Monday, October 1, 2012

Cool Math Olympics

    International Math Olympiad problems will be presented along with solutions here in "Cool Math Olympics". Olympiad problems from 1959 (first time Math Olympiad started) to current year will be presented and solved here. First set of Olympiad problems and solutions (one day gap) for 1959 Olympiad is here at "1959 Cool Math Olympics". Olympians, or wannabe Olympians, participate in our "Cool Math Olympics". You performance is much appreciated.

1. Prove that the fraction `(21n+4) / (14n+3)` is irreducible for every natural number `n`. (Hint: What is the definition of GCD? And if `a` and `b` have GCD of `1`, what about `a` and `a - b` or, `b` and `a - b`).
2. For what real values of x is

                `\sqrt{(x + \sqrt{2x - 1})} + \sqrt{(x - \sqrt{2x - 1})} = A`,
given (a) `A =
1` (b) `A = 1/2`, (c) `A = 2`
where only non-negative real numbers are admitted for square roots? (Hint: Too many square roots, looks like it requires raising to power two multiple times, or is it?)
3. Let `a, b, c` be real numbers. Consider the quadratic equation in `cos(x)`:
                `a * cos^{2}(x) + b * cos(x) + c = 0`
Using the numbers `a, b, c` form a quadratic equation in `cos(2x)`, whose roots are the same as those of the original equation. Compare the equations in `cos(x)` and `cos(2x)` for `a = 4, b = 2, c = -1`. (Hint: Use formulas for sum and product of roots, also apply same for `cos(2x)`).
    3a. Bouns question: What happened when you used `a = 4, b = 2, c = -1` and why?
4. Construct a right triangle with given hypotenuse `c` such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. (Hint: Parameterize by `\theta`).
    4a. What is the angle `\theta` got? Is it dependent on `c`?
    4b. What is the geometric mean? What is it same as?
5. An arbitrary point `M` is selected in the interior of the segment `AB`. The squares `AMCD` and `MBEF` are constructed on the same side of `AB`; with the segments `AM` and `MB` as their respective bases. The circles circumscribed about these squares, with centers `P` and `Q`; intersect at `M` and also at another point `N`. Let `N^{1}` denote the point of intersection of the straight lines `AF` and `BC`.
    (a) Prove that the points `N` and `N^{1}` coincide.
    (b) Prove that the straight lines `MN` pass through a fixed point `S` independent of the choice of `M`.
    (c) Find the locus of the midpoints of the segments `PQ` as `M` varies between `A` and `B`.
(Hint: Use Cartesian Co-ordinates. Like many such problems, choose origin and axes to minimize the paraphernalia).
    5d. Bonus problem. There is a way to solve 5a, using analytical geometry.
    5e. Prove that `AF` and `BC` intersect making an angle of `90^{0}`.
    5f. Prove also using analytical geometry, (use `\theta` where `tan(\theta) = (AM)/(MB)` as well) that there is such a fixed point like `S`.
    5g. Argue that such a fixed point, would be on the perpendicular bisector of `AB` (use symmetry arguments, possibly?)
6. Two planes, `P` and `Q` intersect along the line `p`. The point A is given in the plane `P` and the point `C` in the plane `Q`; neither of these points lies on the straight line `p`. Construct an isosceles trapezoid `ABCD` (with `AB` parallel to `CD`) in which a circle can be inscribed, and with vertices `B` and `D` lying in the planes `P` and `Q` respectively. (Don't know what this problem is about. Whats the idea of this problem, anyway? What is "in which a circle can be inscribed", totally vague).

    1959 Cool Math Olympics complete. Boy, these things take time, Phew. I personally prefer analytical geometry approach to problems than Co-ordinate geom approach, that's just dry, what do you people think? Though these problems are way back in 1959, Olympics solving took time for me, partly because I dealt with these math subjects, long long long time ago (15 years). However, these problems rank as "brilliant" for their formulation, where the results are pretty surprising (Especially 4 and 5). Hats-off to those people who had such great geometric insights. (Spoiler: Pretty much everything in geometry has an interesting locus, or passes through a fixed point I think. What do you people think?)

    People who could crack these (with or without hints) can start telling their friends that they are good enough to do it. Start the celebrations, and the geekery of you telling others about these problems. Next time, with 1960 Cool Math Olympics. 

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