2021年AIME I 真题:
Problem 1
Zou and Chou are practicing their -meter sprints by running
races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is
if they won the previous race but only
if they lost the previous race. The probability that Zou will win exactly
of the
races is
, where
and
are relatively prime positive integers. Find
.
Problem 2
In the diagram below, is a rectangle with side lengths
and
, and
is a rectangle with side lengths
and
as shown. The area of the shaded region common to the interiors of both rectangles is
, where
and
are relatively prime positive integers. Find
.
Problem 3
Find the number of positive integers less than that can be expressed as the difference of two integral powers of
Problem 4
Find the number of ways identical coins can be separated into three nonempty piles so that there are fewer coins in the first pile than in the second pile and fewer coins in the second pile than in the third pile.
Problem 5
Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences.
Problem 6
Segments and
are edges of a cube and
is a diagonal through the center of the cube. Point
satisfies
,
,
, and
. Find
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