Difference between revisions of "2021 AIME II Problems"

(Problem 1)
(Problem 2)
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==Problem 2==
 
==Problem 2==
These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.
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In the diagram below, <math>ABCD</math> is a rectangle with side lengths <math>AB=3</math> and <math>BC=11</math>, and <math>AECF</math> is a rectangle with side lengths <math>AF=7</math> and <math>FC=9,</math> as shown. The area of the shaded region common to the interiors of both rectangles is <math>\frac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
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[asy]
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pair A, B, C, D, E, F;
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A = (0,3);
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B=(0,0);
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C=(11,0);
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D=(11,3);
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E=foot(C, A, (9/4,0));
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F=foot(A, C, (35/4,3));
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draw(A--B--C--D--cycle);
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draw(A--E--C--F--cycle);
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filldraw(A--(9/4,0)--C--(35/4,3)--cycle,gray*0.5+0.5*lightgray);
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dot(A^^B^^C^^D^^E^^F);
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label("<math>A</math>", A, W);
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label("<math>B</math>", B, W);
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label("<math>C</math>", C, (1,0));
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label("<math>D</math>", D, (1,0));
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label("<math>F</math>", F, N);
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label("<math>E</math>", E, S);
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[/asy]
  
 
[[2021 AIME II Problems/Problem 2|Solution]]
 
[[2021 AIME II Problems/Problem 2|Solution]]

Revision as of 13:43, 22 March 2021

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2021 AIME II (Answer Key)
Printable version | AoPS Contest CollectionsPDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Problem 1

Zou and Chou are practicing their 100-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Problem 2

In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[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]

Solution

Problem 3

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 4

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 5

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 6

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 7

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 8

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 9

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 10

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 11

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 12

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 13

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 14

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

Problem 15

These problems will not be available until the 2021 AIME II is released on Thursday, March 18, 2021.

Solution

See also

2021 AIME II (ProblemsAnswer KeyResources)
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

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