1986 AIME Problems
1986 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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Contents
Problem 1
What is the sum of the solutions to the equation ?
Problem 2
Evaluate the product .
Problem 3
If and , what is ?
Problem 4
Determine if , , , , and satisfy the system of equations below.
Problem 5
What is that largest positive integer for which is divisible by ?
Problem 6
The pages of a book are numbered through . When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of . What was the number of the page that was added twice?
Problem 7
The increasing sequence consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the term of this sequence.
Problem 8
Let be the sum of the base logarithms of all the proper divisors of . What is the integer nearest to ?
Problem 9
In , , , and . An interior point is then drawn, and segments are drawn through parallel to the sides of the triangle. If these three segments are of an equal length , find .
Problem 10
In a parlor game, the magician asks one of the participants to think of a three digit number (abc) where a, b, and c represent digits in base 10 in the order indicated. The magician then asks this person to form the numbers (acb), (bca), (bac), (cab), and (cba), to add these five numbers, and to reveal their sum, . If told the value of , the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if .
Problem 11
The polynomial may be written in the form , where and the 's are constants. Find the value of .
Problem 12
Let the sum of a set of numbers be the sum of its elements. Let be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of have the same sum. What is the largest sum a set with these properties can have?
Problem 13
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, three HT, four TH, and five TT subsequences. How many different sequences of 15 coin tosses will contain exactly two HH, three HT, four TH, and five TT subsequences?
Problem 14
The shortest distances between an interior diagonal of a rectangular parallelepiped, , and the edges it does not meet are , , and . Determine the volume of .
Problem 15
Let triangle be a right triangle in the xy-plane with a right angle at . Given that the length of the hypotenuse is , and that the medians through and lie along the lines and respectively, find the area of triangle .