Roses are red, but life is not blue…

Image for post
Image for post

So this is my first, not math-related topic but I am pretty sure that it is not the last. How can I be so sure? Because I changed my direction and adjusted my life a little bit. As a result, this story will tell you how you can do it as well. Now, let me start with my case. I sharply changed my mind a week ago and am glad that it happened before the new year because the year wishes give me the motivation to keep going with this change. …


A very entertaining way to solve problems which seems hard otherwise…

Image for post
Image for post

A Simple Search…

Let’s have a look at some easygoing questions which you can solve by trying every possible case. But we should find more efficient solutions to those questions. You know what I mean… Rather than counting by fingers, we are going to find a solution which would give the direct answer without any counting. First, let me give an example of that kind of questions.

Suppose you are flipping a coin. You write down the result in every single flipping as heads or tails. …


Such an impressive way of handling hard problems in geometry!

Image for post
Image for post

Definition

Given two circles ω1 and ω2 with distinct centres, the radical axis of the circles is the set of points P such that,

Powω1 (P) = Powω2 (P).

This might seem rather confusing definition as P seems to be a phantom point. However, in other words, we can define the radical axis as follows.

Let ω(1) and ω(2) be circles with distinct centres O(1) and O(2).
The radical axis of ω1 and ω2 is a straight line perpendicular to O(1)O(2).


Temmuz ayının sorusu ile karşınızdayız.

Image for post
Image for post

Bu ay, geçen aydan farklı olarak bir sonlu matematik sorusuyla değil, bir sayılar teorisi sorusuyla karşınızdayız.

Sorumuz oldukça kısa:

a ve b tamsayılar olmak üzere

(a² + b² + 3)/ab ifadesinin tamsayı olmasını sağlayan tüm (a,b) tamsayı ikilileri nelerdir?

Sorunun cevabını 1 hafta sonra yayınlanacak olan cevap videosunda ve Medium sayfamızda yayınlayacağımız makalede bulabilirsiniz!


Bu makalede Temmuz ayının sorusunun çözümünü inceleyeceğiz.

Image for post
Image for post

A² + b² + 3, ab ye bölünecek şekilde kaç (a,b) tamsayı ikilisi vardır?

Şimdi önce negatiflik, pozitiflik durumuna bakalım. Eğer bir (a,b) ikilisi şartı sağlarsa, (-a,b), (a,-b) ve

(-a,-b) de verilen şartı sağlar. Bundan dolayı sadece pozitif (a,b) ikililerini bulsak yeter. (a,b) bir çözüm ise (b,a) da bir çözüm olduğundan genelliği bozmadan a>=b kabul edelim.

Öncelikle b=1 durumunu inceleyelim.

a1 I. a² + 1² + 3 = a² + 4

Buradan (2,1), (1,1), ve (4,1) ikilileri gelir.

S kümesi şöyle tanımlansın:

S = { (a,b) : (a² + b² + 3) / ab bir tamsayı ve a>b, b>1. }


Two outstanding solutions in a mindblowing competition: International Mathematical Olympiad

Image for post
Image for post

IMO (International Mathematical Olympiad) is a competition to which a lot of high school students from different countries attend. The students who attend to IMO are qualified to be a member of their country’s national math team. The Olympiad takes 2 days and each day students are given four and a half hours to solve 3 problems selected by the Problem Selection Committee. The participating countries except the host country are expected to propose up to six questions, with solutions.

From Regulations IMO:

The aims of the IMO are:

 to discover, encourage and challenge mathematically gifted young people in…


Let’s learn about the concept of directed angles which enables us to write much shorter solutions.

Image for post
Image for post

Let’s denote such angles with . Since this is not a standard notation, if you would like to use it somewhere, do not forget to mention this in your first sentences.

Here is how it works:

First, we consider ABC to be positive if the vertices A, B, C appear in clockwise order, and negative otherwise. In particular, ABC ≠CBA; they are negatives.

Then, we are taking the angles modulo 180◦. For example,

−150◦ = 30◦ = 210◦

(From “Euclidian Geometry In Mathematical Olympiads” by Evan Chen)

For example,


Image for post
Image for post

Haziran ayı itibariyle başlattığımız “Ayın Sorusu” sorularının ilki ile karşınızdayız. Ayın sorusunun cevabını ise her ayın 8’inde sayfamızda bulabilirsiniz!

Hemen soruya geçelim.

15 voleybol takımından oluşan bir eleme grubunda her takım diğer takımlardan her biriyle tam olarak 1 kez karşılaşıyor. Oyunların hiçbiri beraberlikle sonuçlanmıyor. Toplam yenilgi sayısı N’yi aşmayan bütün takımlar bir sonraki tura geçiyor. En az 7 takımın tur atlamasını olanaklı kılan N sayılarından en küçüğünü kaçtır?

Anlaşılması için bir örnek verelim. Diyelim ki N sayısı 10, o zaman yenilgi sayısı 10'u aşmayan yani yenilgi sayısı 0 ile 10 arasında değişen tüm takımlar bir sonraki tura geçebiliyor. Bu şekilde bir sonraki tura geçebilen takım sayısının 7'den büyük eşit olmasını istiyoruz. N sayısını en fazla ne kadar küçük tutabiliriz?

Umarım sizin için uğraşması keyifli bir soru olur!


How does this geometry work? Have you ever thought about that? Well, there are pretty much theorems and axioms that helps Euclidean geometry work. Stewart’s Theorem is one of them. Let’s learn what it is!

Image for post
Image for post

Lets’s jump straight into the theorem and the proof.

Theorem

Let a, b and c be the length of the sides of the triangle ABC. Let x be the length of the cevian to the side of length a, a = BC . If the cevian of If the cevian divides the side of length a, into two segments of length m and n, with m adjacent to c and n adjacent to b, then Stewart’s Theorem states that:

(mb² + nc²) / (m + n) — mn = x²

Proof

In this proof, we will be using the Cosine Theorem to…


If you are trying to understand trigonometry you should definitely read this. This theorem is a commonly used trick for some problems that require pretty genius and creative geometric problems.

Image for post
Image for post

In this article, I will be proving the law of cosines. If you know a little about triangles and angles you can do it yourself! So, before reading the proof, you had better try to prove it.

The Law of Cosines

Let’s start with the theorem. In any type of triangle, the following equation holds.

Ceren Şahin

Co-founder at Betamat | Dart & Python

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store