Difference between revisions of "2012 AIME I Problems"
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| − | Let <math>z</math> and <math>w</math> be complex numbers such that <math>z^{13} = w</math> and <math>w^{11} = z</math>. If the imaginary part of <math>z</math> can be written as <math> \sin {\frac{m\pi}{n}}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>n</math>. | + | Let <math>z</math> and <math>w</math> be complex numbers such that <math>z^{13} = w</math> and <math>w^{11} = z</math>. If the imaginary part of <math>z</math> can be written as <math> \sin ({\frac{m\pi}{n}})</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>n</math>. |
[[2012 AIME I Problems/Problem 6|Solution]] | [[2012 AIME I Problems/Problem 6|Solution]] | ||
Revision as of 22:14, 16 March 2012
| 2012 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
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Instructions
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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
Find the number of positive integers with three not necessarily distinct digits, 
, with 
and 
such that both and 
are multiples of 
.
Problem 2
The terms of an arithmetic sequence add to 
. The first term of the sequence is increased by 
, the second term is increased by 
, the third term is increased by 
, and in general, the 
th term is increased by the th odd positive integer. The terms of the new sequence add to 
. Find the sum of the first, last, and middle terms of the original sequence.
Problem 3
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person.
Problem 4
Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at 
, , and 
miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are 
miles from Dodge, and they have been traveling for 
minutes. Find 
.
Problem 5
Let 
be the set of all binary integers that can be written using exactly zeros and 
ones where leading zeros are allowed. If all possible subtractions are performed in which one element of is subtracted from another, find the number of times the answer is obtained.
Problem 6
Let 
and 
be complex numbers such that 
and 
. If the imaginary part of can be written as 
, where 
and are relatively prime positive integers, find .
Problem 7
Problem 8
Problem 9
Problem 10
Let 
be the set of all perfect squares whose rightmost three digits in base 
are 
. Let 
be the set of all numbers of the form 
, where 
is in . In other words, is the set of numbers that result when the last three digits of each number in are truncated. Find the remainder when the tenth smallest element of is divided by 
.
