Page 1 of 1
USAMO #1 1998
Posted:
Tue Apr 17, 2007 5:45 pm
by stupidityismygam
Suppose that the set [unparseable or potentially dangerous latex formula] has been partitioned into disjoint pairs [unparseable or potentially dangerous latex formula] so that for all [unparseable or potentially dangerous latex formula] equals [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]. Prove that the sum
[unparseable or potentially dangerous latex formula]
ends in the digit [unparseable or potentially dangerous latex formula]
Posted:
Tue Apr 17, 2007 7:35 pm
by kashhustler
yayyy we did this problem at camp. in fact, i just worked it earlier today when i was looking through my packets from camp.
i have to go to dinner now but here's a hint, we worked it in the parity and games lecture
Posted:
Tue Apr 17, 2007 8:17 pm
by stupidityismygam
Assume all [unparseable or potentially dangerous latex formula]
so we just have to find the units[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula] which is odd
if the sum is odd, then the difference is odd (prove it yourself)
so we have [unparseable or potentially dangerous latex formula]
where 999-y=x
since the difference is odd, there have to be an odd number of x's, so then an even number of 6's
so let y=2z, so we need to find the units of [unparseable or potentially dangerous latex formula]
take this mod 10, and you get [unparseable or potentially dangerous latex formula] QED
Posted:
Wed Apr 18, 2007 4:09 am
by makashi
Nice!