2

Part of 2012 JBMO TST - Turkey

Problems(2)

Partition of a set

Source: Turkey JBMO Team Selection Test Problem 2

S=\{1,2,3,\ldots,2012\}.

We want to partition

S

into two disjoint sets such that both sets do not contain two different numbers whose sum is a power of

2.

Find the number of such partitions.

inductionfloor functionlogarithmscombinatorics proposedcombinatorics

Find all numbers satisfying a condition

Source: Turkey JBMO Team Selection Test Problem 6

Find all positive integers

m,n

and prime numbers

p

\frac{5^m+2^np}{5^m-2^np}

is a perfect square.

number theorygreatest common divisormodular arithmeticnumber theory proposedGCD