2016年AIME II 真题:
Problem 1
Initially Alex, Betty, and Charlie had a total of peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats
of his peanuts, Betty eats
of her peanuts, and Charlie eats
of his peanuts. Now the three numbers of peanuts each person has forms an arithmetic progression. Find the number of peanuts Alex had initially.
Problem 2
There is a chance of rain on Saturday and a
chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is
, where
and
are relatively prime positive integers. Find
.
Problem 3
Let and
be real numbers satisfying the system
Find the value of
.
Problem 4
An rectangular box is built from
unit cubes. Each unit cube is colored red, green, or yellow. Each of the
layers of size
parallel to the
faces of the box contains exactly
red cubes, exactly
green cubes, and some yellow cubes. Each of the
layers of size
parallel to the
faces of the box contains exactly
green cubes, exactly
yellow cubes, and some red cubes. Find the smallest possible volume of the box.
Problem 5
Triangle has a right angle at
. Its side lengths are pairwise relatively prime positive integers, and its perimeter is
. Let
be the foot of the altitude to
, and for
, let
be the foot of the altitude to
in
. The sum
. Find
.
Problem 6
For polynomial , define
. Then
, where
and
are relatively prime positive integers. Find
.
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