抽屉原则知，至少有两组顶点所填数之和相等，

则此两组顶点即为所求的四个顶点．

单项选择题

I will never forget the year I was about twelve years old. My mother told us that we would not be (16) Christmas gifts because there was not enough money. I felt sad and thought, "What would I say when the other kids asked what I’ d (17) " Just when I started to (18) that there would not be a Christmas that year, three women (19) at our house with gifts for all of us. For me they brought a doll. I felt such a sense of (20) that I would no longer have to be embarrassed when I returned to school. I wasn’t (21) . Somebody had thought (22) of me to bring me a gift.

Years later, when I stood in the kitchen of my new house, thinking how I wanted to make my (23) Christmas there special and memorable, I (24) remembered the women’s visit. I decided that I wanted to create that same feeling of (25) for as many children as I could possibly reach.

So I (26) a plan and gathered forty people from my company to help. We gathered about 125 orphans (孤儿) at the Christmas party. For every child, we wrapped colorful packages filled with

toys, clothes, and school supplies, (27) with a child’ s name. We wanted all of them to know they were (28) Before I called out their names and handed them their gifts, I (29) them that they couldn’t open their presents (30) every child had come forward. Finally the (31) they had been waiting for came as I called out, "One, two, three. Open your presents!" As the children opened their packages, their faces beamed and their bright smiles (32) up the room. The (33) in the room was obvious, and (34) wasn’ t just about toys. It was a feeling, and the feeling I knew (35) that Christmas so long ago when the women came to visit. I wasn’t forgotten. Somebody thought of me. I matter.

29().

A. reminded

B. guaranteed

C. convinced

D. promised
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