Problem 1 Let a square lattice of dimensions \(n \times n\) be given.Calculate the number \(R\) of different rectangles which can be drawn, with the vertices in the lattice points.Two rectangles are considered different if they have different sizes or Read more
Problem 1 Calculate the maximum number of parts in which a plane can be divided by \(n\) lines. Solution 1.1A necessary condition is that the \(n\) lines must intersect two by two and no three lines intersect at the same point. Read more
1) Algebra of matrices A matrix \(A(m,n)\) defined on the field of real numbers \(\mathbb{R}\) is a collection of real numbers \((a_{ij})\), indexed by natural numbers \(i, j\), with \(1\le i\le m\) and \( 1\le j\le n\). We can represent Read more
Exercise 1 A square and a triangle have the same area. Which shape has the greatest perimeter? HintIt may be useful to remember the following formula: Arithmetic Mean-Geometric Mean InequalityLet \(x_{1},x_{2}, \dots x_{n}\) be non-negative real numbers; then: \[ \dfrac Read more
Problem Let \((a_{1},a_{2}, \cdots ,a_{n})\) be a permutation of the set \(\{1,2, \cdots,n \}\). Compute the average value, indicated with \(M_{n}\) , of the following sum: \[ (a_{1}- a_{2})^2 + (a_{2}-a_{3})^2 + \cdots + (a_{n-1} – a_{n})^2 \] taken on Read more
Euclidean geometry studies geometric objects such as lines, triangles, rectangles, circles, etc. Fractals are also geometric objects; however, they have specific properties that distinguish them and cannot be classified as objects of classical geometry. Although Mandelbrot (1924-2010) is generally considered the father of Read more
Problem There are four people A, B, C, D. A box with a red ball inside it is given to A, who can leave things unchanged with probability \(p\) or replace the red ball with a white one with probability Read more
Video game programming requires extensive use of mathematics and physics. Some typical examples are the following: controlling the motion of objects in space, drawing geometric shapes on the scene, computing the effects after a collision of objects in accordance with Newton’s Read more
1) Gauss’s Modular Arithmetic Given a positive integer \(m \), we say that two integers \( a\) and \(b \) are congruent modulo \(m\) if they give the same remainder when divided by \(m \). We use the following notation Read more
The study of integers and their properties is the fundamental object of Number Theory. The properties of prime numbers and divisors of an integer were first studied extensively during the period of ancient Greece (Pythagoras, Euclid, etc.); the study resumed Read more