2009 十月 « Mathematics in HKCEE, HKAL, and Beyond

Archive for 十月, 2009

A nice coordinate geometry problem (suitable for Form 7 pure)

十月 28, 2009

Let $A(a, a^3 + pa + q)$ be a point on the curve $C: y = x^3 + px + q.$ Read the rest of this entry ?

Posted in HKAL, Pure Mathematics, Teaching | 1 Comment »

For my IMO Students (Basic and Advanced)

十月 28, 2009

Here is good problem for my IMO students.

Given a real number $R > 0$ . Find the greatest area of triangle ABC with circum-radius $R.$

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For my Form 7 Pure students: A proof of the Rational Root Theorem

十月 19, 2009

Let $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \in \mathbb{Z}[x].$ Suppose $p/q$ , where $\gcd(p, q) = 1$ is a rational root.

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To my Form 7 students: I am impressed!

十月 15, 2009

Just finished my CB class tonight and I am impressed by two students. They both expressed interests in the following problem:

Suppose $p(x)$ is a polynomial of degree $n$ . Suppose further that

(i) $p(x) - p(x - 1) = x^{100}$ for all $x,$ and

(ii) $p(1) = 1.$

Find the leading coefficient of $p(x)$ and the degree of $p(x).$

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Posted in HKAL, Pure Mathematics, Teaching | 2 意見»

Selected solutions for my F7 Pure (Polynomials)

十月 3, 2009

Book 2, pg 14. (00IQ12)(c)
(i) Let y = x^2, then we have ay^2 – by + a.
Considering the discriminant yields: b^2 – 4a^2 > 0. Therefore, it has two distinct roots: $\alpha, \beta.$ If they are both positive, then we are done. Otherwise, they must be both negative since $\alpha \beta = a/a = 1.$ (Note $a \not = 0$ since $ab > 0$ .)
Suppose $\alpha, \beta < 0.$ Then we have $x = \pm i \sqrt{|\alpha|}, \pm i \sqrt{|\beta|}.$