HMMT November 2013

What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?

Plot points $A, B, C$ at coordinates $(0,0),(0,1)$, and $(1,1)$ in the plane, respectively. Let $S$ denote the union of the two line segments $A B$ and $B C$. Let $X_{1}$ be the area swept out when Bobby rotates $S$ counterclockwise 45 degrees about point $A$. Let $X_{2}$ be the area swept out when Calvin rotates $S$ clockwise 45 degrees about point $A$. Find $\frac{X_{1}+X_{2}}{2}$.

A 24-hour digital clock shows times $h: m: s$, where $h, m$, and $s$ are integers with $0 \leq h \leq 23$, $0 \leq m \leq 59$, and $0 \leq s \leq 59$. How many times $h: m: s$ satisfy $h+m=s$ ?

A 50 -card deck consists of 4 cards labeled " $i$ " for $i=1,2, \ldots, 12$ and 2 cards labeled " 13 ". If Bob randomly chooses 2 cards from the deck without replacement, what is the probability that his 2 cards have the same label?

Let $A B C$ be an isosceles triangle with $A B=A C$. Let $D$ and $E$ be the midpoints of segments $A B$ and $A C$, respectively. Suppose that there exists a point $F$ on ray $\overrightarrow{D E}$ outside of $A B C$ such that triangle $B F A$ is similar to triangle $A B C$. Compute $\frac{A B}{B C}$.

Find the number of positive integer divisors of 12 ! that leave a remainder of 1 when divided by 3 .

Find the largest real number $\lambda$ such that $a^{2}+b^{2}+c^{2}+d^{2} \geq a b+\lambda b c+c d$ for all real numbers $a, b, c, d$.

How many of the first 1000 positive integers can be written as the sum of finitely many distinct numbers from the sequence $3^{0}, 3^{1}, 3^{2}, \ldots$ ?

Let $A B C$ be a triangle and $D$ a point on $B C$ such that $A B=\sqrt{2}, A C=\sqrt{3}, \angle B A D=30^{\circ}$, and $\angle C A D=45^{\circ}$. Find $A D$.

How many functions $f:\{1,2, \ldots, 2013\} \rightarrow\{1,2, \ldots, 2013\}$ satisfy $f(j)<f(i)+j-i$ for all integers $i, j$ such that $1 \leq i<j \leq 2013 ?$

Two cars are driving directly towards each other such that one is twice as fast as the other. The distance between their starting points is 4 miles. When the two cars meet, how many miles is the faster car from its starting point?

You are standing at a pole and a snail is moving directly away from the pole at $1 \mathrm{~cm} / \mathrm{s}$. When the snail is 1 meter away, you start "Round 1 ". In Round $n(n \geq 1)$, you move directly toward the snail at $n+1 \mathrm{~cm} / \mathrm{s}$. When you reach the snail, you immediately turn around and move back to the starting pole at $n+1 \mathrm{~cm} / \mathrm{s}$. When you reach the pole, you immediately turn around and Round $n+1$ begins. At the start of Round 100, how many meters away is the snail?

Let $A B C$ be a triangle with $A B=5, B C=4$, and $C A=3$. Initially, there is an ant at each vertex. The ants start walking at a rate of 1 unit per second, in the direction $A \rightarrow B \rightarrow C \rightarrow A$ (so the ant starting at $A$ moves along ray $\overrightarrow{A B}$, etc.). For a positive real number $t$ less than 3 , let $A(t)$ be the area of the triangle whose vertices are the positions of the ants after $t$ seconds have elapsed. For what positive real number $t$ less than 3 is $A(t)$ minimized?

There are 2 runners on the perimeter of a regular hexagon, initially located at adjacent vertices. Every second, each of the runners independently moves either one vertex to the left, with probability $\frac{1}{2}$, or one vertex to the right, also with probability $\frac{1}{2}$. Find the probability that after a 2013 second run (in which the runners switch vertices 2013 times each), the runners end up at adjacent vertices once again.

Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Company XYZ wants to locate their base at the point $P$ in the plane minimizing the total distance to their workers, who are located at vertices $A, B$, and $C$. There are 1,5 , and 4 workers at $A, B$, and $C$, respectively. Find the minimum possible total distance Company XYZ's workers have to travel to get to $P$.

Let $b(n)$ be the number of digits in the base -4 representation of $n$. Evaluate $\sum_{i=1}^{2013} b(i)$.

Let $N$ be the largest positive integer that can be expressed as a 2013-digit base -4 number. What is the remainder when $N$ is divided by 210 ?

Find the sum of all positive integers $n$ such that there exists an integer $b$ with $|b| \neq 4$ such that the base -4 representation of $n$ is the same as the base $b$ representation of $n$.

Tim the Beaver can make three different types of geometrical figures: squares, regular hexagons, and regular octagons. Tim makes a random sequence $F_{0}, F_{1}, F_{2}, F_{3}, \ldots$ of figures as follows:

• $F_{0}$ is a square.
• For every positive integer $i, F_{i}$ is randomly chosen to be one of the 2 figures distinct from $F_{i-1}$ (each chosen with equal probability $\frac{1}{2}$ ).
• Tim takes 4 seconds to make squares, 6 to make hexagons, and 8 to make octagons. He makes one figure after another, with no breaks in between.

Suppose that exactly 17 seconds after he starts making $F_{0}$, Tim is making a figure with $n$ sides. What is the expected value of $n$ ?

Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between 1 and $n$, inclusive: if $n=1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In particular, if he gets $k=1$, he will stop rolling the die.) If he starts out with a 6 -sided die, what is the expected number of rolls he makes?

The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\overline{A B C}$ and $N=\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \cdot N$.

Consider triangle $A B C$ with side lengths $A B=4, B C=7$, and $A C=8$. Let $M$ be the midpoint of segment $A B$, and let $N$ be the point on the interior of segment $A C$ that also lies on the circumcircle of triangle $M B C$. Compute $B N$.

In triangle $A B C, \angle B A C=60^{\circ}$. Let $\omega$ be a circle tangent to segment $A B$ at point $D$ and segment $A C$ at point $E$. Suppose $\omega$ intersects segment $B C$ at points $F$ and $G$ such that $F$ lies in between $B$ and $G$. Given that $A D=F G=4$ and $B F=\frac{1}{2}$, find the length of $C G$.

Points $A, B, C$ lie on a circle $\omega$ such that $B C$ is a diameter. $A B$ is extended past $B$ to point $B^{\prime}$ and $A C$ is extended past $C$ to point $C^{\prime}$ such that line $B^{\prime} C^{\prime}$ is parallel to $B C$ and tangent to $\omega$ at point $D$. If $B^{\prime} D=4$ and $C^{\prime} D=6$, compute $B C$.

In equilateral triangle $A B C$, a circle $\omega$ is drawn such that it is tangent to all three sides of the triangle. A line is drawn from $A$ to point $D$ on segment $B C$ such that $A D$ intersects $\omega$ at points $E$ and $F$. If $E F=4$ and $A B=8$, determine $|A E-F D|$.

Define the sequence $\left\{x_{i}\right\}_{i \geq 0}$ by $x_{0}=x_{1}=x_{2}=1$ and $x_{k}=\frac{x_{k-1}+x_{k-2}+1}{x_{k-3}}$ for $k>2$. Find $x_{2013}$.

For an integer $n \geq 0$, let $f(n)$ be the smallest possible value of $|x+y|$, where $x$ and $y$ are integers such that $3 x-2 y=n$. Evaluate $f(0)+f(1)+f(2)+\cdots+f(2013)$.

Let $\omega=\cos \frac{2 \pi}{727}+i \sin \frac{2 \pi}{727}$. The imaginary part of the complex number

$$\prod_{k=8}^{13}\left(1+\omega^{3^{k-1}}+\omega^{2 \cdot 3^{k-1}}\right)$$

is equal to $\sin \alpha$ for some angle $\alpha$ between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$, inclusive. Find $\alpha$.

Two fair six-sided dice are rolled. What is the probability that their sum is at least 10 ?

A square is inscribed in a circle of radius 1. Find the perimeter of the square.

Find the minimum possible value of $\left(x^{2}+6 x+2\right)^{2}$ over all real numbers $x$.

How many positive integers less than 100 are relatively prime to 200? (Two numbers are relatively prime if their greatest common factor is 1 .)

A right triangle has area 5 and a hypotenuse of length 5 . Find its perimeter.

Marty and three other people took a math test. Everyone got a non-negative integer score. The average score was 20. Marty was told the average score and concluded that everyone else scored below average. What was the minimum possible score Marty could have gotten in order to definitively reach this conclusion?

Evaluate the expression where the digit 2 appears 2013 times.

Find the remainder when $1^{2}+3^{2}+5^{2}+\cdots+99^{2}$ is divided by 1000 .

How many pairs of real numbers $(x, y)$ satisfy the equation

$$y^{4}-y^{2}=x y^{3}-x y=x^{3} y-x y=x^{4}-x^{2}=0 ?$$

David has a unit triangular array of 10 points, 4 on each side. A looping path is a sequence $A_{1}, A_{2}, \ldots, A_{10}$ containing each of the 10 points exactly once, such that $A_{i}$ and $A_{i+1}$ are adjacent (exactly 1 unit apart) for $i=1,2, \ldots, 10$. (Here $A_{11}=A_{1}$.) Find the number of looping paths in this array.

Given that $62^{2}+122^{2}=18728$, find positive integers $(n, m)$ such that $n^{2}+m^{2}=9364$.

Let $S=\{1,2, \ldots, 2013\}$. Find the number of ordered triples $(A, B, C)$ of subsets of $S$ such that $A \subseteq B$ and $A \cup B \cup C=S$.

Find all triples of positive integers $(x, y, z)$ such that $x^{2}+y-z=100$ and $x+y^{2}-z=124$.

Find all real numbers $x$ between 0 and 360 such that $\sqrt{3} \cos 10^{\circ}=\cos 40^{\circ}+\sin x^{\circ}$.

A bug is on one exterior vertex of solid $S$, a $3 \times 3 \times 3$ cube that has its center $1 \times 1 \times 1$ cube removed, and wishes to travel to the opposite exterior vertex. Let $O$ denote the outer surface of $S$ (formed by the surface of the $3 \times 3 \times 3$ cube). Let $L(S)$ denote the length of the shortest path through $S$. (Note that such a path cannot pass through the missing center cube, which is empty space.) Let $L(O)$ denote the length of the shortest path through $O$. What is the ratio $\frac{L(S)}{L(O)}$ ?

Find the sum of $\frac{1}{n}$ over all positive integers $n$ with the property that the decimal representation of $\frac{1}{n}$ terminates.

The rightmost nonzero digit in the decimal expansion of 101 ! is the same as the rightmost nonzero digit of $n!$, where $n$ is an integer greater than 101. Find the smallest possible value of $n$.

Let $p, q, r, s$ be distinct primes such that $p q-r s$ is divisible by 30 . Find the minimum possible value of $p+q+r+s$.

There exist unique nonnegative integers $A, B$ between 0 and 9 , inclusive, such that

$$(1001 \cdot A+110 \cdot B)^{2}=57,108,249$$

Find $10 \cdot A+B$.

Suppose $A, B, C$, and $D$ are four circles of radius $r>0$ centered about the points $(0, r),(r, 0)$, $(0,-r)$, and $(-r, 0)$ in the plane. Let $O$ be a circle centered at $(0,0)$ with radius $2 r$. In terms of $r$, what is the area of the union of circles $A, B, C$, and $D$ subtracted by the area of circle $O$ that is not contained in the union of $A, B, C$, and $D$ ? (The union of two or more regions in the plane is the set of points lying in at least one of the regions.)

Let $S$ be a subset of $\{1,2,3, \ldots, 12\}$ such that it is impossible to partition $S$ into $k$ disjoint subsets, each of whose elements sum to the same value, for any integer $k \geq 2$. Find the maximum possible sum of the elements of $S$.

The number $989 \cdot 1001 \cdot 1007+320$ can be written as the product of three distinct primes $p, q, r$ with $p<q<r$. Find $(p, q, r)$.

Find the number of subsets $S$ of $\{1,2, \ldots 6\}$ satisfying the following conditions:

• $S$ is non-empty.
• No subset of $S$ has the property that the sum of its elements is 10 .

Let $a, b$ be positive reals with $a>b>\frac{1}{2} a$. Place two squares of side lengths $a, b$ next to each other, such that the larger square has lower left corner at $(0,0)$ and the smaller square has lower left corner at $(a, 0)$. Draw the line passing through $(0, a)$ and $(a+b, 0)$. The region in the two squares lying above the line has area 2013. If $(a, b)$ is the unique pair maximizing $a+b$, compute $\frac{a}{b}$.

Find all triples of real numbers $(a, b, c)$ such that $a^{2}+2 b^{2}-2 b c=16$ and $2 a b-c^{2}=16$.

Triangle $A B C$ has $A B=4, B C=3$, and a right angle at $B$. Circles $\omega_{1}$ and $\omega_{2}$ of equal radii are drawn such that $\omega_{1}$ is tangent to $A B$ and $A C, \omega_{2}$ is tangent to $B C$ and $A C$, and $\omega_{1}$ is tangent to $\omega_{2}$. Find the radius of $\omega_{1}$. of $\omega_{1}$ and $\omega_{2}$ respectively. Suppose $\omega_{i}$ hits $A C$ at $B_{i}$ for $i=1,2$, so that $O_{1} O_{2}=B_{1} B_{2}=2 r$.

Extend angle bisector $A O_{1}$ to hit $B C$ at $P$. By the angle bisector theorem and triangle similarity $\triangle A B_{1} O_{1} \sim \triangle A B P$, we deduce $\frac{r}{A B_{1}}=\frac{B P}{A B}=\frac{3}{4+5}$. Similarly, $\frac{r}{C B_{2}}=\frac{4}{3+5}$, so

$$5=A C=A B_{1}+B_{1} B_{2}+B_{2} C=3 r+2 r+2 r=7 r$$

or $r=\frac{5}{7}$.

Let $\triangle X Y Z$ be a right triangle with $\angle X Y Z=90^{\circ}$. Suppose there exists an infinite sequence of equilateral triangles $X_{0} Y_{0} T_{0}, X_{1} Y_{1} T_{1}, \ldots$ such that $X_{0}=X, Y_{0}=Y, X_{i}$ lies on the segment $X Z$ for all $i \geq 0, Y_{i}$ lies on the segment $Y Z$ for all $i \geq 0, X_{i} Y_{i}$ is perpendicular to $Y Z$ for all $i \geq 0, T_{i}$ and $Y$ are separated by line $X Z$ for all $i \geq 0$, and $X_{i}$ lies on segment $Y_{i-1} T_{i-1}$ for $i \geq 1$. Let $\mathcal{P}$ denote the union of the equilateral triangles. If the area of $\mathcal{P}$ is equal to the area of $X Y Z$, find $\frac{X Y}{Y Z}$.

Find the number of ordered triples of integers $(a, b, c)$ with $1 \leq a, b, c \leq 100$ and $a^{2} b+b^{2} c+c^{2} a=$ $a b^{2}+b c^{2}+c a^{2}$.

Chords $\overline{A B}$ and $\overline{C D}$ of circle $\omega$ intersect at $E$ such that $A E=8, B E=2, C D=10$, and $\angle A E C=90^{\circ}$. Let $R$ be a rectangle inside $\omega$ with sides parallel to $\overline{A B}$ and $\overline{C D}$, such that no point in the interior of $R$ lies on $\overline{A B}, \overline{C D}$, or the boundary of $\omega$. What is the maximum possible area of $R$ ?

Suppose that $x$ and $y$ are chosen randomly and uniformly from $(0,1)$. What is the probability that $\left\lfloor\sqrt{\frac{x}{y}}\right\rfloor$ is even?

On each side of a 6 by 8 rectangle, construct an equilateral triangle with that side as one edge such that the interior of the triangle intersects the interior of the rectangle. What is the total area of all regions that are contained in exactly 3 of the 4 equilateral triangles?

Find the number of positive integers less than 1000000 that are divisible by some perfect cube greater than 1. Your score will be $\max \left\{0,\left\lfloor 20-200\left|1-\frac{k}{S}\right|\right\rfloor\right\}$, where $k$ is your answer and $S$ is the actual answer.

Consider the following 4 by 4 grid with one corner square removed:

You may start at any square in this grid and at each move, you may either stop or travel to an adjacent square (sharing a side, not just a corner) that you have not already visited (the square you start at is automatically marked as visited). Determine the distinct number of paths you can take. Your score will be max $\left\{0,\left\lfloor 20-200\left|1-\frac{k}{S}\right|\right\rfloor\right\}$, where $k$ is your answer and $S$ is the actual answer.

Pick a subset of at least four of the following seven numbers, order them from least to greatest, and write down their labels (corresponding letters from A through G) in that order: (A) $\pi$; (B) $\sqrt{2}+\sqrt{3} ;(\mathrm{C}) \sqrt{10}$; (D) $\frac{355}{113}$; (E) $16 \tan ^{-1} \frac{1}{5}-4 \tan ^{-1} \frac{1}{240} ;(\mathrm{F}) \ln (23)$; and (G) $2^{\sqrt{e}}$. If the ordering of the numbers you picked is correct and you picked at least 4 numbers, then your score for this problem will be $(N-2)(N-3)$, where $N$ is the size of your subset; otherwise, your score is 0 .