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Problem #PRU-60359

Problemas Teoría de Números Sistemas Numéricos Sistema de Números Decimales Métodos Métodos Algebraicos Particiones en pares y grupos; biyecciones Principio de la Casilla Principio de la Casilla (otro)

14–16 4.0

We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.

Problem #PRU-60362

Problemas Métodos Métodos algebraicos Particiones en pares y grupos; biyecciones Principio del casillero Principio del casillero (otro)

13–15 4.0

You are given 1002 different integers that are no greater than 2000. Prove that it is always possible to choose three of the given numbers so that the sum of two of them is equal to the third.

Will this still always be possible if we are given 1001 integers rather than 1002?

Problem #PRU-60366

Problemas Teoría de Números Sistemas numéricos Sistema de números decimales Métodos Métodos algebraicos Particiones en pares y grupos; biyeciones Principio del casillero Principio del casillero (otro)

13–14 4.0

Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide (are the same) an infinite number of times.

Problem #PRU-60489

Problemas Teoría de números Divisibilidad Ecuaciones en números enteros Métodos Principio extremal Principio extremal (otro) Métodos de geometría Interpretaciones geométricas en álgebra Principio del casillero Principio del casillero (otro) El máximo común divisor (MCD) y el mínimo común múltiplo (MCM). Números mutuamente primos

13–15 4.0

Let \(a\), \(b\), \(c\) be integers; where \(a\) and \(b\) are not equal to zero.

Prove that the equation \(ax + by = c\) has integer solutions if and only if \(c\) is divisible by \(d = \mathrm{GCD} (a, b)\).

Problem #PRU-60667

Problemas Probabilidad y estadística Teoría de la probabilidad Distribución continua Teoría de números Sistemas numéricos Sistema decimal

14–16 4.0

Prove that in a three-digit number, that is divisible by 37, you can always rearrange the numbers so that the new number will also be divisible by 37.

Problem #PRU-60850

Problemas Métodos Ejemplos y contraejemplos. Pruebas constructivas Principio del casillero Principio del casillero (otro) Matemáticas discretas Teoría de algoritmos Teoría de algoritmos (otro)

13–16 4.0

Prove that there are infinitely many composite numbers among the numbers \(\lfloor 2^k \sqrt{2}\rfloor\) (\(k = 0, 1, \dots\)).

Problem #PRU-60851

Problemas Cálculo Números reales Números racionales e irracionales Álgebra Trigonometría Trigonometría (otros)

13–15 4.0

Prove the irrationality of the following numbers:

a) \(\sqrt{3}{17}\)

b) \(\sqrt{2} + \sqrt{3}\)

c) \(\sqrt{2} + \sqrt{3} + \sqrt{5}\)

d) \(\sqrt{3}{3} - \sqrt{2}\)

e) \(\cos 10^{\circ}\)

f) \(\tan 10^{\circ}\)

g) \(\sin 1^{\circ}\)

h) \(\log_{2}3\)

Problem #PRU-60854

Problemas Cálculo Números reales Números racionales e irracionales

13–16 4.0

Is it possible for

a) the sum of two rational numbers irrational?

b) the sum of two irrational numbers rational?

c) an irrational number with an irrational degree to be rational?

Problem #PRU-60874

Problemas Cálculo Secuencias numéricas Número e Matemáticas discretas Combinatoria Problemas de coloreado

14–16 4.0

\(N\) points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the \(k\) colours. Prove that if \(N > \lfloor k!e\rfloor\), then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.

Problem #PRU-61021

Problemas Cálculo Derivada Derivada y raíces múltiples

15–16 4.0

Prove that the following polynomial does not have any identical roots: \(P(x) = 1 + x + x^2/2! + \dots + x^n/n!\)

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