Solving Problems Using Vieta’s Formula
Problem: Solution: By Vieta s formula, Consider the following, hence, Now, consider the following two identities, and hence the following, Now,
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A Handy Way to Find the Equation of a Tangent Line to a Circle
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 [ ]
Integrating 1/(sin(x)+sec(x)) from 0 to pi/2
Solution Let u = sin x + cos x and v = sin x cos x, Done.
MIT Integration Bee 2023 Semi-final Problem 1
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 [ ]
DSE Math 2023 Paper 1 Answer
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. [ ]
Happy Pie Day
Happy Pi Day! March 14th (3/14) is a day to celebrate the mathematical constant, pi (π), which represents the ratio of the circumference of a circle to its diameter. Pi is an irrational number, meaning it cannot be expressed as a simple fraction, and its decimal representation goes on forever without repeating. Pi Day has [ ]
Find the Hundreds Digit of 2011^2011
This is problem 23 of AMC 10B in 2011. So, here is the problem: I will present three solution here. The first one uses Chinese Remainder Theorem, which is the slowest, takes about 5 to 10 minutes. The second solution I will just use Euler Phi function, which is quicker, about 3 minutes. And the [ ]
Finding a Limit Without L’hôpital Rule or Taylor Expansion
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 [ ]
A Little AIME Problem
Solution In general, let By Pythagoras Identity, Solve this system by adding and subtracting, we get Hence, and Therefore, Let , then
Integrating the 2022nd power of cosine x from 0 to 2Pi
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 [ ]