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 .
![[asy] pair A, B, C, D, E, F; A = (0,3); B=(0,0); C=(11,0); D=(11,3); E=foot(C, A, (9/4,0)); F=foot(A, C, (35/4,3)); draw(A--B--C--D--cycle); draw(A--E--C--F--cycle); filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray); dot(A^^B^^C^^D^^E^^F); label("$A$", A, W); label("$B$", B, W); label("$C$", C, (1,0)); label("$D$", D, (1,0)); label("$F$", F, N); label("$E$", E, S); [/asy]](https://latex.artofproblemsolving.com/c/e/7/ce7ef019f55e9d0cf7364f8d93782a020489c947.png)
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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