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1990年,墨西哥举行了第一届数学奥林匹克。竞赛分两天进行,每天4.5小时,共有62名中学生参加。这里提供的试题译自俄罗斯《Квант》杂志92年第1期。 1.证明:如果两个不可约分数的和是整数,那么这两个分数的分母相同。证设两个不可约的分数为q/p和n/m,则 q/p+n/m=(qm+pn)/pm是整数。即pm|(qm+pn)。 (1) 由p|(qm+pn)知,p|qm,而(p,q)=1,∴p|m。 (2) 同理,由m|(qm+pn)推得m|p。
In 1990, Mexico held its first Mathematical Olympiad. The competition took place in two days, 4.5 hours a day. A total of 62 middle school students participated. The test questions provided here have been translated from the first issue of the Russian magazine “Квант” in 1992. 1. Proof: If the sum of the two irreducible scores is an integer, the denominators of the two scores are the same. Assuming that the two irreducible scores are q/p and n/m, then q/p+n/m=(qm+pn)/pm is an integer. That is, pm|(qm+pn). (1) From p|(qm+pn), p|qm, and (p,q)=1, ∴p|m. (2) Similarly, m|(p) is derived from m|(qm+pn).