Our olympic course professor gave us some exercises, and among these there is one which resolution is particularly instructive. Find the sum of the fifth powers of the roots of: To solve it, we need to write the recursion which is associated to the polynomial, which is: This recursion will give us the sum of […]
Category: Maths
Old Problem With Generating Functions
A similar problem to this one is the one that initiated me to the art of using the generating functions, I’ll tell you a version which has less calculations to make. If you roll a complete set of DnD dice, in how many ways can you obtain 13 as sum of their results? Just for […]
A Very Notorious Problem, But In 3D
I’ve already made an article on the 2D version of this problem, if you want to check it out before this one, you can visit https://www.mattiagiuri.com/2020/04/21/a-very-notorious-problem/. Now, the 3D version says: if you have 3 reals x, y, z chosen randomly between 0 and 1, what’s the probability that It’s important to notice that any […]
Problem For A Particular Occasion
I proposed this problem in a birthday team competition I wrote, but at the time I asked for its equivalent in 18 dimension. Today I’ll explain it in 3 dimensions, and I want to point out that 3 is the perfect number for today’s occurrence. Now, how many parts of space (finite or infinite) can […]
Using Symmetry
I put this problem in an old team competition I organized. It is not simple, and I asked only for the case where n is even, now I’ll show you the complete version. Let be reals in an interval of lenght 1. Find the maximum value of Now, let’s treat as constants and let be […]
A Problem From Three Days Ago’s Simulation
This problem took me a while to solve because of its calculations, but it’s actually very instructive. It makes you see how few conditions can determine a lot. All of the data is in the figure above, we have to find the area of the rectangle. Now let where alpha is the biggest of the […]
Hearts Equations In Math
In this article, I’ll show you five different equations you can use to plot your own heart. Let’s start with the simplest: I think this is the most classic of the five I’m going to show you. If you prefer one which is a little bit thinner, there is this one: On the other hand, […]
Algebra Problem From Yesterday’s Simulation
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 […]