HMMT November 2014

Two circles $\omega$ and $\gamma$ have radii 3 and 4 respectively, and their centers are 10 units apart. Let $x$ be the shortest possible distance between a point on $\omega$ and a point on $\gamma$, and let $y$ be the longest possible distance between a point on $\omega$ and a point on $\gamma$. Find the product $x y$.

Let $A B C$ be a triangle with $\angle B=90^{\circ}$. Given that there exists a point $D$ on $A C$ such that $A D=D C$ and $B D=B C$, compute the value of the ratio $\frac{A B}{B C}$.

Compute the greatest common divisor of $4^{8}-1$ and $8^{12}-1$.

In rectangle $A B C D$ with area 1 , point $M$ is selected on $\overline{A B}$ and points $X, Y$ are selected on $\overline{C D}$ such that $A X<A Y$. Suppose that $A M=B M$. Given that the area of triangle $M X Y$ is $\frac{1}{2014}$, compute the area of trapezoid $A X Y B$.

Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by 2 and add 1 or he may multiply the stored value by 4 and add 3 . The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at 1 and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move? (By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)

Let $A B C$ be a triangle with $A B=5, A C=4, B C=6$. The angle bisector of $C$ intersects side $A B$ at $X$. Points $M$ and $N$ are drawn on sides $B C$ and $A C$, respectively, such that $\overline{X M} \| \overline{A C}$ and $\overline{X N} \| \overline{B C}$. Compute the length $M N$.

Consider the set of 5 -tuples of positive integers at most 5 . We say the tuple ( $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ ) is perfect if for any distinct indices $i, j, k$, the three numbers $a_{i}, a_{j}, a_{k}$ do not form an arithmetic progression (in any order). Find the number of perfect 5 -tuples.

Let $a, b, c, x$ be reals with $(a+b)(b+c)(c+a) \neq 0$ that satisfy

$$\frac{a^{2}}{a+b}=\frac{a^{2}}{a+c}+20, \frac{b^{2}}{b+c}=\frac{b^{2}}{b+a}+14, \text { and } \frac{c^{2}}{c+a}=\frac{c^{2}}{c+b}+x$$

Compute $x$.

For any positive integers $a$ and $b$, define $a \oplus b$ to be the result when adding $a$ to $b$ in binary (base 2), neglecting any carry-overs. For example, $20 \oplus 14=10100_{2} \oplus 1110_{2}=11010_{2}=26$. (The operation $\oplus$ is called the exclusive or.) Compute the sum

$$\sum_{k=0}^{2^{2014}-1}\left(k \oplus\left\lfloor\frac{k}{2}\right\rfloor\right)$$

Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.

Suppose that $m$ and $n$ are integers with $1 \leq m \leq 49$ and $n \geq 0$ such that $m$ divides $n^{n+1}+1$. What is the number of possible values of $m$ ?

Find the probability that the townspeople win if there are initially two townspeople and one goon.

Find the smallest positive integer $n$ such that, if there are initially $2 n$ townspeople and 1 goon, then the probability the townspeople win is greater than $50 \%$.

Find the smallest positive integer $n$ such that, if there are initially $n+1$ townspeople and $n$ goons, then the probability the townspeople win is less than $1 \%$.

Suppose there are initially 1001 townspeople and two goons. What is the probability that, when the game ends, there are exactly 1000 people in jail?

Suppose that there are initially eight townspeople and one goon. One of the eight townspeople is named Jester. If Jester is sent to jail during some morning, then the game ends immediately in his sole victory. (However, the Jester does not win if he is sent to jail during some night.) Find the probability that only the Jester wins.

Let $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$ be pairwise distinct parabolas in the plane. Find the maximum possible number of intersections between two or more of the $\mathcal{P}_{i}$. In other words, find the maximum number of points that can lie on two or more of the parabolas $\mathcal{P}_{1}, \mathcal{P}_{2}, \mathcal{P}_{3}$.

Let $\mathcal{P}$ be a parabola with focus $F$ and directrix $\ell$. A line through $F$ intersects $\mathcal{P}$ at two points $A$ and $B$. Let $D$ and $C$ be the feet of the altitudes from $A$ and $B$ onto $\ell$, respectively. Given that $A B=20$ and $C D=14$, compute the area of $A B C D$.

Consider the parabola consisting of the points $(x, y)$ in the real plane satisfying

$$(y+x)=(y-x)^{2}+3(y-x)+3 .$$

Find the minimum possible value of $y$.

In equilateral triangle $A B C$ with side length 2 , let the parabola with focus $A$ and directrix $B C$ intersect sides $A B$ and $A C$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $C A$ intersect sides $B C$ and $B A$ at $B_{1}$ and $B_{2}$, respectively. Finally, let the parabola with focus $C$ and directrix $A B$ intersect sides $C A$ and $C B$ at $C_{1}$ and $C_{2}$, respectively. Find the perimeter of the triangle formed by lines $A_{1} A_{2}, B_{1} B_{2}, C_{1} C_{2}$.

Let $z$ be a complex number and $k$ a positive integer such that $z^{k}$ is a positive real number other than 1. Let $f(n)$ denote the real part of the complex number $z^{n}$. Assume the parabola $p(n)=a n^{2}+b n+c$ intersects $f(n)$ four times, at $n=0,1,2,3$. Assuming the smallest possible value of $k$, find the largest possible value of $a$.

What is the smallest positive integer $n$ which cannot be written in any of the following forms?

• $n=1+2+\cdots+k$ for a positive integer $k$.
• $n=p^{k}$ for a prime number $p$ and integer $k$.
• $n=p+1$ for a prime number $p$.

Let $f(x)=x^{2}+6 x+7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x$.

The side lengths of a triangle are distinct positive integers. One of the side lengths is a multiple of 42 , and another is a multiple of 72 . What is the minimum possible length of the third side?

How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.

Let $A, B, C, D, E$ be five points on a circle; some segments are drawn between the points so that each of the $\binom{5}{2}=10$ pairs of points is connected by either zero or one segments. Determine the number of sets of segments that can be drawn such that:

• It is possible to travel from any of the five points to any other of the five points along drawn segments.
• It is possible to divide the five points into two nonempty sets $S$ and $T$ such that each segment has one endpoint in $S$ and the other endpoint in $T$.

Find the number of strictly increasing sequences of nonnegative integers with the following properties:

• The first term is 0 and the last term is 12 . In particular, the sequence has at least two terms.
• Among any two consecutive terms, exactly one of them is even.

Sammy has a wooden board, shaped as a rectangle with length $2^{2014}$ and height $3^{2014}$. The board is divided into a grid of unit squares. A termite starts at either the left or bottom edge of the rectangle, and walks along the gridlines by moving either to the right or upwards, until it reaches an edge opposite the one from which the termite started. Depicted below are two possible paths of the termite.

The termite's path dissects the board into two parts. Sammy is surprised to find that he can still arrange the pieces to form a new rectangle not congruent to the original rectangle. This rectangle has perimeter $P$. How many possible values of $P$ are there?

Let $\mathcal{H}$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $\mathcal{H}$, then draws the largest circle with center $P$ that is contained in $\mathcal{H}$. What is this probability that the radius of this circle is less than $\frac{1}{2}$ ?

How many lines pass through exactly two points in the following hexagonal grid?

Let $A B C D E F$ be a convex hexagon with the following properties. (a) $\overline{A C}$ and $\overline{A E}$ trisect $\angle B A F$. (b) $\overline{B E} \| \overline{C D}$ and $\overline{C F} \| \overline{D E}$. (c) $A B=2 A C=4 A E=8 A F$.

Suppose that quadrilaterals $A C D E$ and $A D E F$ have area 2014 and 1400, respectively. Find the area of quadrilateral $A B C D$.

Solve for $x$ in the equation $20 \cdot 14+x=20+14 \cdot x$.

Find the area of a triangle with side lengths 14,48 , and 50.

Victoria wants to order at least 550 donuts from Dunkin' Donuts for the HMMT 2014 November contest. However, donuts only come in multiples of twelve. Assuming every twelve donuts cost $\$ 7.49$, what is the minimum amount Victoria needs to pay, in dollars? (Because HMMT is affiliated with MIT, the purchase is tax exempt. Moreover, because of the size of the order, there is no delivery fee.)

How many two-digit prime numbers have the property that both digits are also primes?

Suppose that $x, y, z$ are real numbers such that

$$x=y+z+2, y=z+x+1, \text { and } z=x+y+4$$

Compute $x+y+z$.

In the octagon COMPUTER exhibited below, all interior angles are either $90^{\circ}$ or $270^{\circ}$ and we have $C O=O M=M P=P U=U T=T E=1$.

Point $D$ (not to scale in the diagram) is selected on segment $R E$ so that polygons COMPUTED and $C D R$ have the same area. Find $D R$.

Let $A B C D$ be a quadrilateral inscribed in a circle with diameter $\overline{A D}$. If $A B=5, A C=6$, and $B D=7$, find $C D$.

Find the number of digits in the decimal representation of $2^{41}$.

Let $f$ be a function from the nonnegative integers to the positive reals such that $f(x+y)=f(x) \cdot f(y)$ holds for all nonnegative integers $x$ and $y$. If $f(19)=524288 k$, find $f(4)$ in terms of $k$.

Let $A B C$ be a triangle with $C A=C B=5$ and $A B=8$. A circle $\omega$ is drawn such that the interior of triangle $A B C$ is completely contained in the interior of $\omega$. Find the smallest possible area of $\omega$.

How many integers $n$ in the set $\{4,9,14,19, \ldots, 2014\}$ have the property that the sum of the decimal digits of $n$ is even?

Sindy writes down the positive integers less than 200 in increasing order, but skips the multiples of 10. She then alternately places + and - signs before each of the integers, yielding an expression $+1-2+3-4+5-6+7-8+9-11+12-\cdots-199$. What is the value of the resulting expression?

Let $A B C$ be a triangle with $A B=A C=\frac{25}{14} B C$. Let $M$ denote the midpoint of $\overline{B C}$ and let $X$ and $Y$ denote the projections of $M$ onto $\overline{A B}$ and $\overline{A C}$, respectively. If the areas of triangle $A B C$ and quadrilateral $A X M Y$ are both positive integers, find the minimum possible sum of these areas.

How many ways can the eight vertices of a three-dimensional cube be colored red and blue such that no two points connected by an edge are both red? Rotations and reflections of a given coloring are considered distinct.

Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex (each with probability $\frac{1}{2}$ ) and walks along the edge to it. What is the probability that after 10 minutes, he ends up where he had started?

A particular coin has a $\frac{1}{3}$ chance of landing on heads (H), $\frac{1}{3}$ chance of landing on tails ( T ), and $\frac{1}{3}$ chance of landing vertically in the middle (M). When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT?

Let $A B C$ be a triangle with $A B=A C=5$ and $B C=6$. Denote by $\omega$ the circumcircle of $A B C$. We draw a circle $\Omega$ which is externally tangent to $\omega$ as well as to the lines $A B$ and $A C$ (such a circle is called an $A$-mixtilinear excircle). Find the radius of $\Omega$.

For any positive integer $x$, define $\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \in$ $\{0,2,4,5,7,9,11\}$ and $t \in\{1,3,6,8,10\}$ such that $x+s-t$ is divisible by 12 . For any nonnegative integer $i$, let $a_{i}$ denote the number of $x \in\{0,1, \ldots, 11\}$ for which $|\operatorname{Accident}(x)|=i$. Find

$$a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}$$

Let a sequence $\left\{a_{n}\right\}_{n=0}^{\infty}$ be defined by $a_{0}=\sqrt{2}, a_{1}=2$, and $a_{n+1}=a_{n} a_{n-1}^{2}$ for $n \geq 1$. The sequence of remainders when $a_{0}, a_{1}, a_{2}, \cdots$ are divided by 2014 is eventually periodic with some minimal period $p$ (meaning that $a_{m}=a_{m+p}$ for all sufficiently large integers $m$, and $p$ is the smallest such positive integer). Find $p$.

Determine the number of sequences of sets $S_{1}, S_{2}, \ldots, S_{999}$ such that

$$S_{1} \subseteq S_{2} \subseteq \cdots \subseteq S_{999} \subseteq\{1,2, \ldots, 999\}$$

Here $A \subseteq B$ means that all elements of $A$ are also elements of $B$.

If you flip a fair coin 1000 times, what is the expected value of the product of the number of heads and the number of tails?

Evaluate the infinite sum

$$\sum_{n=2}^{\infty} \log _{2}\left(\frac{1-\frac{1}{n}}{1-\frac{1}{n+1}}\right)$$

Seven little children sit in a circle. The teacher distributes pieces of candy to the children in such a way that the following conditions hold.

• Every little child gets at least one piece of candy.
• No two little children have the same number of pieces of candy.
• The numbers of candy pieces given to any two adjacent little children have a common factor other than 1.
• There is no prime dividing every little child's number of candy pieces.

What is the smallest number of pieces of candy that the teacher must have ready for the little children?

Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. We construct isosceles right triangle $A C D$ with $\angle A D C=90^{\circ}$, where $D, B$ are on the same side of line $A C$, and let lines $A D$ and $C B$ meet at $F$. Similarly, we construct isosceles right triangle $B C E$ with $\angle B E C=90^{\circ}$, where $E, A$ are on the same side of line $B C$, and let lines $B E$ and $C A$ meet at $G$. Find $\cos \angle A G F$.

What is the smallest positive integer $n$ which cannot be written in any of the following forms?

• $n=1+2+\cdots+k$ for a positive integer $k$.
• $n=p^{k}$ for a prime number $p$ and integer $k$.
• $n=p+1$ for a prime number $p$.
• $n=p q$ for some distinct prime numbers $p$ and $q$

Consider a permutation $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ of $\{1,2,3,4,5\}$. We say the tuple $\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ is flawless if for all $1 \leq i<j<k \leq 5$, the sequence $\left(a_{i}, a_{j}, a_{k}\right)$ is not an arithmetic progression (in that order). Find the number of flawless 5 -tuples.

In triangle $A B C$, let the parabola with focus $A$ and directrix $B C$ intersect sides $A B$ and $A C$ at $A_{1}$ and $A_{2}$, respectively. Similarly, let the parabola with focus $B$ and directrix $C A$ intersect sides $B C$ and $B A$ at $B_{1}$ and $B_{2}$, respectively. Finally, let the parabola with focus $C$ and directrix $A B$ intersect sides $C A$ and $C B$ at $C_{1}$ and $C_{2}$, respectively.

If triangle $A B C$ has sides of length 5,12 , and 13 , find the area of the triangle determined by lines $A_{1} C_{2}, B_{1} A_{2}$ and $C_{1} B_{2}$.

Let $x$ be a complex number such that $x+x^{-1}$ is a root of the polynomial $p(t)=t^{3}+t^{2}-2 t-1$. Find all possible values of $x^{7}+x^{-7}$.

Let $\omega$ be a fixed circle with radius 1 , and let $B C$ be a fixed chord of $\omega$ such that $B C=1$. The locus of the incenter of $A B C$ as $A$ varies along the circumference of $\omega$ bounds a region $\mathcal{R}$ in the plane. Find the area of $\mathcal{R}$.

Suppose we keep rolling a fair 2014-sided die (whose faces are labelled $1,2, \ldots, 2014$ ) until we obtain a value less than or equal to the previous roll. Let $E$ be the expected number of times we roll the die. Find the nearest integer to $100 E$.

Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm ; the opening of the cup corresponds to the base, which points upwards. Every minute after the game begins, the following takes place: if $n$ minutes have elapsed, Albert stirs his drink vigorously and takes a sip of height $\frac{1}{n^{2}} \mathrm{~cm}$. Shortly afterwards, while Albert is busy watching the game, Mike adds cranberry juice to the cup until it's once again full in an attempt to create Mike's cranberry lemonade. Albert takes sips precisely every minute, and his first sip is exactly one minute after the game begins. After an infinite amount of time, let $A$ denote the amount of cranberry juice that has been poured (in square centimeters). Find the integer nearest $\frac{27}{\pi^{2}} A$.

Let $f(x)=x^{2}-2$, and let $f^{n}$ denote the function $f$ applied $n$ times. Compute the remainder when $f^{24}(18)$ is divided by 89.

How many ways can you remove one tile from a $2014 \times 2014$ grid such that the resulting figure can be tiled by $1 \times 3$ and $3 \times 1$ rectangles?

Let $M$ denote the number of positive integers which divide 2014!, and let $N$ be the integer closest to $\ln (M)$. Estimate the value of $N$. If your answer is a positive integer $A$, your score on this problem will be the larger of 0 and $\left\lfloor 20-\frac{1}{8}|A-N|\right\rfloor$. Otherwise, your score will be zero.

Ten points are equally spaced on a circle. A graph is a set of segments (possibly empty) drawn between pairs of points, so that every two points are joined by either zero or one segments. Two graphs are considered the same if we can obtain one from the other by rearranging the points. Let $N$ denote the number of graphs with the property that for any two points, there exists a path from one to the other among the segments of the graph. Estimate the value of $N$. If your answer is a positive integer $A$, your score on this problem will be the larger of 0 and $\lfloor 20-5|\ln (A / N)|\rfloor$. Otherwise, your score will be zero.

Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named. C. (Ceva) Three cevians $A D, B E, C F$ of a triangle $A B C$ are concurrent if and only if $\frac{B D}{D C} \frac{C E}{E A} \frac{A F}{F B}=1$. E. (Euler) In a triangle $A B C$ with incenter $I$ and circumcenter $O$, we have $I O^{2}=R(R-2 r)$, where $r$ is the inradius and $R$ is the circumradius of $A B C$. H. (Heron) The area of a triangle $A B C$ is $\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{1}{2}(a+b+c)$. M. (Menelaus) If $D, E, F$ lie on lines $B C, C A, A B$, then they are collinear if and only if $\frac{B D}{D C} \frac{C E}{E A} \frac{A F}{F B}=$ -1 , where the ratios are directed. P. (Pascal) Intersections of opposite sides of cyclic hexagons are collinear. S. (Stewart) Let $A B C$ be a triangle and $D$ a point on $B C$. Set $m=B D, n=C D, d=A D$. Then $m a n+d a d=b m b+c n c$. V. (Varignon) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram.

If your answer is a list of $4 \leq N \leq 7$ labels in a correct order, your score will be $(N-2)(N-3)$. Otherwise, your score will be zero.