HMMT November 2020

In the Cartesian plane, a line segment with midpoint $(2020,11)$ has one endpoint at $(a, 0)$ and the other endpoint on the line $y=x$. Compute $a$.

Let $T$ be a trapezoid with two right angles and side lengths $4,4,5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into 4 regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240 d$.

Jody has 6 distinguishable balls and 6 distinguishable sticks, all of the same length. How many ways are there to use the sticks to connect the balls so that two disjoint non-interlocking triangles are formed? Consider rotations and reflections of the same arrangement to be indistinguishable.

Nine fair coins are flipped independently and placed in the cells of a 3 by 3 square grid. Let $p$ be the probability that no row has all its coins showing heads and no column has all its coins showing tails. If $p=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

Compute the sum of all positive integers $a \leq 26$ for which there exist integers $b$ and $c$ such that $a+23 b+15 c-2$ and $2 a+5 b+14 c-8$ are both multiples of 26 .

A sphere is centered at a point with integer coordinates and passes through the three points $(2,0,0)$, $(0,4,0),(0,0,6)$, but not the origin $(0,0,0)$. If $r$ is the smallest possible radius of the sphere, compute $r^{2}$.

In triangle $A B C$ with $A B=8$ and $A C=10$, the incenter $I$ is reflected across side $A B$ to point $X$ and across side $A C$ to point $Y$. Given that segment $X Y$ bisects $A I$, compute $B C^{2}$. (The incenter $I$ is the center of the inscribed circle of triangle $A B C$.)

A bar of chocolate is made of 10 distinguishable triangles as shown below:

How many ways are there to divide the bar, along the edges of the triangles, into two or more contiguous pieces?

In the Cartesian plane, a perfectly reflective semicircular room is bounded by the upper half of the unit circle centered at $(0,0)$ and the line segment from $(-1,0)$ to $(1,0)$. David stands at the point $(-1,0)$ and shines a flashlight into the room at an angle of $46^{\circ}$ above the horizontal. How many times does the light beam reflect off the walls before coming back to David at $(-1,0)$ for the first time?

A sequence of positive integers $a_{1}, a_{2}, a_{3}, \ldots$ satisfies

$$a_{n+1}=n\left\lfloor\frac{a_{n}}{n}\right\rfloor+1$$

for all positive integers $n$. If $a_{30}=30$, how many possible values can $a_{1}$ take? (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is not greater than $x$.)

Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among $n$ bags, so that each bag contains at most 13 pounds of fruit. What is the minimum $n$ for which this is possible?

In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is 160 meters tall, the Stata Center is 90 meters tall, and the two buildings are 120 meters apart. Furthermore, both zipline at 10 meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100 a$.

Harvard has recently built a new house for its students consisting of $n$ levels, where the $k$ th level from the top can be modeled as a 1-meter-tall cylinder with radius $k$ meters. Given that the area of all the lateral surfaces (i.e. the surfaces of the external vertical walls) of the building is 35 percent of the total surface area of the building (including the bottom), compute $n$.

Points $G$ and $N$ are chosen on the interiors of sides $E D$ and $D O$ of unit square $D O M E$, so that pentagon $G N O M E$ has only two distinct side lengths. The sum of all possible areas of quadrilateral $N O M E$ can be expressed as $\frac{a-b \sqrt{c}}{d}$, where $a, b, c, d$ are positive integers such that $\operatorname{gcd}(a, b, d)=1$ and $c$ is square-free (i.e. no perfect square greater than 1 divides $c$ ). Compute $1000 a+100 b+10 c+d$.

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign.

The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $\left[\frac{a}{b}, \frac{c}{d}\right)$ where $a, b, c, d$ are positive integers with $\operatorname{gcd}(a, b)=\operatorname{gcd}(c, d)=1$. Compute $1000 a+100 b+10 c+d$.

The elevator buttons in Harvard's Science Center form a $3 \times 2$ grid of identical buttons, and each button lights up when pressed. One day, a student is in the elevator when all the other lights in the elevator malfunction, so that only the buttons which are lit can be seen, but one cannot see which floors they correspond to. Given that at least one of the buttons is lit, how many distinct arrangements can the student observe? (For example, if only one button is lit, then the student will observe the same arrangement regardless of which button it is.)

While waiting for their food at a restaurant in Harvard Square, Ana and Banana draw 3 squares $\square_{1}, \square_{2}, \square_{3}$ on one of their napkins. Starting with Ana, they take turns filling in the squares with integers from the set $\{1,2,3,4,5\}$ such that no integer is used more than once. Ana's goal is to minimize the minimum value $M$ that the polynomial $a_{1} x^{2}+a_{2} x+a_{3}$ attains over all real $x$, where $a_{1}, a_{2}, a_{3}$ are the integers written in $\square_{1}, \square_{2}, \square_{3}$ respectively. Banana aims to maximize $M$. Assuming both play optimally, compute the final value of $100 a_{1}+10 a_{2}+a_{3}$.

After viewing the John Harvard statue, a group of tourists decides to estimate the distances of nearby locations on a map by drawing a circle, centered at the statue, of radius $\sqrt{n}$ inches for each integer $2020 \leq n \leq 10000$, so that they draw 7981 circles altogether. Given that, on the map, the Johnston Gate is 10 -inch line segment which is entirely contained between the smallest and the largest circles, what is the minimum number of points on this line segment which lie on one of the drawn circles? (The endpoint of a segment is considered to be on the segment.)

While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a 2020-term strictly increasing geometric sequence with an integer common ratio $r$. Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_{0}$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_{0}$, what is the closest integer to $\log _{2} d$ ?

Sean enters a classroom in the Memorial Hall and sees a 1 followed by 2020 0's on the blackboard. As he is early for class, he decides to go through the digits from right to left and independently erase the $n$th digit from the left with probability $\frac{n-1}{n}$. (In particular, the 1 is never erased.) Compute the expected value of the number formed from the remaining digits when viewed as a base- 3 number. (For example, if the remaining number on the board is 1000 , then its value is 27 .)

For how many positive integers $n \leq 1000$ does the equation in real numbers

$$x^{\lfloor x\rfloor}=n$$

have a positive solution for $x$ ? (For a real number $x,\lfloor x\rfloor$ denotes the largest integer that is not greater than $x$.)

How many ways are there to arrange the numbers $\{1,2,3,4,5,6,7,8\}$ in a circle so that every two adjacent elements are relatively prime? Consider rotations and reflections of the same arrangement to be indistinguishable.

Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius 1 . Compute $\frac{120 A}{\pi}$.

Marisa has two identical cubical dice labeled with the numbers $\{1,2,3,4,5,6\}$. However, the two dice are not fair, meaning that they can land on each face with different probability. Marisa rolls the two dice and calculates their sum. Given that the sum is 2 with probability 0.04 , and 12 with probability 0.01 , the maximum possible probability of the sum being 7 is $p$. Compute $\lfloor 100 p\rfloor$.

For each positive integer $n$, let $a_{n}$ be the smallest nonnegative integer such that there is only one positive integer at most $n$ that is relatively prime to all of $n, n+1, \ldots, n+a_{n}$. If $n<100$, compute the largest possible value of $n-a_{n}$.

Regular hexagon $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ has side length 2 . For $1 \leq i \leq 6$, let $C_{i}$ be a unit circle centered at $P_{i}$ and $\ell_{i}$ be one of the internal common tangents of $C_{i}$ and $C_{i+2}$, where $C_{7}=C_{1}$ and $C_{8}=C_{2}$. Assume that the lines $\left\{\ell_{1}, \ell_{2}, \ell_{3}, \ell_{4}, \ell_{5}, \ell_{6}\right\}$ bound a regular hexagon. The area of this hexagon can be expressed as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Roger the ant is traveling on a coordinate plane, starting at ( 0,0 ). Every second, he moves from one lattice point to a different lattice point at distance 1, chosen with equal probability. He will continue to move until he reaches some point $P$ for which he could have reached $P$ more quickly had he taken a different route. For example, if he goes from $(0,0)$ to $(1,0)$ to $(1,1)$ to $(1,2)$ to $(0,2)$, he stops at $(0,2)$ because he could have gone from $(0,0)$ to $(0,1)$ to $(0,2)$ in only 2 seconds. The expected number of steps Roger takes before he stops can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Altitudes $B E$ and $C F$ of acute triangle $A B C$ intersect at $H$. Suppose that the altitudes of triangle $E H F$ concur on line $B C$. If $A B=3$ and $A C=4$, then $B C^{2}=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Alice and Bob take turns removing balls from a bag containing 10 black balls and 10 white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the expected number of black balls which Bob has can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let $x$ and $y$ be non-negative real numbers that sum to 1 . Compute the number of ordered pairs $(a, b)$ with $a, b \in\{0,1,2,3,4\}$ such that the expression $x^{a} y^{b}+y^{a} x^{b}$ has maximum value $2^{1-a-b}$.

Two hexagons are attached to form a new polygon $P$. Compute the minimum number of sides that $P$ can have.

Let $a$ be a positive integer such that $2 a$ has units digit 4. What is the sum of the possible units digits of $3 a$ ?

How many six-digit multiples of 27 have only 3,6 , or 9 as their digits?

Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3 , and Buddy wins otherwise. If the probability that Ainsley wins is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.

The points $(0,0),(1,2),(2,1),(2,2)$ in the plane are colored red while the points $(1,0),(2,0),(0,1),(0,2)$ are colored blue. Four segments are drawn such that each one connects a red point to a blue point and each colored point is the endpoint of some segment. The smallest possible sum of the lengths of the segments can be expressed as $a+\sqrt{b}$, where $a, b$ are positive integers. Compute $100 a+b$.

If $x, y, z$ are real numbers such that $x y=6, x-z=2$, and $x+y+z=9$, compute $\frac{x}{y}-\frac{z}{x}-\frac{z^{2}}{x y}$. Proposed by: Ragulan Sivakumar

Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism.

A small village has $n$ people. During their yearly elections, groups of three people come up to a stage and vote for someone in the village to be the new leader. After every possible group of three people has voted for someone, the person with the most votes wins.

This year, it turned out that everyone in the village had the exact same number of votes! If $10 \leq n \leq$ 100 , what is the number of possible values of $n$ ?

A fair coin is flipped eight times in a row. Let $p$ be the probability that there is exactly one pair of consecutive flips that are both heads and exactly one pair of consecutive flips that are both tails. If $p=\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100 a+b$.

The number 3003 is the only number known to appear eight times in Pascal's triangle, at positions

$$\binom{3003}{1},\binom{3003}{3002},\binom{a}{2},\binom{a}{a-2},\binom{15}{b},\binom{15}{15-b},\binom{14}{6},\binom{14}{8}$$

Compute $a+b(15-b)$.

Two diameters and one radius are drawn in a circle of radius 1 , dividing the circle into 5 sectors. The largest possible area of the smallest sector can be expressed as $\frac{a}{b} \pi$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.

In a single-elimination tournament consisting of $2^{9}=512$ teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy is then given the results of all 511 matches and must create a list of teams such that she can guarantee that the third-best team is on the list. What is the minimum possible length of Joy's list?

Wendy is playing darts with a circular dartboard of radius 20 . Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board again. What is the expected number of darts she has to throw before all the darts are within 10 units of the center?

A point $(x, y)$ is selected uniformly at random from the unit square $S=\{(x, y) \mid 0 \leq x \leq 1,0 \leq$ $y \leq 1\}$. If the probability that $(3 x+2 y, x+4 y)$ is in $S$ is $\frac{a}{b}$, where $a, b$ are relatively prime positive integers, compute $100 a+b$.

For a real number $r$, the quadratics $x^{2}+(r-1) x+6$ and $x^{2}+(2 r+1) x+22$ have a common real root. The sum of the possible values of $r$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.

Three players play tic-tac-toe together. In other words, the three players take turns placing an "A", "B", and "C", respectively, in one of the free spots of a $3 \times 3$ grid, and the first player to have three of their label in a row, column, or diagonal wins. How many possible final boards are there where the player who goes third wins the game? (Rotations and reflections are considered different boards, but the order of placement does not matter.)

Let $\mathbb{N}_{>1}$ denote the set of positive integers greater than 1 . Let $f: \mathbb{N}_{>1} \rightarrow \mathbb{N}_{>1}$ be a function such that $f(m n)=f(m) f(n)$ for all $m, n \in \mathbb{N}_{>1}$. If $f(101!)=101$ !, compute the number of possible values of $f(2020 \cdot 2021)$.

Suppose Harvard Yard is a $17 \times 17$ square. There are 14 dorms located on the perimeter of the Yard. If $s$ is the minimum distance between two dorms, the maximum possible value of $s$ can be expressed as $a-\sqrt{b}$ where $a, b$ are positive integers. Compute $100 a+b$.

Three distinct vertices of a regular 2020-gon are chosen uniformly at random. The probability that the triangle they form is isosceles can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

Let $\omega_{1}$ be a circle of radius 5 , and let $\omega_{2}$ be a circle of radius 2 whose center lies on $\omega_{1}$. Let the two circles intersect at $A$ and $B$, and let the tangents to $\omega_{2}$ at $A$ and $B$ intersect at $P$. If the area of $\triangle A B P$ can be expressed as $\frac{a \sqrt{b}}{c}$, where $b$ is square-free and $a, c$ are relatively prime positive integers, compute $100 a+10 b+c$.

Let $f(n)$ be the number of distinct prime divisors of $n$ less than 6 . Compute

$$\sum_{n=1}^{2020} f(n)^{2}$$

In triangle $A B C, A B=32, A C=35$, and $B C=x$. What is the smallest positive integer $x$ such that $1+\cos ^{2} A, \cos ^{2} B$, and $\cos ^{2} C$ form the sides of a non-degenerate triangle?

Two points are chosen inside the square $\{(x, y) \mid 0 \leq x, y \leq 1\}$ uniformly at random, and a unit square is drawn centered at each point with edges parallel to the coordinate axes. The expected area of the union of the two squares can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.

Compute the number of positive integers less than 10 ! which can be expressed as the sum of at most 4 (not necessarily distinct) factorials.

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive integers where $a_{1}=\sum_{i=0}^{100} i$ ! and $a_{i}+a_{i+1}$ is an odd perfect square for all $i \geq 1$. Compute the smallest possible value of $a_{1000}$.

Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e. there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of the players can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.

In $\triangle A B C, D$ and $E$ are the midpoints of $B C$ and $C A$, respectively. $A D$ and $B E$ intersect at $G$. Given that $G E C D$ is cyclic, $A B=41$, and $A C=31$, compute $B C$.

Bernie has 2020 marbles and 2020 bags labeled $B_{1}, \ldots, B_{2020}$ in which he randomly distributes the marbles (each marble is placed in a random bag independently). If $E$ the expected number of integers $1 \leq i \leq 2020$ such that $B_{i}$ has at least $i$ marbles, compute the closest integer to $1000 E$.

In acute triangle $A B C$, let $H$ be the orthocenter and $D$ the foot of the altitude from $A$. The circumcircle of triangle $B H C$ intersects $A C$ at $E \neq C$, and $A B$ at $F \neq B$. If $B D=3, C D=7$, and $\frac{A H}{H D}=\frac{5}{7}$, the area of triangle $A E F$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.

Let $a_{1}, a_{2}, a_{3}, \ldots$ be a sequence of positive real numbers that satisfies

$$\sum_{n=k}^{\infty}\binom{n}{k} a_{n}=\frac{1}{5^{k}}$$

for all positive integers $k$. The value of $a_{1}-a_{2}+a_{3}-a_{4}+\cdots$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.

For some positive real $\alpha$, the set $S$ of positive real numbers $x$ with $\{x\}>\alpha x$ consists of the union of several intervals, with total length 20.2. The value of $\alpha$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.) Proposed by: Daniel Zhu

The numbers $1,2, \ldots, 10$ are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. $1,2,3,4,5$, or $8,9,10,1,2$ ). If the probability that there exists some number that was not selected by any of the four people is $p$, compute $10000 p$.

In quadrilateral $A B C D$, there exists a point $E$ on segment $A D$ such that $\frac{A E}{E D}=\frac{1}{9}$ and $\angle B E C$ is a right angle. Additionally, the area of triangle $C E D$ is 27 times more than the area of triangle $A E B$. If $\angle E B C=\angle E A B, \angle E C B=\angle E D C$, and $B C=6$, compute the value of $A D^{2}$.

Let $a$ be the proportion of teams that correctly answered problem 1 on the Guts round. Estimate $A=\lfloor 10000 a\rfloor$. An estimate of $E$ earns $\max (0,\lfloor 20-|A-E| / 20\rfloor)$ points. If you have forgotten, question 1 was the following:

Two hexagons are attached to form a new polygon $P$. What is the minimum number of sides that $P$ can have?

Estimate $A$, the number of times an 8-digit number appears in Pascal's triangle. An estimate of $E$ earns $\max (0,\lfloor 20-|A-E| / 200\rfloor)$ points.

Let $p_{i}$ be the $i$ th prime. Let

$$f(x)=\sum_{i=1}^{50} p_{i} x^{i-1}=2+3 x+\cdots+229 x^{49}$$

If $a$ is the unique positive real number with $f(a)=100$, estimate $A=\lfloor 100000 a\rfloor$. An estimate of $E$ will earn $\max (0,\lfloor 20-|A-E| / 250\rfloor)$ points.