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Singapore Math

Sunday, April 18, 2010

Math for Girls ONLY


Since it started participating in the International Mathematical Olympiad (IMO) in 1985, when it informally sent only two contestants, the China team of six contestants has been Number One fourteen times. An outstanding achievement, considering that the most populated nation is relatively new to this high-level mathematical competition among high-school students, as compared to other countries like Russia and other ex-communist countries, with their rich culture of decades-old Olympiad and competitive mathematics.


Note that China, which hosted the 31st IMO in 1990, didn’t send a team to the IMO when the event was held in Taiwan in 1998. The first IMO, which was held in Romania in 1959, has since been held annually, except in 1980.

Since 1986, the China team has never had a female student. To encourage more female mathletes, the China Mathematical Olympiad Committee launched the China Girls’ Mathematical Olympiad in 2002. The top two winners will be admitted directly into the national training team of about 20 to 30 students, from which six students will be finally selected to form the China IMO team.

In 2007, the first girl who was winner of China Girls’ Mathematical Olympiad was selected to enter the 2008 China national team and won a gold medal at the 49th IMO, in Hanoi, Vietnam.

A Sample of Questions

Let’s look at a sample of these high-school contests problems, which were posed in recent China Girls’ Mathematical Olympiads.

Eight persons join a party.
(1) If there exist three persons who know each other in any group of five, prove that we can find that four persons know each other.
(2) If there exist three persons in a group of six who know each other in a cyclical manner, can we find four persons who know each other in a cyclical manner? [2006, #4]

A positive integer m is called good, if there is a positive integer n such that m is the quotient of n over the number of positive integer divisors of n (including 1 and n itself). Prove that 1, 2,… ⋯ , 17 are good numbers and that 18 is not a good number. [2007, #1]

Find all positive integers n such that 20n + 2 can divide 2003n + 2002. [2002, #1]

Let ABC be an obtuse triangle inscribed in a circle of radius 1. Prove that triangle ABC can be covered by an isosceles right-angled triangle with hypotenuse √2 + 1. [2004, #3]

Find all pairs of positive integers (x, y) satisfying xy = yx–y. [2002, #6]

Given that a × b rectangle with a > b > 0, determine the minimum length of a square that covers the rectangle. (A square covers the rectangle if each point in the rectangle lies inside the square.)

Let x and y be positive real numbers with x3 + y3 = xy. Prove that x2 + 4y2 < 1. [2005, #5]

Let a, b, c be integers each with absolute value less than or equal to 10.
The cubic polynomial
f(x) = x3 + ax2 + bx + c
satisfies the property
f(2 + √3)< 0.0001.
Determine if 2 + √3 is a root of f. [2007, #7]

If China had taken the lead to host a Girls’ Mathematical Olympiad to support its female mathletes, it wouldn’t too late for other countries to follow suit to encourage more girls to take part in math contests and competitions. This could only have a positive effect on the enrollment of both female undergrads and postgraduate students, who would be motivated to take up more advanced math courses in university.

A Girls’ IMO

Due to some countries’ high pools of talented and gifted mathletes, perhaps the IMO committee would re-look at its invitation guidelines, by inviting more than one team from countries like China, Russia and the United States, to give more opportunities for high-school students to be a medalist at the IMO.

An Individual Category

There could be an open individual category so that potential medalists-mathletes who don’t make it to their national team (due to the limit number of six contestants per country) could still compete on their own. And, to encourage more girls to compete, there could also be an IMO for female mathletes, in the spirit of the China Girls’ Mathematical Olympiad.

References

Xiong, B. & Lee, P. Y. (eds.) (2009). Mathematical olympiad in China (2007-2009): Problems and solutions. East China Normal University Press & World Scientific.

Xiong, B. & Lee, P. Y.  (eds.) (2007). Mathematical olympiad in China: Problems and solutions. East China Normal University Press & World Scientific.

© Yan Kow Cheong, April 18 2010.