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1981年5月,上海市数学会主办了一九八一年上海市中学生数学竞赛。这次竞赛以本届高中毕业班学生为对象,也吸收了少数高中一年级学生参加。竞赛分两轮进行,第一轮以区县为单位,由市竞赛命题组统一命题,各区县分别同时举行,试题考查范围规定不超出现行中学数学教学大纲(修订草案)的要求。参加竞赛的全市共有4537个同学,根据竞赛成绩,由各区县选拔出375个同学参加第二轮市一级的竞赛。根据事前宣布的竞赛办法,第二轮竞赛的考试范围,可以不受大纲的约束。所以这次试题与往年比较,难度稍有提高。竞赛结果,有四人获得一等奖,六人获得二等奖,三十一人获得三等奖。很可喜的是获得一等奖的这四位同学,总分都在93分以上,而且其中有二人还获得了满分,另外还有两个高中一年级的在校学生,也进入了竞赛优胜者的行列。为了提供参考,我们特约请参加这次竞赛命题和阅卷工作的部分同志,对这次竞赛的试题,给出参考解答,并作扼要的分析。为了节省篇幅,在下面给出某些试题的解答中,我们删去了一些中间步骤,在正式作出解答时,宜于把一些必要的中间步骤,加以叙述。
In May 1981, the Shanghai Mathematical Society hosted the 1981 Shanghai High School Mathematics Competition. The competition was aimed at students in this year’s high school graduating class, and also absorbed a small number of first-year senior high school students. The competition is divided into two rounds. The first round is divided into districts and counties. The city’s competition proposition group has unified propositions. The districts and counties are held at the same time. The examination scope of the questions does not exceed the requirements of the current middle school mathematics syllabus (revision draft). There were 4537 students in the city participating in the competition. According to the results of the competition, 375 students from all districts and counties selected and participated in the second round of city level competitions. According to the previously announced competition method, the scope of the second round of competition examination may not be bound by the outline. Therefore, the difficulty of this test is slightly higher than that of previous years. As a result of the competition, four people won the first prize, six won the second prize, and thirty-one won the third prize. Very gratifying is that the four students who won the first prize have a total score of more than 93 points, and two of them have also obtained full scores. In addition, there are two students who are in the first year of high school and have also entered the competition. The ranks of the people. In order to provide reference, we specially invited some of the comrades who participated in the competition’s proposition and marking work to give answers to the questions of the competition and make a brief analysis. In order to save space, in the answers to some of the questions given below, we have omitted some intermediate steps. In formulating answers, it is appropriate to describe some necessary intermediate steps.