HMMT 1999

If $a @ b=\frac{a^{3}-b^{3}}{a-b}$, for how many real values of $a$ does $a @ 1=0$ ?

For what single digit $n$ does 91 divide the 9 -digit number $12345 n 789$ ?

Alex is stuck on a platform floating over an abyss at $1 \mathrm{ft} / \mathrm{s}$. An evil physicist has arranged for the platform to fall in (taking Alex with it) after traveling 100ft. One minute after the platform was launched, Edward arrives with a second platform capable of floating all the way across the abyss. He calculates for 5 seconds, then launches the second platform in such a way as to maximize the time that one end of Alex's platform is between the two ends of the new platform, thus giving Alex as much time as possible to switch. If both platforms are 5 ft long and move with constant velocity once launched, what is the speed of the second platform (in $\mathrm{ft} / \mathrm{s}$ )?

Find all possible values of $\frac{d}{a}$ where $a^{2}-6 a d+8 d^{2}=0, a \neq 0$.

You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out you must open all the doors and disarm all the mines. In the room is a panel with 3 buttons, which conveniently contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially 3 doors are closed and 3 mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?

Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne and Carl in 5. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back? Express your answer as a fraction in lowest terms.

Matt has somewhere between 1000 and 2000 pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2,3,4,5,6,7$, and 8 piles but ends up with one sheet left over each time. How many piles does he need?

If $f(x)$ is a monic quartic polynomial such that $f(-1)=-1, f(2)=-4, f(-3)=-9$, and $f(4)=-16$, find $f(1)$.

How many ways are there to cover a $3 \times 8$ rectangle with 12 identical dominoes?

Pyramid $E A R L Y$ is placed in $(x, y, z)$ coordinates so that $E=(10,10,0), A=(10,-10,0), R=$ $(-10,-10,0), L=(-10,10,0)$, and $Y=(0,0,10)$. Tunnels are drilled through the pyramid in such a way that one can move from $(x, y, z)$ to any of the 9 points $(x, y, z-1),(x \pm 1, y, z-1)$, $(x, y \pm 1, z-1),(x \pm 1, y \pm 1, z-1)$. Sean starts at $Y$ and moves randomly down to the base of the pyramid, choosing each of the possible paths with probability $\frac{1}{9}$ each time. What is the probability that he ends up at the point $(8,9,0)$ ?

Two $10 \times 24$ rectangles are inscribed in a circle as shown. Find the shaded area.

A semicircle is inscribed in a semicircle of radius 2 as shown. Find the radius of the smaller semicircle.

In a cube with side length 6 , what is the volume of the tetrahedron formed by any vertex and the three vertices connected to that vertex by edges of the cube?

A cross-section of a river is a trapezoid with bases 10 and 16 and slanted sides of length 5. At this section the water is flowing at $\pi \mathrm{mph}$. A little ways downstream is a dam where the water flows through 4 identical circular holes at 16 mph . What is the radius of the holes?

In triangle $B E N$ shown below with its altitudes intersecting at $X, N A=7, E A=3, A X=4$, and $N S=8$. Find the area of $B E N$.

A sphere of radius 1 is covered in ink and rolling around between concentric spheres of radii 3 and 5. If this process traces a region of area 1 on the larger sphere, what is the area of the region traced on the smaller sphere?

A dart is thrown at a square dartboard of side length 2 so that it hits completely randomly. What is the probability that it hits closer to the center than any corner, but within a distance 1 of a corner?

Squares $A B K L, B C M N, C A O P$ are drawn externally on the sides of a triangle $A B C$. The line segments $K L, M N, O P$, when extended, form a triangle $A^{\prime} B^{\prime} C^{\prime}$. Find the area of $A^{\prime} B^{\prime} C^{\prime}$ if $A B C$ is an equilateral triangle of side length 2.

A regular tetrahedron has two vertices on the body diagonal of a cube with side length 12 . The other two vertices lie on one of the face diagonals not intersecting that body diagonal. Find the side length of the tetrahedron.

In the figure below, $A B=15, B D=18, A F=15, D F=12, B E=24$, and $C F=17$. Find $B G: F G$.

Find all twice differentiable functions $f(x)$ such that $f^{\prime \prime}(x)=0, f(0)=19$, and $f(1)=99$.

A rectangle has sides of length $\sin x$ and $\cos x$ for some $x$. What is the largest possible area of such a rectangle?

Find

$$\int_{-4 \pi \sqrt{2}}^{4 \pi \sqrt{2}}\left(\frac{\sin x}{1+x^{4}}+1\right) d x$$

$f$ is a continuous real-valued function such that $f(x+y)=f(x) f(y)$ for all real $x, y$. If $f(2)=5$, find $f(5)$.

Let $f(x)=x+\frac{1}{2 x+\frac{1}{2 x+\frac{1}{2 x+}}}$ for $x>0$. Find $f(99) f^{\prime}(99)$.

Evaluate $\frac{d}{d x}\left(\sin x-\frac{4}{3} \sin ^{3} x\right)$ when $x=15$.

If a right triangle is drawn in a semicircle of radius $1 / 2$ with one leg (not the hypotenuse) along the diameter, what is the triangle's maximum possible area?

A circle is randomly chosen in a circle of radius 1 in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?

What fraction of the Earth's volume lies above the 45 degrees north parallel? You may assume the Earth is a perfect sphere. The volume in question is the smaller piece that we would get if the sphere were sliced into two pieces by a plane.

Let $A_{n}$ be the area outside a regular $n$-gon of side length 1 but inside its circumscribed circle, let $B_{n}$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\frac{A_{n}}{B_{n}}$.

One of the receipts for a math tournament showed that 72 identical trophies were purchased for $\$-99.9-$, where the first and last digits were illegible. How much did each trophy cost?

Stacy has $d$ dollars. She enters a mall with 10 shops and a lottery stall. First she goes to the lottery and her money is doubled, then she goes into the first shop and spends 1024 dollars. After that she alternates playing the lottery and getting her money doubled (Stacy always wins) then going into a new shop and spending $\$ 1024$. When she comes out of the last shop she has no money left. What is the minimum possible value of $d$ ?

An unfair coin has the property that when flipped four times, it has the same probability of turning up 2 heads and 2 tails (in any order) as 3 heads and 1 tail (in any order). What is the probability of getting a head in any one flip?

You are given 16 pieces of paper numbered $16,15, \ldots, 2,1$ in that order. You want to put them in the order $1,2, \ldots, 15,16$ switching only two adjacent pieces of paper at a time. What is the minimum number of switches necessary?

Find an ordered pair $(a, b)$ of real numbers for which $x^{2}+a x+b$ has a non-real root whose cube is 343 .

Let $C$ be a circle with two diameters intersecting at an angle of 30 degrees. A circle $S$ is tangent to both diameters and to $C$, and has radius 1 . Find the largest possible radius of $C$.

As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces 1 counterfeit coin for every 99 real ones. Alex isn't very good at this, so none of the counterfeit coins are the right weight. Since the mint is not perfect, each coin is weighed before leaving. If the coin is not the right weight, then it is sent to a lab for testing. The scale is accurate $95 \%$ of the time, $5 \%$ of all the coins minted are sent to the lab, and the lab's test is accurate $90 \%$ of the time. If the lab says a coin is counterfeit, what is the probability that it really is?

Find the minimum possible value of the largest of $x y, 1-x-y+x y$, and $x+y-2 x y$ if $0 \leq x \leq y \leq 1$.

A combination lock has a 3 number combination, with each number an integer between 0 and 39 inclusive. Call the numbers $n_{1}, n_{2}$, and $n_{3}$. If you know that $n_{1}$ and $n_{3}$ leave the same remainder when divided by 4 , and $n_{2}$ and $n_{1}+2$ leave the same remainder when divided by 4 , how many possible combinations are there?

A ladder is leaning against a house with its lower end 15 feet from the house. When the lower end is pulled 9 feet farther from the house, the upper end slides 13 feet down. How long is the ladder (in feet)?

How many non-empty subsets of $\{1,2,3,4,5,6,7,8\}$ have exactly $k$ elements and do not contain the element $k$ for some $k=1,2, \ldots, 8$.

Consider the equation $F O R T Y+T E N+T E N=S I X T Y$, where each of the ten letters represents a distinct digit from 0 to 9 . Find all possible values of SIXTY.

If $a$ and $b$ are randomly selected real numbers between 0 and 1 , find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd.

Reduce the number $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}$.

Let $\frac{1}{1-x-x^{2}-x^{3}}=\sum_{i=0}^{\infty} a_{n} x^{n}$, for what positive integers $n$ does $a_{n-1}=n^{2}$ ?

Find all the roots of $\left(x^{2}+3 x+2\right)\left(x^{2}-7 x+12\right)\left(x^{2}-2 x-1\right)+24=0$.

Evaluate $\sum_{n=2}^{17} \frac{n^{2}+n+1}{n^{4}+2 n^{3}-n^{2}-2 n}$.

If 5 points are placed in the plane at lattice points (i.e. points $(x, y)$ where $x$ and $y$ are both integers) such that no three are collinear, then there are 10 triangles whose vertices are among these points. What is the minimum possible number of these triangles that have area greater than $1 / 2$ ?

Circles $C_{1}, C_{2}, C_{3}$ have radius 1 and centers $O, P, Q$ respectively. $C_{1}$ and $C_{2}$ intersect at $A, C_{2}$ and $C_{3}$ intersect at $B, C_{3}$ and $C_{1}$ intersect at $C$, in such a way that $\angle A P B=60^{\circ}, \angle B Q C=36^{\circ}$, and $\angle C O A=72^{\circ}$. Find angle $A B C$ (degrees).

A fair coin is flipped every second and the results are recorded with 1 meaning heads and 0 meaning tails. What is the probability that the sequence 10101 occurs before the first occurance of the sequence 010101?

Start with an angle of $60^{\circ}$ and bisect it, then bisect the lower $30^{\circ}$ angle, then the upper $15^{\circ}$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^{\circ}$ angle into two angles. Find the measure (degrees) of the smaller angle.

Alex, Pei-Hsin, and Edward got together before the contest to send a mailing to all the invited schools. Pei-Hsin usually just stuffs the envelopes, but if Alex leaves the room she has to lick them as well and has a $25 \%$ chance of dying from an allergic reaction before he gets back. Licking the glue makes Edward a bit psychotic, so if Alex leaves the room there is a $20 \%$ chance that Edward will kill Pei-Hsin before she can start licking envelopes. Alex leaves the room and comes back to find Pei-Hsin dead. What is the probability that Edward was responsible?

If $x, y$, and $z$ are distinct positive integers such that $x^{2}+y^{2}=z^{3}$, what is the smallest possible value of $x+y+z$.

Evaluate $\sum_{n=0}^{\infty} \frac{\cos n \theta}{2^{n}}$, where $\cos \theta=\frac{1}{5}$.

Let $r$ be the radius of the inscribed circle of triangle $A B C$. Take a point $D$ on side $B C$, and let $r_{1}$ and $r_{2}$ be the inradii of triangles $A B D$ and $A C D$. Prove that $r, r_{1}$, and $r_{2}$ can always be the side lengths of a triangle.

You want to sort the numbers 54321 using block moves. In other words, you can take any set of numbers that appear consecutively and put them back in at any spot as a block. For example, $653421 \rightarrow 426531$ is a valid block move for 6 numbers. What is the minimum number of block moves necessary to get 12345 ?

Evaluate $\sum_{n=1}^{\infty} \frac{n^{5}}{n!}$.

What is the smallest square-free composite number that can divide a number of the form $4242 \ldots 42 \pm 1$ ?

You are somewhere on a ladder with 5 rungs. You have a fair coin and an envelope that contains either a double-headed coin or a double-tailed coin, each with probability $1 / 2$. Every minute you flip a coin. If it lands heads you go up a rung, if it lands tails you go down a rung. If you move up from the top rung you win, if you move down from the bottom rung you lose. You can open the envelope at any time, but if you do then you must immediately flip that coin once, after which you can use it or the fair coin whenever you want. What is the best strategy (i.e. on what rung(s) should you open the envelope)?

$A, B, C, D$, and $E$ are relatively prime integers (i.e., have no single common factor) such that the polynomials $5 A x^{4}+4 B x^{3}+3 C x^{2}+2 D x+E$ and $10 A x^{3}+6 B x^{2}+3 C x+D$ together have 7 distinct integer roots. What are all possible values of $A$ ? Your team has been given a sealed envelope that contains a hint for this problem. If you open the envelope, the value of this problem decreases by 20 points. To get full credit, give the sealed envelope to the judge before presenting your solution.