Difference between revisions of "2019 IMO Problems/Problem 4"

Revision as of 00:55, 16 December 2019

Find all pairs $(k,n)$ of positive integers such that

$k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).$

$LHS$ $k$ ! = 1(when $k$ = 1), 2 (when $k$ = 2), 6(when $k$ = 3)

$RHS = 1$ (when $n$ = 1), 6 (when $n$ = 2)

Hence, (1,1), (3,2) satisfy

For $k$ = 2: RHS is strictly increasing, and will never satisfy $k$ = 2 for integer n since RHS = 6 when $n$ = 2.

It remains to prove that these are the only pairs that satisfy.

~flamewavelight and phoenixfire

@@ Line 14: / Line 14: @@
 For <math>k</math> = 2: RHS is strictly increasing, and will never satisfy <math>k</math> = 2 for integer n since RHS = 6 when <math>n</math> = 2.
-For <math>k</math> > 3, <math>n</math> > 2:
+It remains to prove that these are the only pairs that satisfy.
-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