The DSE (click here to see what HKDSE is) is close. The math core exam is less than 2 weeks time. I am going to share a really nice equation to find the tangent of a given point on the circle. Suppose that the general form of a circle is given by then the equation of a tangent line at point will be as follows: Proof: Differentiate the equation of the circle with respect to x implicitly. hence y’ is, then the slope of the tangent at point would be By point slope form of a line, Now, since the given point satisfies the circle, we plug into the circle, Hence, we have the following ..read more

Solution Let u = sin x + cos x and v = sin x – cos x, Done ..read more

I was browsing the net couple days ago and saw this integral. At first it looks quite difficult. I have tried numerous different substitutions and they don’t work. And later I tried using complex number and it seems to work. So, here is my solution. Solution: By Euler’s formula, cosine function is the real part of a complex number. Hence, Re-arrange the exponents we’ll get Now, since We can use integration by parts, By using the double angle identity of sine and cosine ..read more

The actual mass of a packet of cheese is [215 g, 225 g), hence the total mass of 250 packets is within the interval [53750 g, 56250 g) or [53.75 kg, 56.25 kg). 53.6 kg correct to the nearest 0.1 kg lies outside the interval of [53.75 kg, 56.25 kg). Therefore the claim is false. (a) and take the intersection of these two intervals, (b) -4.5 < x < 3 implies negative integer x could be -4, -3, … , -1, a total of 4 negative integers. Let F be the number of female passengers and M be the number of male passengers. The first sentence implies Now, 24 female passengers leave ..read more

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This is problem 23 of AMC 10B in 2011. So, here is the problem: I will present two solution here. The first one uses Chinese Remainder Theorem, which is slow. And the second solution I will use the trick of using Binomial Theorem. SLOW method: By Chinese Remainder Theorem, we have We have . We need to find Since , then So, To find , Next, we will find , Since , then So, Next, we are going to find Therefore, Therefore, the hundreds digit is 6. FAST Method: By Binomial Theorem, Therefore the hundreds digit is 6 ..read more

Saw this limit when I was watching a Korean drama, Melancholia. The drama is about a romance love (and math) story between a math prodigy and his math teacher. In the drama, the school principal and one of the math teachers had been helping out the daughter of a politician cheating in the math tests and exams so she can always be the top of the class. In one of the exam, the main actress (one of the math teachers) has modified one of the questions due to the error or typo in one question. She didn’t notify her colleagues she modified one of the questions and of course her colleague later gave ..read more

Solution In general, let By Pythagoras Identity, Solve this system by adding and subtracting, we get Hence, and Therefore, Let , then ..read more

So here is the definite integral: And here is the solution: The trick here is to use Euler’s formula and the rest is kinda straight forward. In general, for any natural number n, we have the following, If , then So, the integral vanishes unless or . If n = 2k, then Since the integral vanishes when n is odd, we can rewrite it as the following, For example, when n = 1011, then ..read more

The way of solving this problem is to recognize that this is a math olympiad type problem and it probably has an elegant solution. This equation is nice. There is a symmetry in the coefficients and this is a nice hint for us to crack this problem. My idea is coming from squaring the number 111. But 1, 2, 3, 2, 1 is not exactly 1, 7, 14, 7, 1. However I know I am getting there. If I play around with the numbers a bit, it is not difficult to get the following. So, here it is. The hard part is solved! The rest is trivial ..read more