Maths Olympiad Exam Logical Reasoning General Mental Ability

Show that there is no integer a such that a^2 - 3a - 19 is divisible by 289.

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Maths Olympiad Other Question

In a book with page numbers from 1 to 100, some pages are torn off. The sum of
the numbers on the remaining pages is 4949. How many pages are torn off?
Show that 32008 + 42009 can be wriiten as product of two positive integers each of
which is larger than 2009182.