Question bank AIME 2001

Let $N$ be the largest positive integer with the following property: reading from left to right, each pair of consecutive digits of $N$ forms a perfect square. What are the leftmost three digits of $N?$

Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M - m.$

The first four terms are $x_1 = 211,$ $x_2 = 375,$ $x_3 = 420,$ and $x_4 = 523.$ For $n \ge 5,$ $$
\begin{aligned} x_n &= x_{n-1} - x_{n-2} \\ &\quad {}+ x_{n-3} - x_{n-4}. \end{aligned}
$$ Find the value of $x_{531} + x_{753} + x_{975}.$

Let $R = (8, 6).$ The lines whose equations are $8y = 15x$ and $10y = 3x$ contain points $P$ and $Q,$ respectively, such that $R$ is the midpoint of $\overline{PQ}.$ The length $PQ$ equals $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

A set of positive numbers has the *triangle property* if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\{4, 5, 6, \ldots, n\}$ of consecutive positive integers, all of whose ten-element subsets have the triangle property. What is the largest possible value of $n?$

Square $ABCD$ is inscribed in a circle. Square $EFGH$ has vertices $E$ and $F$ on $\overline{CD}$ and vertices $G$ and $H$ on the circle. The ratio of the area of square $EFGH$ to the area of square $ABCD$ can be expressed as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers and $m \lt n.$ Find $10n + m.$

Let $\triangle PQR$ be a right triangle with $PQ = 90,$ $PR = 120,$ and $QR = 150.$ Let $C_1$ be the inscribed circle. Construct $\overline{ST},$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR},$ such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_1.$ Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_1.$ Let $C_2$ be the inscribed circle of $\triangle RST$ and $C_3$ the inscribed circle of $\triangle QUV.$ The distance between the centers of $C_2$ and $C_3$ can be written as $\sqrt{10n}.$ What is $n?$

A certain function $f$ has the properties that $f(3x) = 3f(x)$ for all positive real values of $x,$ and that $f(x) = 1 - |x - 2|$ for $1 \le x \le 3.$ Find the smallest $x$ for which $f(x) = f(2001).$

Each unit square of a $3$-by-$3$ unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a $2$-by-$2$ red square is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

How many positive integer multiples of $1001$ can be expressed in the form $10^{j} - 10^{i},$ where $i$ and $j$ are integers and $0 \le i \lt j \le 99?$

Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its $6$ matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}.$ The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Given a triangle, its *midpoint triangle* is obtained by joining the midpoints of its sides. A sequence of polyhedra $\mathcal{P}_i$ is defined recursively as follows: $\mathcal{P}_0$ is a regular tetrahedron whose volume is $1.$ To obtain $\mathcal{P}_{i+1},$ replace the midpoint triangle of every face of $\mathcal{P}_i$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $\mathcal{P}_3$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

In quadrilateral $ABCD,$ $\angle BAD \cong \angle ADC$ and $\angle ABD \cong \angle BCD,$ $AB = 8,$ $BD = 10,$ and $BC = 6.$ The length $CD$ may be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1.$ These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m},$ where $0 \le \theta_{1} \lt \theta_{2} \lt \ldots \lt \theta_{2n} \lt 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \cdots + \theta_{2n}.$

Let $EFGH,$ $EFDC,$ and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8,$ and let $A$ be the eighth vertex of the cube. Let $I,$ $J,$ and $K$ be points on $\overline{EF},$ $\overline{EH},$ and $\overline{EC},$ respectively, so that $EI = EJ = EK = 2.$ A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE},$ and containing the edges $\overline{IJ},$ $\overline{JK},$ and $\overline{KI}.$ The surface area of $S,$ including the walls of the tunnel, is $m + n\sqrt{p},$ where $m,$ $n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m + n + p.$

Find the sum of all positive two-digit integers that are divisible by each of their digits.

A finite set $\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\mathcal{S} \cup \{1\}$ is $13$ less than the mean of $\mathcal{S},$ and the mean of $\mathcal{S} \cup \{2001\}$ is $27$ more than the mean of $\mathcal{S}.$ Find the mean of $\mathcal{S}.$

Find the sum of the roots, real and non-real, of the equation $$
x^{2001} + \left(\tfrac{1}{2} - x\right)^{2001} = 0,
$$ given that there are no multiple roots.

In triangle $ABC,$ angles $A$ and $B$ measure $60$ degrees and $45$ degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T,$ and $AT = 24.$ The area of triangle $ABC$ can be written in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a + b + c.$

An equilateral triangle is inscribed in the ellipse whose equation is $x^2 + 4y^2 = 4.$ One vertex of the triangle is $(0, 1),$ one altitude is contained in the $y$-axis, and the length of each side is $\sqrt{\frac{m}{n}},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$