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Problem #PRU-58080

Problemas Matemáticas Discretas Combinatoria Geometría en papel cuadriculado Problemas de pintura Métodos Principio del casillero Principio del casillero (número finito de puntos, líneas, etc.)

12–13 3.0

A ream of squared paper is shaded in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

Problem #PRU-58081

Problemas Geometría Geometría plana Triángulos Tipos de triángulos Triángulo equilátero

12–14 3.0

There are 5 points inside an equilateral triangle with side of length 1. Prove that the distance between some two of them is less than 0.5.

Problem #PRU-58082

Problemas Geometría Geometría plana Desigualdades geométricas Desigualdades en un triángulo Un segmento dentro del triángulo es menor que el lado más largo, Problemas Geometría Geometría plana Triángulos Tipos de triángulos Triángulos rectángulos Teorema de Pitágoras y su recíproco

12–14 4.0

A \(3\times 4\) rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than \(\sqrt5\).

Problem #PRU-58084

Problemas Geometría Geometría plana Círculos Círculo, sector, segmento, etc. Métodos Principio del casillero Principio del casillero (ángulos y longitudes) Principio del casillero (número finito de puntos, líneas, etc.)

12–13 4.0

There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.

Problem #PRU-58089

Problemas Geometría Geometría plana Figuras convexas y no convexas Polígonos convexos Métodos Principio del palomar Principio del palomar (número finito de puntos, líneas, etc.)

12–15 4.0

What is the minimum number of points necessary to mark inside a convex \(n\)-gon, so that at least one marked point always lies inside any triangle whose vertices are the vertices of the polygon?

Problem #PRU-58313

Problemas Álgebra Ecuaciones algebraicas y sistemas de ecuaciones Ecuaciones de orden superior. Ecuaciones polinómicas palindrómicas Ecuaciones de orden superior (otros) Cálculo Números reales Partes entera y fraccionaria. Propiedad arquimediana

12–14 2.0

Ten circles are marked on the circle. How many non-closed non-self-intersecting nine-point broken lines exist with vertices at these points?

Problem #PRU-60335

Problemas Matemáticas Discretas Teoría de Grafos

11–13 2.0

a) In Wonderland, there are three cities \(A\), \(B\) and \(C\). 6 roads lead from city \(A\) to city \(B\), and 4 roads lead from city \(B\) to city \(C\). How many ways can you travel from \(A\) to \(C\)?

b) In Wonderland, another city \(D\) was built as well as several new roads – two from \(A\) to \(D\) and two from \(D\) to \(C\). In how many ways can you now get from city \(A\) to city \(C\)?

Problem #PRU-60336

Problemas Teoría de Números Sistemas Numéricos Sistema Decimal

11–13 2.0

How many distinct seven-digit numbers exist? It is assumed that the first digit cannot be zero.

Problem #PRU-60337

Problemas Matemáticas Discretas Combinatoria

11–13 2.0

A car registration number consists of three letters of the Russian alphabet (that is, 30 letters are used) and three digits: first we have a letter, then three digits followed by two more letters. How many different car registration numbers are there?

Problem #PRU-60345

Problemas Geometría Geometría sólida Geometría visual en el espacio

11–13 2.0

A passenger left his things in an automatic storage room, and when he came to get his things, it turned out that he had forgotten the code. He only remembers that in the code there were the numbers 23 and 37. To open the room, you need to correctly type a five-digit number. What is the least number of codes you need to sort through in order to open the room for sure?

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