This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads.
“31. Prove that if n is a natural number, then we have
(√2 – 1)n = √m – √(m – 1),
where m is a natural number.”
Here, natural numbers are 1, 2, 3, …
I found it to be quite challenging, as all the Polish Math Olympiad problems seem to be.
See the Amazing Identity
Here is surprising problem from the 1875 The Analyst
“81. By G. W. Hill, Nyack Turnpike, N. Y. — Prove that, identically,
By “identically” the proposer means for all n = 1, 2, 3, ….
See the Surprising Identity
(Update 8/20/2021) James Propp at his website has an informative, extensive article on mathematical induction and its variations.
Virtually the very first “math” problem I got interested in involved a 7th grade homework problem in 2005 that a colleague at work said her son had been given. I ended up commenting and helping on a number of further problems, which gave me some insight into the state of current public school teaching in mathematics. It was both encouraging and discouraging at the same time. I will join the math education commentary at a later date.
The problem was not that bad: What is the largest power of 2 that divides 800! without a remainder? (where “!” means “factorial”, for example, 5! = 5 x 4 x 3 x 2 x 1). I solved it in my usual pedestrian way. I showed it to a friend of mine (an algebraist!) and he of course had a nifty approach. He showed it to a colleague of his at NSF (a physicist) and he had the niftiest solution of all! (Most humbling.) See the Power of 2 Problem.