Given two polynomials and , let a, b, c, d be the roots of , find First, notice that We know that plugging a, b, c, or d into gives 0, so we have to find the sum of squares of the roots minus the sum of the roots plus 4. By Vieta’s Theorem, the […]
Geometry Problem From Today’s Simulation
Today there was a national maths team competition simulation on phiquadro.it and, although I was really on fire, I was really upset I had no time to solve this geometry problem, so I decided to publish it here. In the figure above, the equilateral triangle ABC has circumradius = 10. An isosceles trapezoid is drawn […]
A Strange Recursion
This is a really nice recursion I wanted to share. We have to find the general formula for the nth term of the succession. We have: Also: By subtracting the two expressions and simplifying we get: Now, since we can further simplify to get the final form: By solving the recursion’s equation we can find […]
Plotting Waves In Python
I really like plotting waves in Python: there are many ways you can show interesting patterns both in 2D and 3D. I’m obviously using the numpy and matplotlib libraries to create and plot the functions. 2D Wave Plot I really wanted to try to reproduce Arctic Monkeys’ AM cover image, and that’s what I was […]
Plotting A Torus With Python
In geometry, a torus is a surface generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. To make it simple, it has the form of a ring or of a donut. How can we plot it with Python? Well, matplotlib’s 3D plotting needs three coordinate arrays, so […]
Simson Lines In USA TST
Given a point P on a triangle’s ABC’s circumcircle, then its projections on the sides of ABC are collinear (Simson Line). Using the notation from the figure above, notice that perpendiculars produce cyclic quadrilaterals: P is on the circumcircle of triangles YZA, ZXB and XYC. So we compute: due to all the cyclic quadrilaterals, and […]
A Python Biology Dictionary
Last year I had to prepare for a biology test, so I decided to combine business with pleasure and write my own biology dictionary on Python: it provided a brief definition and the pages on the book I had to see to find what I needed. Python Implementation from fuzzywuzzy import process import pandas as […]
Double Counting: Part 1
Double counting is a widely used technique of combinatorics, which consists of counting the same quantity with two different methods so that you can create an equation. Today I’m going to show you some of its application. Double Counting In A Polyhedron Does a polyhedron exist with an odd number of faces each having an […]
Old But Gold Polynomials
This article is about a couple of problems about polynomials which I regard as highly instructive. Seems A Demon But It’s Really Not I first saw this problem in 2018, and I initially thought it was a very hard problem, except it wasn’t. But, at first sight, it could make you feel a bit lost. […]
Cauchy-Schwarz And Titu Inequalities
Today I’m going to show two very nice inequalities to know, which really are useful in the mathematical olympiads. What They Say And Their Proof The Cauchy-Schwarz inequality asserts that, for real tuples we have: Here’s the proof: consider the expression It’s obviously true since each term of the sum is non negative. Hence we […]