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文[1]介绍了2011年在莫斯科举行的全俄中学生第38届奥数试题及解答.本文摘译部分试题及解答,供数学教育工作者参考.一、试题1.(8年级)如果在两个数的积中,第一个因数增加1,而第二个因数减少1,那么它们的积增加2011.如果相反,第一个因数减少1,而第二个因数增加1,这时原来的积怎样变化?2.(8年级)若a、b、c>0,且满足等式a~2(a-1)+b~2(b-1)+c~2(c-1)=a(a-1)+b(b-1)+c(c-1).求所有a、b、c的值.3.(8年级)在△ABC的边BC和BA上分别取点D和E,使DE∥AC.∠AED和∠EDC的平分线交AC于点F.证明△ABC的内心
Paper [1] introduced the 2011 All-Russian high school students held in Moscow 38th Mathematical Olympiad test questions and answers.This paper extracts part of the questions and answers for math education workers reference .Test questions 1 (grade 8) In the product of the numbers, the first factor increases by 1 and the second factor decreases by 1, then their product increases by 2011. If, on the contrary, the first factor decreases by 1 and the second factor by 1, then the original (8th grade) If a, b, c> 0 and the equations a ~ 2 (a-1) + b ~ 2 (b-1) + c ~ 2 (c-1) = a (a-1) + b (b-1) + c (c-1). Find all the values of a, b and c 3. (Grade 8) Take points D And E, so that the bisecting line of DE∥AC.∠AED and ∠ EDC AC at point F. Prove △ ABC’s heart