剑已出鞘,谁与争锋

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博客“数学第一剑”近来小有名气。我去CIBC谈贷款,吃惊地发现那个贷款专家竟然知道我的博客,说我在华人社区也算是个“名人”了。上周,又有一位开地产公司的李先生近日开起了补习学校;他访遍大多地区的这个校那个院,想找人合作;直到有一天,朋友提醒他:搞数学辅导,不看“数学第一剑”怎么行?看后过于感慨,便找到真人搞起了合作。

李先生说,在多伦多,在奥数及大中小学数学教育界,敢称数学第一剑的,非我莫属。“数学第一剑”是谁?中科院数学研究所数学博士,硕士导师,数学建模竞赛导师,北京市高等数学竞赛导师,加拿大Carleton University高级访问学者,数学数论专家。2003年起,在多伦多地区开办各种数学班,涵盖大学,中小学,及奥数竞赛各个方面,一周7,每天至少授课6小时,10多年如一日,对加拿大的中小学数学教育,可谓了如指掌。结合高深的数学造诣和在加拿大丰富的教学经验,创建了一套从小学五年级到大学年级(涵盖IMO)的系统数学教材和针对加拿大本地学生的创新的教学方法. 经不断使用完善, 效果显著10多年来,有很多教授的学生进入美加名校,在各种数学竞赛上不断拿奖。

难得有人赏识本人的才华,我便答应了他的合作建议。我们的第一个目标是:把有才有志的高中生(911年级)送进数学奥林匹克国家队。只要做做后面的几道测试题,写清你的思路,我便能看出你是不是英雄好汉;再加上你愿意学习的决心,我保证让你成为数学高手,受益终生。现在,我的剑已经出鞘,你敢接招吗?!

  1. Solve the equation x2 + x2/(x + 1)2 = 8.

  2. Let A be the sum of the digits of 44444444, and B is the sum of the digits of A. Find the sum of the digits of B.

  3. Find the last digit of [(sqrt(3) + sqrt(2))^2015], where [x] is the integer part of x.

  4. Prove that logx is not a rational function of x.

  5. Solve (2cos8x – 1)(2cos4x – 1)(2cos2x – 1)(2cosx – 1) = 1.

  6. Determine all functions f : R → R for which f(2f(x) + f(y)) = 2x + y for every x and y.

  7. Sketch the graph of x3 + 3xy + y3 = 1.

  8. The solutions of the equation z6 + 6iz5 – 15z4 – 20iz3 + 15z2 + 6iz = 33 are the vertices of a convex polygon in the complex plane. What is the area of the polygon?

  9. If three points are chosen randomly on the circumferences of a given circle, calculate the probability that the triangle determined by the three points is acute.

  10. The medians of a tetrahedron are the line segments joining a vertex to the centroid of its opposite face. Show that the four medians interest at a single point, and divides each other in the ratio 3:1.

  11. Let f(n, k) be the number of ways of distributing k candies to n children so that each child receives at most 3 candies. Determine the value of f(2016, 1) + f(2016, 4) + f(2016, 7) +…+ f(2016, 1006) + f(2016, 1009).

  12. Two people take turns to withdraw apples from three baskets which contain 3, 4, and 5 apples respectively. On each turn, one must take at least 1 apple from the same basket. The person who gets the last apple wins the game. Who has a certain strategy to win; the one who start first or the other?

解答可以发送到 : [email protected] [email protected].

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