We solve problems

Problem #PRU-58089

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines 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-58093

Problemas Métodos Principio del palomar Principio del palomar (ángulos y longitudes)

13–14 2.0

A plane contains \(n\) straight lines, of which no two are parallel. Prove that some of the angles will be smaller than \(180^\circ/n\).

Problem #PRU-58094

Problemas Geometría Geometría plana Círculos Ángulo central. Longitud de arco y circunferencia Conexión entre la longitud del arco, el ángulo y el acorde Métodos Principio del casillero Principio del casillero (ángulos y longitudes)

13–15 4.0

Several chords are drawn through a unit circle. Prove that if each diameter intersects with no more than \(k\) chords, then the total length of all the chords is less than \(\pi k\).

Problem #PRU-58097

Problemas Geometría Geometría plana Círculos Ángulo central. Longitud del arco y circunferencia Transformaciones planas Proyección sobre una línea Proyección ortogonal Métodos Principio del casillero Principio del casillero (ángulos y longitudes)

13–16 3.0

Several circles, whose total length of circumferences is 10, are placed inside a square of side 1. Prove that there will always be some straight line that crosses at least four of the circles.

Problem #PRU-58220

Problems Dissections

13–14 2.0

Cut an arbitrary triangle into 3 parts and out of these pieces construct a rectangle.

Problem #PRU-58275

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

13–14 2.0

Fill an ordinary chessboard \(8\times 8\) with the tiles shown in the figure.

Problem #PRU-60299

Problems Number Theory Divisibility Divisibility of a number. General properties Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

From the set of numbers 1 to \(2n\), \(n + 1\) numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

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

Problems Number Theory Numeral systems Decimal number system

11–13 2.0

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

Problem #PRU-60337

Problems Discrete Mathematics Combinatorics

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?