Difference between revisions of "2021 AIME II Problems"
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==Problem 3== | ==Problem 3== | ||
| − | + | Find the number of positive integers less than <math>1000</math> that can be expressed as the difference of two integral powers of <math>2.</math> | |
[[2021 AIME II Problems/Problem 3|Solution]] | [[2021 AIME II Problems/Problem 3|Solution]] | ||
Revision as of 12:45, 22 March 2021
These are not the problems you are looking for.
| 2021 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
| ||
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
Problem 1
Zou and Chou are practicing their 100-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, W);
label("
", B, W);
label("
", C, (1,0));
label("
", D, (1,0));
label("
", F, N);
label("
", E, S);
[/asy]
Someone please help with the diagram
Problem 3
Find the number of positive integers less than 
that can be expressed as the difference of two integral powers of 
Problem 4
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 5
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 6
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 7
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 8
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 9
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 10
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 11
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 12
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 13
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 14
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
Problem 15
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
See also
| 2021 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by 2021 AIME I |
Followed by 2022 AIME I | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
