HMMT 2019

What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits are the same when read backwards.)

Let $N=2^{\left(2^{2}\right)}$ and $x$ be a real number such that $N^{\left(N^{N}\right)}=2^{\left(2^{x}\right)}$. Find $x$.

Let $x$ and $y$ be positive real numbers. Define $a=1+\frac{x}{y}$ and $b=1+\frac{y}{x}$. If $a^{2}+b^{2}=15$, compute $a^{3}+b^{3}$.

Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function satisfying

• $f(1)=1$;
• for $n \in \mathbb{N}, f(2 n)=2 f(n)$ and $f(2 n+1)=2 f(n)-1$.

Determine the sum of all positive integer solutions to $f(x)=19$ that do not exceed 2019.

Let $a_{1}, a_{2}, \ldots$ be an arithmetic sequence and $b_{1}, b_{2}, \ldots$ be a geometric sequence. Suppose that $a_{1} b_{1}=20$, $a_{2} b_{2}=19$, and $a_{3} b_{3}=14$. Find the greatest possible value of $a_{4} b_{4}$.

For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \sqrt{2}+b \sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \leq 20$ and $r_{1}+r_{2}=\sqrt{2}$.

Find the value of

$$\sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \sum_{c=1}^{\infty} \frac{a b(3 a+c)}{4^{a+b+c}(a+b)(b+c)(c+a)}$$

There is a unique function $f: \mathbb{N} \rightarrow \mathbb{R}$ such that $f(1)>0$ and such that

$$\sum_{d \mid n} f(d) f\left(\frac{n}{d}\right)=1$$

for all $n \geq 1$. What is $f\left(2018^{2019}\right)$ ?

Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube $k$-harmonic if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S}|k|$.

The sequence of integers $\left\{a_{i}\right\}_{i=0}^{\infty}$ satisfies $a_{0}=3, a_{1}=4$, and

$$a_{n+2}=a_{n+1} a_{n}+\left\lceil\sqrt{a_{n+1}^{2}-1} \sqrt{a_{n}^{2}-1}\right\rceil$$

for $n \geq 0$. Evaluate the sum

$$\sum_{n=0}^{\infty}\left(\frac{a_{n+3}}{a_{n+2}}-\frac{a_{n+2}}{a_{n}}+\frac{a_{n+1}}{a_{n+3}}-\frac{a_{n}}{a_{n+1}}\right) .$$

Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$ ?

In rectangle $A B C D$, points $E$ and $F$ lie on sides $A B$ and $C D$ respectively such that both $A F$ and $C E$ are perpendicular to diagonal $B D$. Given that $B F$ and $D E$ separate $A B C D$ into three polygons with equal area, and that $E F=1$, find the length of $B D$.

Let $A B$ be a line segment with length 2 , and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $A B$ with $A X=2 P X$. Find the area of $S$.

Convex hexagon $A B C D E F$ is drawn in the plane such that $A C D F$ and $A B D E$ are parallelograms with area 168. $A C$ and $B D$ intersect at $G$. Given that the area of $A G B$ is 10 more than the area of $C G B$, find the smallest possible area of hexagon $A B C D E F$.

Isosceles triangle $A B C$ with $A B=A C$ is inscribed in a unit circle $\Omega$ with center $O$. Point $D$ is the reflection of $C$ across $A B$. Given that $D O=\sqrt{3}$, find the area of triangle $A B C$.

Six unit disks $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}, C_{6}$ are in the plane such that they don't intersect each other and $C_{i}$ is tangent to $C_{i+1}$ for $1 \leq i \leq 6$ (where $C_{7}=C_{1}$ ). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R-r$.

Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $H$ be the orthocenter of $A B C$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $A H B, B H C, C H A$.

In triangle $A B C$ with $A B<A C$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $O H$ lies on $B C, B C=1$, and the perimeter of $A B C$ is 6 , find the area of $A B C$.

In a rectangular box $A B C D E F G H$ with edge lengths $A B=A D=6$ and $A E=49$, a plane slices through point $A$ and intersects edges $B F, F G, G H, H D$ at points $P, Q, R, S$ respectively. Given that $A P=A S$ and $P Q=Q R=R S$, find the area of pentagon $A P Q R S$.

In triangle $A B C, A B=13, B C=14, C A=15$. Squares $A B B_{1} A_{2}, B C C_{1} B_{2}, C A A_{1} C_{2}$ are constructed outside the triangle. Squares $A_{1} A_{2} A_{3} A_{4}, B_{1} B_{2} B_{3} B_{4}, C_{1} C_{2} C_{3} C_{4}$ are constructed outside the hexagon $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$. Squares $A_{3} B_{4} B_{5} A_{6}, B_{3} C_{4} C_{5} B_{6}, C_{3} A_{4} A_{5} C_{6}$ are constructed outside the hexagon $A_{4} A_{3} B_{4} B_{3} C_{4} C_{3}$. Find the area of the hexagon $A_{5} A_{6} B_{5} B_{6} C_{5} C_{6}$.

How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \ldots, 20$ on its sides). He conceals the results but tells you that at least half of the rolls are 20. Suspicious, you examine the first two dice and find that they show 20 and 19 in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20 ?$

Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?

Yannick is playing a game with 100 rounds, starting with 1 coin. During each round, there is a $n \%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?

Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right 1 unit with equal probability at each step.) If she lands on a point of the form $(6 m, 6 n)$ for $m, n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6 m+3,6 n+3)$ for $m, n \in \mathbb{Z}$, she descends to hell. What is the probability that she ascends to heaven?

A point $P$ lies at the center of square $A B C D$. A sequence of points $\left\{P_{n}\right\}$ is determined by $P_{0}=P$, and given point $P_{i}$, point $P_{i+1}$ is obtained by reflecting $P_{i}$ over one of the four lines $A B, B C, C D, D A$, chosen uniformly at random and independently for each $i$. What is the probability that $P_{8}=P$ ?

In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\frac{2}{3}$ probability of voting for Alice and a $\frac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$ ) with four colors. A coloring is called multichromatic if whenever $a, b$ and $\operatorname{gcd}(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?

How many ways can one fill a $3 \times 3$ square grid with nonnegative integers such that no nonzero integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7 ?

Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $\left(a_{1}, a_{2}, a_{3}, a_{4}\right)$ to some position $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ with integer coordinates satisfying $\left(x_{1}-a_{1}\right)^{2}+\left(x_{2}-a_{2}\right)^{2}+\left(x_{3}-a_{3}\right)^{2}+\left(x_{4}-a_{4}\right)^{2}=4$ and $\left|\left(x_{1}+x_{2}+x_{3}+x_{4}\right)-\left(a_{1}+a_{2}+a_{3}+a_{4}\right)\right|=2$. In how many ways can Fred reach $(10,10,10,10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?

Let $A B C D$ be a parallelogram. Points $X$ and $Y$ lie on segments $A B$ and $A D$ respectively, and $A C$ intersects $X Y$ at point $Z$. Prove that

$$\frac{A B}{A X}+\frac{A D}{A Y}=\frac{A C}{A Z}$$

Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \ldots, f(n))$ is a permutation of $(1,2, \ldots, n)$ ?

For any angle $0<\theta<\pi / 2$, show that

$$0<\sin \theta+\cos \theta+\tan \theta+\cot \theta-\sec \theta-\csc \theta<1$$

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n$ !.

Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.

Scalene triangle $A B C$ satisfies $\angle A=60^{\circ}$. Let the circumcenter of $A B C$ be $O$, the orthocenter be $H$, and the incenter be $I$. Let $D, T$ be the points where line $B C$ intersects the internal and external angle bisectors of $\angle A$, respectively. Choose point $X$ on the circumcircle of $\triangle I H O$ such that $H X \| A I$. Prove that $O D \perp T X$.

A convex polygon on the plane is called wide if the projection of the polygon onto any line in the same plane is a segment with length at least 1 . Prove that a circle of radius $\frac{1}{3}$ can be placed completely inside any wide polygon.

Can the set of lattice points $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \leq x, y \leq 252, x \neq y\}$ be colored using 10 distinct colors such that for all $a \neq b, b \neq c$, the colors of $(a, b)$ and $(b, c)$ are distinct?

Let $p>2$ be a prime number. $\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{p}$ for each nonnegative integer $k$. For example, $(x+2)(2 x+3)=2 x^{2}+2 x+1$ in $\mathbb{F}_{5}[x]$ because the corresponding coefficients are equal modulo 5 . Let $f, g \in \mathbb{F}_{p}[x]$. The pair $(f, g)$ is called compositional if

$$f(g(x)) \equiv x^{p^{2}}-x$$

in $\mathbb{F}_{p}[x]$. Find, with proof, the number of compositional pairs (in terms of $p$ ).

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial

$$P(x)=(2 n) x^{2 n}+(2 n-1) x^{2 n-1}+\cdots+(n+1) x^{n+1}+n x^{n}+(n+1) x^{n-1}+\cdots+(2 n-1) x+2 n$$

lie on the unit circle (i.e. $|r|=1$ ).

Find the sum of all real solutions to $x^{2}+\cos x=2019$.

There are 100 people in a room with ages $1,2, \ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?

Let $S(x)$ denote the sum of the digits of a positive integer $x$. Find the maximum possible value of $S(x+2019)-S(x)$.

Tessa has a figure created by adding a semicircle of radius 1 on each side of an equilateral triangle with side length 2 , with semicircles oriented outwards. She then marks two points on the boundary of the figure. What is the greatest possible distance between the two points?

Call a positive integer $n$ weird if $n$ does not divide $(n-2)$ !. Determine the number of weird numbers between 2 and 100 inclusive.

The pairwise products $a b, b c, c d$, and $d a$ of positive integers $a, b, c$, and $d$ are $64,88,120$, and 165 in some order. Find $a+b+c+d$.

For any real number $\alpha$, define

$$\operatorname{sign}(\alpha)= \begin{cases}+1 & \text { if } \alpha>0 \\ 0 & \text { if } \alpha=0 \\ -1 & \text { if } \alpha<0\end{cases}$$

How many triples $(x, y, z) \in \mathbb{R}^{3}$ satisfy the following system of equations

$$\begin{aligned} & x=2018-2019 \cdot \operatorname{sign}(y+z) \\ & y=2018-2019 \cdot \operatorname{sign}(z+x) \\ & z=2018-2019 \cdot \operatorname{sign}(x+y) ? \end{aligned}$$

A regular hexagon $P R O F I T$ has area 1. Every minute, greedy George places the largest possible equilateral triangle that does not overlap with other already-placed triangles in the hexagon, with ties broken arbitrarily. How many triangles would George need to cover at least $90 \%$ of the hexagon's area?

Define $P=\{\mathrm{S}, \mathrm{T}\}$ and let $\mathcal{P}$ be the set of all proper subsets of $P$. (A proper subset is a subset that is not the set itself.) How many ordered pairs $(\mathcal{S}, \mathcal{T})$ of proper subsets of $\mathcal{P}$ are there such that (a) $\mathcal{S}$ is not a proper subset of $\mathcal{T}$ and $\mathcal{T}$ is not a proper subset of $\mathcal{S}$; and (b) for any sets $S \in \mathcal{S}$ and $T \in \mathcal{T}, S$ is not a proper subset of $T$ and $T$ is not a proper subset of $S$ ?

Let

$$\begin{aligned} & A=(1+2 \sqrt{2}+3 \sqrt{3}+6 \sqrt{6})(2+6 \sqrt{2}+\sqrt{3}+3 \sqrt{6})(3+\sqrt{2}+6 \sqrt{3}+2 \sqrt{6})(6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6}) \\ & B=(1+3 \sqrt{2}+2 \sqrt{3}+6 \sqrt{6})(2+\sqrt{2}+6 \sqrt{3}+3 \sqrt{6})(3+6 \sqrt{2}+\sqrt{3}+2 \sqrt{6})(6+2 \sqrt{2}+3 \sqrt{3}+\sqrt{6}) \end{aligned}$$

Compute the value of $A / B$.

In the Year 0 of Cambridge there is one squirrel and one rabbit. Both animals multiply in numbers quickly. In particular, if there are $m$ squirrels and $n$ rabbits in Year $k$, then there will be $2 m+2019$ squirrels and $4 n-2$ rabbits in Year $k+1$. What is the first year in which there will be strictly more rabbits than squirrels?

Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer.

Reimu has 2019 coins $C_{0}, C_{1}, \ldots, C_{2018}$, one of which is fake, though they look identical to each other (so each of them is equally likely to be fake). She has a machine that takes any two coins and picks one that is not fake. If both coins are not fake, the machine picks one uniformly at random. For each $i=1,2, \ldots, 1009$, she puts $C_{0}$ and $C_{i}$ into the machine once, and machine picks $C_{i}$. What is the probability that $C_{0}$ is fake?

Let $A B C$ be a triangle where $A B=9, B C=10, C A=17$. Let $\Omega$ be its circumcircle, and let $A_{1}, B_{1}, C_{1}$ be the diametrically opposite points from $A, B, C$, respectively, on $\Omega$. Find the area of the convex hexagon with the vertices $A, B, C, A_{1}, B_{1}, C_{1}$.

Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ :

• If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends;
• If $A$ and $B$ are enemies and $B$ and $C$ are enemies, then $A$ and $C$ are friends;
• If $A$ and $B$ are friends and $B$ and $C$ are enemies, then $A$ and $C$ are enemies.

How many possible relationship configurations are there among the five people?

Let $\mathbb{R}$ be the set of real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that for all real numbers $x$ and $y$, we have

$$f\left(x^{2}\right)+f\left(y^{2}\right)=f(x+y)^{2}-2 x y$$

Let $S=\sum_{n=-2019}^{2019} f(n)$. Determine the number of possible values of $S$.

Let $A B C$ be a triangle with $A B=3, B C=4$, and $C A=5$. Let $A_{1}, A_{2}$ be points on side $B C$, $B_{1}, B_{2}$ be points on side $C A$, and $C_{1}, C_{2}$ be points on side $A B$. Suppose that there exists a point $P$ such that $P A_{1} A_{2}, P B_{1} B_{2}$, and $P C_{1} C_{2}$ are congruent equilateral triangles. Find the area of convex hexagon $A_{1} A_{2} B_{1} B_{2} C_{1} C_{2}$.

2019 points are chosen independently and uniformly at random on the interval $[0,1]$. Tairitsu picks 1000 of them randomly and colors them black, leaving the remaining ones white. Hikari then computes the sum of the positions of the leftmost white point and the rightmost black point. What is the probability that this sum is at most 1?

Complex numbers $a, b, c$ form an equilateral triangle with side length 18 in the complex plane. If $|a+b+c|=36$, find $|b c+c a+a b|$.

On floor 0 of a weird-looking building, you enter an elevator that only has one button. You press the button twice and end up on floor 1. Thereafter, every time you press the button, you go up by one floor with probability $\frac{X}{Y}$, where $X$ is your current floor, and $Y$ is the total number of times you have pressed the button thus far (not including the current one); otherwise, the elevator does nothing.

Between the third and the $100^{\text {th }}$ press inclusive, what is the expected number of pairs of consecutive presses that both take you up a floor?

A regular hexagon $A B C D E F$ has side length 1 and center $O$. Parabolas $P_{1}, P_{2}, \ldots, P_{6}$ are constructed with common focus $O$ and directrices $A B, B C, C D, D E, E F, F A$ respectively. Let $\chi$ be the set of all distinct points on the plane that lie on at least two of the six parabolas. Compute

$$\sum_{X \in \chi}|O X|$$

(Recall that the focus is the point and the directrix is the line such that the parabola is the locus of points that are equidistant from the focus and the directrix.)

Determine the number of subsets $S$ of $\{1,2, \ldots, 1000\}$ that satisfy the following conditions:

• $S$ has 19 elements, and
• the sum of the elements in any non-empty subset of $S$ is not divisible by 20 .

Find the smallest positive integer $n$ such that

$$\underbrace{2^{2^{2} \omega^{2}}}_{n 2^{\prime} \mathrm{s}}>\underbrace{((\cdots((100!)!)!\cdots)!)!}_{100 \text { factorials }}$$

Let $S$ be the set of all positive factors of 6000 . What is the probability of a random quadruple $(a, b, c, d) \in S^{4}$ satisfies

$$\operatorname{lcm}(\operatorname{gcd}(a, b), \operatorname{gcd}(c, d))=\operatorname{gcd}(\operatorname{lcm}(a, b), \operatorname{lcm}(c, d)) ?$$

A 5 by 5 grid of unit squares is partitioned into 5 pairwise incongruent rectangles with sides lying on the gridlines. Find the maximum possible value of the product of their areas.

Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $I_{A}, I_{B}, I_{C}$ be the $A, B, C$ excenters of this triangle, and let $O$ be the circumcenter of the triangle. Let $\gamma_{A}, \gamma_{B}, \gamma_{C}$ be the corresponding excircles and $\omega$ be the circumcircle. $X$ is one of the intersections between $\gamma_{A}$ and $\omega$. Likewise, $Y$ is an intersection of $\gamma_{B}$ and $\omega$, and $Z$ is an intersection of $\gamma_{C}$ and $\omega$. Compute

$$\cos \angle O X I_{A}+\cos \angle O Y I_{B}+\cos \angle O Z I_{C} .$$

Consider the eighth-sphere $\left\{(x, y, z) \mid x, y, z \geq 0, x^{2}+y^{2}+z^{2}=1\right\}$. What is the area of its projection onto the plane $x+y+z=1$ ?

How many positive integers $2 \leq a \leq 101$ have the property that there exists a positive integer $N$ for which the last two digits in the decimal representation of $a^{2^{n}}$ is the same for all $n \geq N$ ?

Yannick picks a number $N$ randomly from the set of positive integers such that the probability that $n$ is selected is $2^{-n}$ for each positive integer $n$. He then puts $N$ identical slips of paper numbered 1 through $N$ into a hat and gives the hat to Annie. Annie does not know the value of $N$, but she draws one of the slips uniformly at random and discovers that it is the number 2 . What is the expected value of $N$ given Annie's information?

Three points are chosen inside a unit cube uniformly and independently at random. What is the probability that there exists a cube with side length $\frac{1}{2}$ and edges parallel to those of the unit cube that contains all three points?

Let $A B C$ be a triangle with $A B=6, A C=7, B C=8$. Let $I$ be the incenter of $A B C$. Points $Z$ and $Y$ lie on the interior of segments $A B$ and $A C$ respectively such that $Y Z$ is tangent to the incircle. Given point $P$ such that

$$\angle Z P C=\angle Y P B=90^{\circ}$$

find the length of $I P$.

For positive integers $a$ and $b$ such that $a$ is coprime to $b$, define $\operatorname{ord}_{b}(a)$ as the least positive integer $k$ such that $b \mid a^{k}-1$, and define $\varphi(a)$ to be the number of positive integers less than or equal to $a$ which are coprime to $a$. Find the least positive integer $n$ such that

$$\operatorname{ord}_{n}(m)<\frac{\varphi(n)}{10}$$

for all positive integers $m$ coprime to $n$.