HMMT 2003

Find the smallest value of $x$ such that $a \geq 14 \sqrt{a}-x$ for all nonnegative $a$.

Compute $\frac{\tan ^{2}\left(20^{\circ}\right)-\sin ^{2}\left(20^{\circ}\right)}{\tan ^{2}\left(20^{\circ}\right) \sin ^{2}\left(20^{\circ}\right)}$.

Find the smallest $n$ such that $n$ ! ends in 290 zeroes.

Simplify: $2 \sqrt{1.5+\sqrt{2}}-(1.5+\sqrt{2})$.

Several positive integers are given, not necessarily all different. Their sum is 2003. Suppose that $n_{1}$ of the given numbers are equal to $1, n_{2}$ of them are equal to $2, \ldots$, $n_{2003}$ of them are equal to 2003 . Find the largest possible value of

$$n_{2}+2 n_{3}+3 n_{4}+\cdots+2002 n_{2003} .$$

Let $a_{1}=1$, and let $a_{n}=\left\lfloor n^{3} / a_{n-1}\right\rfloor$ for $n>1$. Determine the value of $a_{999}$.

Let $a, b, c$ be the three roots of $p(x)=x^{3}+x^{2}-333 x-1001$. Find $a^{3}+b^{3}+c^{3}$.

Find the value of $\frac{1}{3^{2}+1}+\frac{1}{4^{2}+2}+\frac{1}{5^{2}+3}+\cdots$.

For how many integers $n$, for $1 \leq n \leq 1000$, is the number $\frac{1}{2}\binom{2 n}{n}$ even?

Suppose $P(x)$ is a polynomial such that $P(1)=1$ and

$$\frac{P(2 x)}{P(x+1)}=8-\frac{56}{x+7}$$

for all real $x$ for which both sides are defined. Find $P(-1)$.

$A D$ and $B C$ are both perpendicular to $A B$, and $C D$ is perpendicular to $A C$. If $A B=4$ and $B C=3$, find $C D$.

As shown, $U$ and $C$ are points on the sides of triangle $M N H$ such that $M U=s$, $U N=6, N C=20, C H=s, H M=25$. If triangle $U N C$ and quadrilateral $M U C H$ have equal areas, what is $s$ ?

A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12 m . What is the area (in $\mathrm{m}^{2}$ ) of the room?

Farmer John is inside of an ellipse with reflective sides, given by the equation $x^{2} / a^{2}+$ $y^{2} / b^{2}=1$, with $a>b>0$. He is standing at the point (3,0), and he shines a laser pointer in the $y$-direciton. The light reflects off the ellipse and proceeds directly toward Farmer Brown, traveling a distance of 10 before reaching him. Farmer John then spins around in a circle; wherever he points the laser, the light reflects off the wall and hits Farmer Brown. What is the ordered pair $(a, b)$ ?

Take a clay sphere of radius 13 , and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius?

Let $R S T U V$ be a regular pentagon. Construct an equilateral triangle $P R S$ with point $P$ inside the pentagon. Find the measure (in degrees) of angle $P T V$.

Let $A B C$ be an equilateral triangle of side length 2 . Let $\omega$ be its circumcircle, and let $\omega_{A}, \omega_{B}, \omega_{C}$ be circles congruent to $\omega$ centered at each of its vertices. Let $R$ be the set of all points in the plane contained in exactly two of these four circles. What is the area of $R$ ?

In triangle $A B C, \angle A B C=50^{\circ}$ and $\angle A C B=70^{\circ}$. Let $D$ be the midpoint of side $B C$. A circle is tangent to $B C$ at $B$ and is also tangent to segment $A D$; this circle instersects $A B$ again at $P$. Another circle is tangent to $B C$ at $C$ and is also tangent to segment $A D$; this circle intersects $A C$ again at $Q$. Find $\angle A P Q$ (in degrees).

Convex quadrilateral $M A T H$ is given with $H M / M T=3 / 4$, and $\angle A T M=\angle M A T=$ $\angle A H M=60^{\circ} . N$ is the midpoint of $M A$, and $O$ is a point on $T H$ such that lines $M T, A H, N O$ are concurrent. Find the ratio $H O / O T$.

You have 2003 switches, numbered from 1 to 2003, arranged in a circle. Initially, each switch is either ON or OFF, and all configurations of switches are equally likely. You perform the following operation: for each switch $S$, if the two switches next to $S$ were initially in the same position, then you set $S$ to ON; otherwise, you set $S$ to OFF. What is the probability that all switches will now be ON?

You are given a $10 \times 2$ grid of unit squares. Two different squares are adjacent if they share a side. How many ways can one mark exactly nine of the squares so that no two marked squares are adjacent?

Daniel and Scott are playing a game where a player wins as soon as he has two points more than his opponent. Both players start at par, and points are earned one at a time. If Daniel has a $60 \%$ chance of winning each point, what is the probability that he will win the game?

In a certain country, there are 100 senators, each of whom has 4 aides. These senators and aides serve on various committees. A committee may consist either of 5 senators, of 4 senators and 4 aides, or of 2 senators and 12 aides. Every senator serves on 5 committees, and every aide serves on 3 committees. How many committees are there altogether?

We wish to color the integers $1,2,3, \ldots, 10$ in red, green, and blue, so that no two numbers $a$ and $b$, with $a-b$ odd, have the same color. (We do not require that all three colors be used.) In how many ways can this be done?

In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e. move one desk forward, back, left or right). In how many ways can this reassignment be made?

You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled $1,2, \ldots$. Each ball has probability $1 / 2^{n}$ of being put into box $n$. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls?

For any subset $S \subseteq\{1,2, \ldots, 15\}$, a number $n$ is called an "anchor" for $S$ if $n$ and $n+|S|$ are both members of $S$, where $|S|$ denotes the number of members of $S$. Find the average number of anchors over all possible subsets $S \subseteq\{1,2, \ldots, 15\}$.

At a certain college, there are 10 clubs and some number of students. For any two different students, there is some club such that exactly one of the two belongs to that club. For any three different students, there is some club such that either exactly one or all three belong to that club. What is the largest possible number of students?

A calculator has a display, which shows a nonnegative integer $N$, and a button, which replaces $N$ by a random integer chosen uniformly from the set $\{0,1, \ldots, N-1\}$, provided that $N>0$. Initially, the display holds the number $N=2003$. If the button is pressed repeatedly until $N=0$, what is the probability that the numbers $1,10,100$, and 1000 will each show up on the display at some point?

A point is chosen randomly with uniform distribution in the interior of a circle of radius 1 . What is its expected distance from the center of the circle?

A particle moves along the $x$-axis in such a way that its velocity at position $x$ is given by the formula $v(x)=2+\sin x$. What is its acceleration at $x=\frac{\pi}{6}$ ?

What is the area of the region bounded by the curves $y=x^{2003}$ and $y=x^{1 / 2003}$ and lying above the $x$-axis?

The sequence of real numbers $x_{1}, x_{2}, x_{3}, \ldots$ satisfies $\lim _{n \rightarrow \infty}\left(x_{2 n}+x_{2 n+1}\right)=315$ and $\lim _{n \rightarrow \infty}\left(x_{2 n}+x_{2 n-1}\right)=2003$. Evaluate $\lim _{n \rightarrow \infty}\left(x_{2 n} / x_{2 n+1}\right)$.

Find the minimum distance from the point $(0,5 / 2)$ to the graph of $y=x^{4} / 8$.

For $n$ an integer, evaluate

$$\lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^{2}-0^{2}}}+\frac{1}{\sqrt{n^{2}-1^{2}}}+\cdots+\frac{1}{\sqrt{n^{2}-(n-1)^{2}}}\right)$$

For what value of $a>1$ is

$$\int_{a}^{a^{2}} \frac{1}{x} \log \frac{x-1}{32} d x$$

minimum?

A right circular cone with a height of 12 inches and a base radius of 3 inches is filled with water and held with its vertex pointing downward. Water flows out through a hole at the vertex at a rate in cubic inches per second numerically equal to the height of the water in the cone. (For example, when the height of the water in the cone is 4 inches, water flows out at a rate of 4 cubic inches per second.) Determine how many seconds it will take for all of the water to flow out of the cone.

Two differentiable real functions $f(x)$ and $g(x)$ satisfy

$$\frac{f^{\prime}(x)}{g^{\prime}(x)}=e^{f(x)-g(x)}$$

for all $x$, and $f(0)=g(2003)=1$. Find the largest constant $c$ such that $f(2003)>c$ for all such functions $f, g$.

Evaluate

$$\int_{-\infty}^{\infty} \frac{1-x^{2}}{1+x^{4}} d x$$

10 people are playing musical chairs with $n$ chairs in a circle. They can be seated in 7 ! ways (assuming only one person fits on each chair, of course), where different arrangements of the same people on chairs, even rotations, are considered different. Find $n$.

$O P E N$ is a square, and $T$ is a point on side $N O$, such that triangle $T O P$ has area 62 and triangle $T E N$ has area 10 . What is the length of a side of the square?

There are 16 members on the Height-Measurement Matching Team. Each member was asked, "How many other people on the team - not counting yourself - are exactly the same height as you?" The answers included six 1's, six 2's, and three 3's. What was the sixteenth answer? (Assume that everyone answered truthfully.)

How many 2-digit positive integers have an even number of positive divisors?

A room is built in the shape of the region between two semicircles with the same center and parallel diameters. The farthest distance between two points with a clear line of sight is 12 m . What is the area (in $\mathrm{m}^{2}$ ) of the room?

In how many ways can 3 bottles of ketchup and 7 bottles of mustard be arranged in a row so that no bottle of ketchup is immediately between two bottles of mustard? (The bottles of ketchup are mutually indistinguishable, as are the bottles of mustard.)

Find the real value of $x$ such that $x^{3}+3 x^{2}+3 x+7=0$.

A broken calculator has the + and $\times$ keys switched. For how many ordered pairs $(a, b)$ of integers will it correctly calculate $a+b$ using the labelled + key?

Consider a 2003-gon inscribed in a circle and a triangulation of it with diagonals intersecting only at vertices. What is the smallest possible number of obtuse triangles in the triangulation?

Bessie the cow is trying to navigate her way through a field. She can travel only from lattice point to adjacent lattice point, can turn only at lattice points, and can travel only to the east or north. (A lattice point is a point whose coordinates are both integers.) $(0,0)$ is the southwest corner of the field. $(5,5)$ is the northeast corner of the field. Due to large rocks, Bessie is unable to walk on the points $(1,1),(2,3)$, or $(3,2)$. How many ways are there for Bessie to travel from $(0,0)$ to $(5,5)$ under these constraints?

A compact disc has the shape of a circle of diameter 5 inches with a 1 -inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the capacity?

You have a list of real numbers, whose sum is 40 . If you replace every number $x$ on the list by $1-x$, the sum of the new numbers will be 20 . If instead you had replaced every number $x$ by $1+x$, what would the sum then be?

How many positive rational numbers less than $\pi$ have denominator at most 7 when written in lowest terms? (Integers have denominator 1.)

In triangle $A B C$ with area 51, points $D$ and $E$ trisect $A B$ and points $F$ and $G$ trisect $B C$. Find the largest possible area of quadrilateral $D E F G$.

The numbers $112,121,123,153,243,313$, and 322 are among the rows, columns, and diagonals of a $3 \times 3$ square grid of digits (rows and diagonals read left-to-right, and columns read top-to-bottom). What 3-digit number completes the list?

If $x \geq 0, y \geq 0$ are integers, randomly chosen with the constraint $x+y \leq 10$, what is the probability that $x+y$ is even?

(a) How many $k$-configurations are there of a set that has $n$ elements?

Suppose $A$ is a set with $n$ elements, and $k$ is a divisor of $n$. Find the number of consistent $k$-configurations of $A$ of order 1 .

(a) Let $A_{n}=\left\{a_{1}, a_{2}, a_{3}, \ldots, a_{n}, b\right\}$, for $n \geq 3$, and let $C_{n}$ be the 2-configuration consisting of $\left\{a_{i}, a_{i+1}\right\}$ for all $1 \leq i \leq n-1,\left\{a_{1}, a_{n}\right\}$, and $\left\{a_{i}, b\right\}$ for $1 \leq i \leq n$. Let $S_{e}(n)$ be the number of subsets of $C_{n}$ that are consistent of order $e$. Find $S_{e}(101)$ for $e=1,2$, and 3 .

(a) Suppose $A$ has $n$ elements, where $n \geq 2$, and $C$ is a 2-configuration of $A$ that is not $m$-separable for any $m<n$. What is (in terms of $n$ ) the smallest number of elements that $C$ can have?

Let $B_{k}(n)$ be the largest possible number of elements in a 2 -separable $k$-configuration of a set with $2 n$ elements $(2 \leq k \leq n)$. Find a closed-form expression (i.e. an expression not involving any sums or products with a variable number of terms) for $B_{k}(n)$.

Prove that any 2 -configuration containing $e$ elements is $m$-separable for some $m \leq$ $\frac{1}{2}+\sqrt{2 e+\frac{1}{4}}$.

Show that, given any 2 -configuration of a set $A$, every element of $A$ belongs to exactly one cell.

(a) Given a set $A$ with $n \geq 1$ elements, find the number of consistent 2-configurations of $A$ of order 1 with exactly 1 cell.

(a) Show that if every cell of a 2-configuration of a finite set $A$ is $m$-separable, then the whole 2-configuration is $m$-separable.

Show that every consistent 2-configuration of order 4 on a finite set $A$ has a subset that is a consistent 2-configuration of order 2.

Simplify $\sqrt[2003]{2 \sqrt{11}-3 \sqrt{5}} \cdot \sqrt[4006]{89+12 \sqrt{55}}$.

The graph of $x^{4}=x^{2} y^{2}$ is a union of $n$ different lines. What is the value of $n$ ?

If $a$ and $b$ are positive integers that can each be written as a sum of two squares, then $a b$ is also a sum of two squares. Find the smallest positive integer $c$ such that $c=a b$, where $a=x^{3}+y^{3}$ and $b=x^{3}+y^{3}$ each have solutions in integers $(x, y)$, but $c=x^{3}+y^{3}$ does not.

Let $z=1-2 i$. Find $\frac{1}{z}+\frac{2}{z^{2}}+\frac{3}{z^{3}}+\cdots$.

Compute the surface area of a cube inscribed in a sphere of surface area $\pi$.

Define the Fibonacci numbers by $F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. For how many $n, 0 \leq n \leq 100$, is $F_{n}$ a multiple of 13 ?

$a$ and $b$ are integers such that $a+\sqrt{b}=\sqrt{15+\sqrt{216}}$. Compute $a / b$.

How many solutions in nonnegative integers $(a, b, c)$ are there to the equation

$$2^{a}+2^{b}=c! ?$$

For $x$ a real number, let $f(x)=0$ if $x<1$ and $f(x)=2 x-2$ if $x \geq 1$. How many solutions are there to the equation

$$f(f(f(f(x))))=x ?$$

Suppose that $A, B, C, D$ are four points in the plane, and let $Q, R, S, T, U, V$ be the respective midpoints of $A B, A C, A D, B C, B D, C D$. If $Q R=2001, S U=2002, T V=$ 2003, find the distance between the midpoints of $Q U$ and $R V$.

Find the smallest positive integer $n$ such that $1^{2}+2^{2}+3^{2}+4^{2}+\cdots+n^{2}$ is divisible by 100 .

As shown in the figure, a circle of radius 1 has two equal circles whose diameters cover a chosen diameter of the larger circle. In each of these smaller circles we similarly draw three equal circles, then four in each of those, and so on. Compute the area of the region enclosed by a positive even number of circles.

If $x y=5$ and $x^{2}+y^{2}=21$, compute $x^{4}+y^{4}$.

A positive integer will be called "sparkly" if its smallest (positive) divisor, other than 1, equals the total number of divisors (including 1). How many of the numbers $2,3, \ldots, 2003$ are sparkly?

The product of the digits of a 5 -digit number is 180 . How many such numbers exist?

What fraction of the area of a regular hexagon of side length 1 is within distance $\frac{1}{2}$ of at least one of the vertices?

There are 10 cities in a state, and some pairs of cities are connected by roads. There are 40 roads altogether. A city is called a "hub" if it is directly connected to every other city. What is the largest possible number of hubs?

Find the sum of the reciprocals of all the (positive) divisors of 144.

Let $r, s, t$ be the solutions to the equation $x^{3}+a x^{2}+b x+c=0$. What is the value of $(r s)^{2}+(s t)^{2}+(r t)^{2}$ in terms of $a, b$, and $c$ ?

What is the smallest number of regular hexagons of side length 1 needed to completely cover a disc of radius 1 ?

$r$ and $s$ are integers such that

$$3 r \geq 2 s-3 \text { and } 4 s \geq r+12 .$$

What is the smallest possible value of $r / s$ ?

There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another painter paints every seventh house blue, and so forth, alternating between red and blue, until 50 painters have been by. After this is finished, how many houses will be red?

How many lattice points are enclosed by the triangle with vertices $(0,99),(5,100)$, and (2003, 500)? Don't count boundary points.

Compute the radius of the inscribed circle of a triangle with sides 15,16 , and 17 .

Let $A B C$ be an isosceles triangle with apex $A$. Let $I$ be the incenter. If $A I=3$ and the distance from $I$ to $B C$ is 2 , then what is the length of $B C$ ?

Find all integers $m$ such that $m^{2}+6 m+28$ is a perfect square.

The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ?

A point in three-space has distances $2,6,7,8,9$ from five of the vertices of a regular octahedron. What is its distance from the sixth vertex?

A palindrome is a positive integer that reads the same backwards as forwards, such as 82328. What is the smallest 5 -digit palindrome that is a multiple of 99 ?

The sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers satisfies the recurrence

$$a_{n+1}=\frac{a_{n}^{2}-a_{n-1}+2 a_{n}}{a_{n-1}+1} .$$

Given that $a_{1}=1$ and $a_{9}=7$, find $a_{5}$.

A cylinder of base radius 1 is cut into two equal parts along a plane passing through the center of the cylinder and tangent to the two base circles. Suppose that each piece's surface area is $m$ times its volume. Find the greatest lower bound for all possible values of $m$ as the height of the cylinder varies.

If $x, y$, and $z$ are real numbers such that $2 x^{2}+y^{2}+z^{2}=2 x-4 y+2 x z-5$, find the maximum possible value of $x-y+z$.

We are given triangle $A B C$, with $A B=9, A C=10$, and $B C=12$, and a point $D$ on $B C$. $B$ and $C$ are reflected in $A D$ to $B^{\prime}$ and $C^{\prime}$, respectively. Suppose that lines $B C^{\prime}$ and $B^{\prime} C$ never meet (i.e., are parallel and distinct). Find $B D$.

$O K R A$ is a trapezoid with $O K$ parallel to $R A$. If $O K=12$ and $R A$ is a positive integer, how many integer values can be taken on by the length of the segment in the trapezoid, parallel to $O K$, through the intersection of the diagonals?

A certain lottery has tickets labeled with the numbers $1,2,3, \ldots, 1000$. The lottery is run as follows: First, a ticket is drawn at random. If the number on the ticket is odd, the drawing ends; if it is even, another ticket is randomly drawn (without replacement). If this new ticket has an odd number, the drawing ends; if it is even, another ticket is randomly drawn (again without replacement), and so forth, until an odd number is drawn. Then, every person whose ticket number was drawn (at any point in the process) wins a prize. You have ticket number 1000. What is the probability that you get a prize?

A teacher must divide 221 apples evenly among 403 students. What is the minimal number of pieces into which she must cut the apples? (A whole uncut apple counts as one piece.)

A quagga is an extinct chess piece whose move is like a knight's, but much longer: it can move 6 squares in any direction (up, down, left, or right) and then 5 squares in a perpendicular direction. Find the number of ways to place 51 quaggas on an $8 \times 8$ chessboard in such a way that no quagga attacks another. (Since quaggas are naturally belligerent creatures, a quagga is considered to attack quaggas on any squares it can move to, as well as any other quaggas on the same square.)

Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \geq 0$. Suppose that there exists a fixed nonnegative integer $m$ such that, for every choice of $a_{0}$ and $a_{1}$, the numbers $a_{m}, a_{m+1}, a_{m+3}$, in this order, form an arithmetic progression. Find all possible values of $y$.

In the figure, if $A E=3, C E=1, B D=C D=2$, and $A B=5$, find $A G$.

All the sequences consisting of five letters from the set $\{T, U, R, N, I, P\}$ (with repetitions allowed) are arranged in alphabetical order in a dictionary. Two sequences are called "anagrams" of each other if one can be obtained by rearranging the letters of the other. How many pairs of anagrams are there that have exactly 100 other sequences between them in the dictionary?

A hotel consists of a $2 \times 8$ square grid of rooms, each occupied by one guest. All the guests are uncomfortable, so each guest would like to move to one of the adjoining rooms (horizontally or vertically). Of course, they should do this simultaneously, in such a way that each room will again have one guest. In how many different ways can they collectively move?

A tightrope walker stands in the center of a rope of length 32 meters. Every minute she walks forward one meter with probability $3 / 4$ and backward one meter with probability $1 / 4$. What is the probability that she reaches the end in front of her before the end behind her?