Posted in PEN Solutions Archive on June 16, 2009 | Leave a Comment »
A37 If is a natural number, prove that the number
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 consecutive integers it is always possible to find one which is relatively prime to all the [...]
Posted in PEN Solutions Archive on April 25, 2009 | Leave a Comment »
E16 Prove that for any prime in the interval , divides
Here is the official solution file (written by Darij Grinberg, Germany): pen19.pdf.
Posted in PEN Solutions Archive on April 8, 2009 | Leave a Comment »
A23 Prove that if
is expressed as a fraction, where is a prime, then divides the numerator.
A24 Let be a prime number and . Prove that
is divisible by .
Here is the official solution file: pen18.pdf
