Posts Tagged ‘math’

This was the challenge I’ve set out before myself, which lead me to creating MathematicaAnki tools (picture below). “My final suggestion is to pick some sort of organisational system and make a real effort to stick to it; a half-hearted system is probably worse than no system at all.” Terence Tao There’s an enormous amount of […]

I was trying to get a matrix for a projection of any point to a line given by the following equotations: After some time of research I’ve ended with the following result: Let then Of course the result isn’t new at all, it was just a big efford for me to do that at midnight […]

Matrixes seem to have a lot of wonderful properties, for example they may help checking if given triangles represented each by three complex numbers (verticles) are simillar. So, two triangles represented by complex numbers and are similar if, and only if: Proof coming soon.

The polynomial argument is a very useful fact which can be used in order to prove, or justify a statement. It bases on simple fact that non-zero polynomial of k degree, can have no more than k solutions, futhermore the difference of two polynomials of k degree can also have max of k solutions if only […]

Today, I decided to post a very eye-catching method for calculating an integral which shows very often in Statistical Mechanics in Physics. Let’s try to calc this First let’s mark it with a name , then What is just a double integral over the whole 2d surface. It can be also written in polar coordinate […]

The angle which arms contain chords or tangents of a circle is the half of the sum (if the verticle of the angle lies inside the circle or on it) or difference (if the verticle lies outside) of middle angles based on the curves, which set this angle. The proof can be illustrated by following […]

Prove that for every natural number there exists different natural numbers such as sum of any two is divisable by their diffrence. Proof: Because number is divisable by , we can write and of course . Adding those two together: Induction: 1. Sequence 1,2 works 2. Assume that sequence works, then the number is divisable […]

A few days ago I were stuck with one exercise which I can’t pushed off :] Meanwhile when I forgotten about it, friend of mine, Ecik, did it. It’s below with his solution. Prove the following: if Solution: I’m open on any suggestions or other solutions of yours. If you’d like to see Ecik’s blog, […]

The solutions can be found here My solution for task number 9 is here: Solution for number 9

Lets calculate more compilated sum further, the last one equals to and now the first of the sums is equal to however the second one is easy, cause its a geometric sequence equal to under the law from previous post. Now we have , in the result of elementary algebraic transformation we recieve It’s easy […]

Asume that we want to calculate Of course, But the sum on the right side is equal to Therefore and so,

Experimental problem 2 Experimental problem 3 Theoretical problem 1 Theoretical problem 2 Theoretical problem 3 Problems from Poland 59 MO 2007/2008 I level/II set. Problem 5 Problem 6 Problem 7  Problem 8

Since my first met with imaginary unit have past about 3 years. I’ve been thinking about it many times. First step forward in understanding was made when I acquainted with Schrödinger equation, it gave me some kind of experience with complex numbers. Meantime I’ve heard a lot of voices like “What is it for if […]

The 8 of November had past, so it’s time to publish my solutions for VIX Polish Mathematical Olympics I level/I set. All available at my Google Documents, written in Office 2007 .docx brilliant format, but converted into awful .doc and from there into .pdf. Hope, that Google will support .docx someday. Solutions are written in […]

Today used all my fuel resources for the next week. But lucky in Katowice I’ve learned something extremely beautiful. Truth is the following identity: Pretty cool, huh? It’s easy to get from the right side to the left side, but the inverse way doesn’t look that easy. But it is J. All you got to […]