LEICESTERSHIRE MATHEMATICS OLYMPIAD
STAGE 2
Saturday 16th February 2008
Problem 1.
Is each statement true or false? Why?
(In this question “powers” means “positive integer powers”, that is, of the form an where n ≥ 1
and n is an integer.)
1) There exist two distinct powers of two whose sum is a power of two.
2) There exists a power of two that equals a power of six.
3) There exist two distinct powers of two that differ by a multiple of 11.
4) The product of any five consecutive integers is divisible by 60.
5) 403 is the sum of two square numbers.
6) There exist three non-zero consecutive integers whose product is a square number.
Problem 2.
Problem 3.
Problem 4.
Problem 5.