Difference between revisions of "2019 IMO Problems/Problem 4"
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Hence, (1,1), (3,2) satisfy | Hence, (1,1), (3,2) satisfy | ||
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| + | When I was born, I was {\small small}. Actually, {\scriptsize I was | ||
| + | very small}. When I got older, I thought some day {\Large I would be | ||
| + | large}, {\Huge maybe even gigantic}. But instead, I'm not even | ||
| + | normalsize. {\small I'm still small.} | ||
Revision as of 14:05, 15 December 2019
Problem
Find all pairs 
of positive integers such that
![\[k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).\]](http://latex.artofproblemsolving.com/7/c/a/7ca521b92d40def7f320702893c01502439174e8.png)
Solution 1
! = 1(when = 1), 2 (when = 2), 6(when = 3)
(when 
= 1), 6 (when = 2)
Hence, (1,1), (3,2) satisfy
When I was born, I was {\small small}. Actually, {\scriptsize I was very small}. When I got older, I thought some day {\Large I would be large}, {\Huge maybe even gigantic}. But instead, I'm not even normalsize. {\small I'm still small.}
{\Large aa large}{\large For = 2: large} RHS is strictly increasing, and will never satisfy = 2 for integer n since RHS = 6 when = 2.
For > 3, > 2:
LHS: Minimum two odd terms other than 1.
RHS: 1st term odd. No other term will be odd. By parity, LHS not equal to RHS.
Hence, (1,1), (3,2) are the only two pairs that satisfy.
~flamewavelight and phoenixfire
