MathDB

3

Part of 2002 AIME Problems

Problems(2)

Dick and Jane

Source:

11/28/2005
Jane is 25 years old. Dick is older than Jane. In nn years, where nn is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let dd be Dick's present age. How many ordered pairs of positive integers (d,n)(d,n) are possible?
modular arithmetic
Geometric Series & Logarithms

Source:

12/28/2006
It is given that log⁡6a+log⁡6b+log⁡6c=6,\log_{6}a+\log_{6}b+\log_{6}c=6, where a,a, b,b, and cc are positive integers that form an increasing geometric sequence and b−ab-a is the square of an integer. Find a+b+c.a+b+c.
logarithmsratiogeometric sequence