Project PEN

20S. Sequences of Consecutive Integers

June 16, 2009 by danielkohen

A37 If $n$ is a natural number, prove that the number

$(n + 1)(n + 2)\cdots(n + 10)$

is not a perfect square.

A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

O51 Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

Here is the official solution file: PEN A9 A37 O51

May 4, 2009 by cpohoata

A37 If $n$ is a natural number, prove that the number

$(n + 1)(n + 2)\cdots(n + 10)$

is not a perfect square.

A9 Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

O51 Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

April 25, 2009 by cpohoata

E16 Prove that for any prime $p$ in the interval $\left]n, \frac {4n}{3}\right]$ , $p$ divides

$\sum^{n}_{j = 0}{{n}\choose{j}}^{4}.$

Here is the official solution file (written by Darij Grinberg, Germany): pen19.pdf.