2017年AIME II 真题:
Problem 1
Find the number of subsets of that are subsets of neither
nor
.
Problem 2
Teams ,
,
, and
are in the playoffs. In the semifinal matches,
plays
, and
plays
. The winners of those two matches will play each other in the final match to determine the champion. When
plays
, the probability that
wins is
, and the outcomes of all the matches are independent. The probability that
will be the champion is
, where
and
are relatively prime positive integers. Find
.
Problem 3
A triangle has vertices ,
, and
. The probability that a randomly chosen point inside the triangle is closer to vertex
than to either vertex
or vertex
can be written as
, where
and
are relatively prime positive integers. Find
.
Problem 4
Find the number of positive integers less than or equal to whose base-three representation contains no digit equal to
.
Problem 5
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are ,
,
,
,
, and
. Find the greatest possible value of
.
Problem 6
Find the sum of all positive integers such that
is an integer.
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