1986 AIME Problems

Part of AIME Problems

Subcontests

(15)

1

Triangle in Cartesian Plane

ABC

be a right triangle in the xy-plane with a right angle at

C

. Given that the length of the hypotenuse

AB

is 60, and that the medians through

A

B

lie along the lines

y=x+3

y=2x+4

respectively, find the area of triangle

ABC

1

Rectangular Prism

The shortest distances between an interior diagonal of a rectangular parallelepiped, P, and the edges it does not meet are

2\sqrt{5}

\frac{30}{\sqrt{13}}

\frac{15}{\sqrt{10}}

. Determine the volume of

P

1

Coin Toss Sequence

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 HHTTHHHHTHHTTTT 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?

1

Set of Elements

Let the sum of a set of numbers be the sum of its elements. Let

S

be a set of positive integers, none greater than 15. Suppose no two disjoint subsets of

S

have the same sum. What is the largest sum a set

S

with these properties can have?

1

Polynomial Coefficients

1-x+x^2-x^3+\cdots+x^{16}-x^{17}

may be written in the form

a_0+a_1y+a_2y^2+\cdots +a_{16}y^{16}+a_{17}y^{17}

y=x+1

a_i

's are constants. Find the value of

a_2

1

A Magic Trick

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,

N

. If told the value of

N

, the magician can identify the original number, (abc). Play the role of the magician and determine the (abc) if

N= 3194

1

Triangle Segments

\triangle ABC

AB= 425

BC=450

AC=510

. An interior point

P

is then drawn, and segments are drawn through

P

parallel to the sides of the triangle. If these three segments are of an equal length

d

d

1

Logarithms and Divisors

S

be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to

S

1

Finding Terms of a Sequence

The increasing sequence

1,3,4,9,10,12,13\cdots

consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the

100^{\text{th}}

term of this sequence.

1

Book Page Errors

The pages of a book are numbered 1 through

n

. When the page numbers of the book were added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of 1986. What was the number of the page that was added twice?

1

Largest Divisor

What is that largest positive integer

n

n^3+100

is divisible by

n+10

1

System of Equations

3x_4+2x_5

x_1

x_2

x_3

x_4

x_5

satisfy the system of equations below.

\begin{array}{l} 2x_1+x_2+x_3+x_4+x_5=6 \\ x_1+2x_2+x_3+x_4+x_5=12 \\ x_1+x_2+2x_3+x_4+x_5=24 \\ x_1+x_2+x_3+2x_4+x_5=48 \\ x_1+x_2+x_3+x_4+2x_5=96 \\ \end{array}

1

Tangents and Cotangents

\tan x+\tan y=25

\cot x + \cot y=30

\tan(x+y)

1

Product Evaluation

Evaluate the product

(\sqrt 5+\sqrt6+\sqrt7)(-\sqrt 5+\sqrt6+\sqrt7)(\sqrt 5-\sqrt6+\sqrt7)(\sqrt 5+\sqrt6-\sqrt7).

1

Find X

What is the sum of the solutions to the equation

\sqrt[4]x =\displaystyle \frac{12}{7-\sqrt[4]x}