We solve problems

Problem #PRU-109685

Problemas Álgebra Desigualdades algebraicas y sistemas de desigualdades Desigualdades lineales y sistemas de desigualdades Métodos Demostración por contradicción Cálculo Números reales Números racionales e irracionales

14–16 4.0

At all rational points of the real line, integers are arranged. Prove that there is a segment such that the sum of the numbers at its ends does not exceed twice the number on its middle.

Problem #PRU-109704

Problemas Álgebra Desigualdades algebraicas y sistemas de desigualdades Desigualdades algebraicas (otras) Métodos Métodos algebraicos Contando de dos maneras Cálculo Números reales Partes enteras y fraccionarias. Propiedad arquimediana

13–15 4.0

Prove that for any positive integer \(n\) the inequality

is true.

Problem #PRU-109707

Problemas Cálculo Funciones de una variable. Continuidad Ciertas propiedades de una función y relaciones de recurrencia.

14–16 4.0

Find all the functions \(f\colon \mathbb {R} \rightarrow \mathbb {R}\) which satisfy the inequality \(f (x + y) + f (y + z) + f (z + x) \geq 3f (x + 2y + 3z)\) for all \(x, y, z\).

Problem #PRU-109715

Problemas Álgebra Secuencias Progresiones Progresión geométrica Cálculo Números reales Partes enteras y fraccionarias. Propiedad arquimediana Sumas de secuencias numéricas y series de diferencias

13–15 4.0

Find the sum \(1/3 + 2/3 + 2^2/3 + 2^3/3 + \dots + 2^{1000}/3\).

Problem #PRU-109748

Problemas Geometría Geometría plana Figuras convexas y no convexas Polígonos convexos Matemáticas discretas Combinatoria Problemas de pintura Métodos Principio del casillero Principio del casillero (número finito de puntos, líneas, etc.) Polígonos Polígonos (otros)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-109754

Problemas Geometría Geometría sólida Geometría visual en el espacio

11–13 2.0

Prove that for all \(x \in (0;\pi /2)\) for \(n > m\), where \(n, m\) are natural, we have the inequality \(2 | \sin^n x-\cos^n x | \leq 3 | \sin^m x-\cos^m x |\);

Problem #PRU-109766

Problemas Problemas divertidos Problemas de palabras Tablas y torneos Tablas numéricas y sus propiedades Métodos Principio del casillero Principio del casillero (otro)

12–14 4.0

Is it possible to arrange natural numbers from 1 to \(2002^2\) in the cells of a \(2002\times2002\) table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?

Problem #PRU-109780

Problemas Cálculo Números reales Matemáticas discretas Teoría de conjuntos y lógica Teoría de conjuntos

14–16 4.0

A number set \(M\) contains \(2003\) distinct positive numbers, such that for any three distinct elements \(a, b, c\) in \(M\), the number \(a^2 + bc\) is rational. Prove that we can choose a natural number \(n\) such that for any \(a\) in \(M\) the number \(a\sqrt{n}\) is rational.

Problem #PRU-109787

Problemas Álgebra Polinomios Identidades algebraicas para polinomios Factorización de polinomios Cálculo Números reales Números racionales e irracionales

14–16 4.0

A numeric set \(M\) containing 2003 distinct numbers is such that for every two distinct elements \(a, b\) in \(M\), the number \(a^2+ b\sqrt 2\) is rational. Prove that for any \(a\) in \(M\) the number \(q\sqrt 2\) is rational.

Problem #PRU-109808

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

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

Problems