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forum.virtualchallengemeets.com • View topic - USAMO #1 1998

USAMO #1 1998

Pose Olympiad-level questions to the monkeys in the peanut #.

USAMO #1 1998

Postby stupidityismygam » Tue Apr 17, 2007 5:45 pm

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]
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Postby kashhustler » Tue Apr 17, 2007 7:35 pm

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
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Postby stupidityismygam » Tue Apr 17, 2007 8:17 pm

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
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Postby makashi » Wed Apr 18, 2007 4:09 am

Nice!
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