The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n
+ x^n = (x+1)^n so what about other solutions for x an integer and
n= 2, 3, 4 or 5?
Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2
Find all 3 digit numbers such that by adding the first digit, the
square of the second and the cube of the third you get the original
number, for example 1 + 3^2 + 5^3 = 135.
This is another solution by Yatir from Maccabim-Reut High
Another challenge is to use the hints given by the two
illustrations in the question and to give alternative proofs that
the sum of a positive number and its reciprocal is greater than or
equal to 2.
Yatir uses this inequality when he sums k fractions and their
reciprocals in the following proof. Can you use a similar method to
give a shorter proof of the result without resorting to
mathematical induction? You will need to expand the expression
given in (1), collect pairs of terms, decide how many pairs there
are and use the inequality for each of the pairs of terms.