5

Part of 2008 India Regional Mathematical Olympiad

Problems(2)

Find all H.P.s with conditions

Source: (Indian) RMO 2008 Problem 5

Three nonzero real numbers

a,b,c

are said to be in harmonic progression if \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}. Find all three term harmonic progressions

a,b,c

of strictly increasing positive integers in which a \equal{} 20 and

b

c

. [17 points out of 100 for the 6 problems]

number theory unsolvednumber theory

How many good integers are there?

Source: Excursion

N

be a ten digit positive integer divisible by

7

. Suppose the first and the last digit of

N

are interchanged and the resulting number (not necessarily ten digit) is also divisible by

7

then we say that

N

is a good integer. How many ten digit good integers are there?

number theoryDivisibilitypositive integer