Question bank AIME 1986

What is the sum of the solutions to the equation $\sqrt[4]{x}=\frac{12}{7-\sqrt[4]{x}}?$

Evaluate the product $$
\begin{gathered} (\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(-\sqrt5+\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5-\sqrt6+\sqrt7)\\ {}\cdot(\sqrt5+\sqrt6-\sqrt7). \end{gathered}
$$

If $\tan x+\tan y=25$ and $\cot x+\cot y=30,$ what is $\tan(x+y)?$

Determine $3x_4+2x_5$ if $x_1,$ $x_2,$ $x_3,$ $x_4,$ and $x_5$ satisfy the system $$
\begin{aligned} 2x_1+x_2+x_3+x_4+x_5&=6,\\ x_1+2x_2+x_3+x_4+x_5&=12,\\ x_1+x_2+2x_3+x_4+x_5&=24,\\ x_1+x_2+x_3+2x_4+x_5&=48,\\ x_1+x_2+x_3+x_4+2x_5&=96. \end{aligned}
$$

What is the largest positive integer $n$ for which $n^3+100$ is divisible by $n+10?$

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?

The increasing sequence $1,$ $3,$ $4,$ $9,$ $10,$ $12,$ $13,$ $\ldots$ consists of all those positive integers which are powers of $3$ or sums of distinct powers of $3.$ Find the $100$th term of this sequence.

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

In $\triangle ABC,$ $AB=425,$ $BC=450,$ and $AC=510.$ An interior point $P$ is drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments have equal length $d,$ find $d.$

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 base-$10$ digits in the indicated order. 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 $N,$ the magician can identify the original number $(abc).$ Play the role of the magician and determine $(abc)$ if $N=3194.$

The polynomial $1-x+x^2-x^3+\cdots$ ${}+x^{16}-x^{17}$ may be written in the form $$
\begin{aligned} &a_0+a_1y+a_2y^2+\cdots\\ &\qquad{}+a_{16}y^{16}+a_{17}y^{17}, \end{aligned}
$$ where $y=x+1$ and the $a_i$’s are constants. Find $a_2.$

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?

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 so on. We denote these by $\mathrm{TH},$ $\mathrm{HH},$ and so on. For example, in the sequence $\mathrm{HHTTHHHHTHHTTTT}$ of $15$ coin tosses, there are two $\mathrm{HH},$ three $\mathrm{HT},$ four $\mathrm{TH},$ and five $\mathrm{TT}$ subsequences. How many different sequences of $15$ coin tosses contain exactly two $\mathrm{HH},$ three $\mathrm{HT},$ four $\mathrm{TH},$ and five $\mathrm{TT}$ subsequences?

The shortest distances between an interior diagonal of a rectangular parallelepiped $P$ and the edges it does not meet are $2\sqrt5,$ $\frac{30}{\sqrt{13}},$ and $\frac{15}{\sqrt{10}}.$ Determine the volume of $P.$

Let triangle $ABC$ be a right triangle in the $xy$-plane with a right angle at $C.$ Given that the hypotenuse $AB$ has length $60,$ and that the medians through $A$ and $B$ lie along the lines $y=x+3$ and $y=2x+4,$ respectively, find the area of $\triangle ABC.$