HMMT 2023

Suppose $P(x)$ is a cubic polynomial with integer coefficients such that $P(\sqrt{5})=5$ and $P(\sqrt[3]{5})=5 \sqrt[3]{5}$. Compute $P(5)$.

Compute the number of positive integers $n \leq 1000$ such that $\operatorname{lcm}(n, 9)$ is a perfect square. (Recall that lcm denotes the least common multiple.)

Suppose $x$ is a real number such that $\sin \left(1+\cos ^{2} x+\sin ^{4} x\right)=\frac{13}{14}$. Compute $\cos \left(1+\sin ^{2} x+\cos ^{4} x\right)$.

Suppose $P(x)$ is a polynomial with real coefficients such that $P(t)=P(1) t^{2}+P(P(1)) t+P(P(P(1)))$ for all real numbers $t$. Compute the largest possible value of $P(P(P(1)))$ ).

Suppose $E, I, L, V$ are (not necessarily distinct) nonzero digits in base ten for which

• the four-digit number $\underline{E} \underline{V} \underline{I} \underline{L}$ is divisible by 73 , and
• the four-digit number $\underline{V} \underline{I} \underline{L} \underline{E}$ is divisible by 74 .

Compute the four-digit number $\underline{L} \underline{I} \underline{V} \underline{E}$.

Suppose $a_{1}, a_{2}, \ldots, a_{100}$ are positive real numbers such that

$$a_{k}=\frac{k a_{k-1}}{a_{k-1}-(k-1)}$$

for $k=2,3, \ldots, 100$. Given that $a_{20}=a_{23}$, compute $a_{100}$.

If $a, b, c$, and $d$ are pairwise distinct positive integers that satisfy $\operatorname{lcm}(a, b, c, d)<1000$ and $a+b=c+d$, compute the largest possible value of $a+b$.

Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\operatorname{gcd}(a, b)=1$. Compute

$$\sum_{(a, b) \in S}\left\lfloor\frac{300}{2 a+3 b}\right\rfloor$$

For any positive integers $a$ and $b$ with $b>1$, let $s_{b}(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that

$$\sum_{i=1}^{\left\lfloor\log _{23} n\right\rfloor} s_{20}\left(\left\lfloor\frac{n}{23^{i}}\right\rfloor\right)=103 \text { and } \sum_{i=1}^{\left\lfloor\log _{20} n\right\rfloor} s_{23}\left(\left\lfloor\frac{n}{20^{i}}\right\rfloor\right)=115$$

Compute $s_{20}(n)-s_{23}(n)$.

Let $\zeta=e^{2 \pi i / 99}$ and $\omega=e^{2 \pi i / 101}$. The polynomial

$$x^{9999}+a_{9998} x^{9998}+\cdots+a_{1} x+a_{0}$$

has roots $\zeta^{m}+\omega^{n}$ for all pairs of integers $(m, n)$ with $0 \leq m<99$ and $0 \leq n<101$. Compute $a_{9799}+a_{9800}+\cdots+a_{9998}$.

Let $A B C D E F$ be a regular hexagon, and let $P$ be a point inside quadrilateral $A B C D$. If the area of triangle $P B C$ is 20 , and the area of triangle $P A D$ is 23 , compute the area of hexagon $A B C D E F$.

Points $X, Y$, and $Z$ lie on a circle with center $O$ such that $X Y=12$. Points $A$ and $B$ lie on segment $X Y$ such that $O A=A Z=Z B=B O=5$. Compute $A B$.

Suppose $A B C D$ is a rectangle whose diagonals meet at $E$. The perimeter of triangle $A B E$ is $10 \pi$ and the perimeter of triangle $A D E$ is $n$. Compute the number of possible integer values of $n$.

Let $A B C D$ be a square, and let $M$ be the midpoint of side $B C$. Points $P$ and $Q$ lie on segment $A M$ such that $\angle B P D=\angle B Q D=135^{\circ}$. Given that $A P<A Q$, compute $\frac{A Q}{A P}$.

Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. Suppose $P Q R S$ is a square such that $P$ and $R$ lie on line $B C, Q$ lies on line $C A$, and $S$ lies on line $A B$. Compute the side length of this square.

Convex quadrilateral $A B C D$ satisfies $\angle C A B=\angle A D B=30^{\circ}, \angle A B D=77^{\circ}, B C=C D$, and $\angle B C D=n^{\circ}$ for some positive integer $n$. Compute $n$.

Quadrilateral $A B C D$ is inscribed in circle $\Gamma$. Segments $A C$ and $B D$ intersect at $E$. Circle $\gamma$ passes through $E$ and is tangent to $\Gamma$ at $A$. Suppose that the circumcircle of triangle $B C E$ is tangent to $\gamma$ at $E$ and is tangent to line $C D$ at $C$. Suppose that $\Gamma$ has radius 3 and $\gamma$ has radius 2. Compute $B D$.

Triangle $A B C$ with $\angle B A C>90^{\circ}$ has $A B=5$ and $A C=7$. Points $D$ and $E$ lie on segment $B C$ such that $B D=D E=E C$. If $\angle B A C+\angle D A E=180^{\circ}$, compute $B C$.

Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$. Point $G$ lies on line $X Z$ such that there exists a triangle $A B C$ with centroid $G$ such that $X$ lies on line $B C, Y$ lies on line $A C$, and $Z$ lies on line $A B$. Compute the largest possible value of $X G$.

Triangle $A B C$ has incenter $I$. Let $D$ be the foot of the perpendicular from $A$ to side $B C$. Let $X$ be a point such that segment $A X$ is a diameter of the circumcircle of triangle $A B C$. Given that $I D=2$, $I A=3$, and $I X=4$, compute the inradius of triangle $A B C$.

There are 800 marbles in a bag. Each marble is colored with one of 100 colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops. Suppose Anna has not stopped after drawing 699 marbles. Compute the probability that she stops immediately after drawing the 700th marble.

Compute the number of ways to tile a $3 \times 5$ rectangle with one $1 \times 1$ tile, one $1 \times 2$ tile, one $1 \times 3$ tile, one $1 \times 4$ tile, and one $1 \times 5$ tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.)

Richard starts with the string HHMMMMTT. A move consists of replacing an instance of HM with MH , replacing an instance of MT with TM, or replacing an instance of TH with HT. Compute the number of possible strings he can end up with after performing zero or more moves.

The cells of a $5 \times 5$ grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through 9 cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly 3 red cells, exactly 3 white cells, and exactly 3 blue cells no matter which route he takes.

Elbert and Yaiza each draw 10 cards from a 20 -card deck with cards numbered $1,2,3, \ldots, 20$. Then, starting with the player with the card numbered 1, the players take turns placing down the lowestnumbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends. Given that Yaiza lost and 5 cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player's hand does not matter.)

Each cell of a $3 \times 3$ grid is labeled with a digit in the set $\{1,2,3,4,5\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once.

Svitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations:

• if $n$ is even, she can replace $n$ with $\frac{n}{2}$;
• if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$; and
• if $n \geq 64$, she can replace $n$ with $n-64$.

Compute the number of possible values that Svitlana can obtain by doing zero or more operations.

A random permutation $a=\left(a_{1}, a_{2}, \ldots, a_{40}\right)$ of $(1,2, \ldots, 40)$ is chosen, with all permutations being equally likely. William writes down a $20 \times 20$ grid of numbers $b_{i j}$ such that $b_{i j}=\max \left(a_{i}, a_{j+20}\right)$ for all $1 \leq i, j \leq 20$, but then forgets the original permutation $a$. Compute the probability that, given the values of $b_{i j}$ alone, there are exactly 2 permutations $a$ consistent with the grid.

There are 100 people standing in a line from left to right. Half of them are randomly chosen to face right (with all $\binom{100}{50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.

Let $x_{0}=x_{101}=0$. The numbers $x_{1}, x_{2}, \ldots, x_{100}$ are chosen at random from the interval $[0,1]$ uniformly and independently. Compute the probability that $2 x_{i} \geq x_{i-1}+x_{i+1}$ for all $i=1,2, \ldots$, 100.

For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4 \tau(n)-n$ over all positive integers $n$.

Prove that there do not exist pairwise distinct complex numbers $a, b, c$, and $d$ such that

$$a^{3}-b c d=b^{3}-c d a=c^{3}-d a b=d^{3}-a b c .$$

Let $A B C D$ be a convex quadrilateral such that $\angle A B C=\angle B C D=\theta$ for some angle $\theta<90^{\circ}$. Point $X$ lies inside the quadrilateral such that $\angle X A D=\angle X D A=90^{\circ}-\theta$. Prove that $B X=X C$.

Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan's pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair ( $a, b$ ) of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan's pair after at most $N$ rounds.

Let $S$ be the set of all points in the plane whose coordinates are positive integers less than or equal to 100 (so $S$ has $100^{2}$ elements), and let $\mathcal{L}$ be the set of all lines $\ell$ such that $\ell$ passes through at least two points in $S$. Find, with proof, the largest integer $N \geq 2$ for which it is possible to choose $N$ distinct lines in $\mathcal{L}$ such that every two of the chosen lines are parallel.

For any odd positive integer $n$, let $r(n)$ be the odd positive integer such that the binary representation of $r(n)$ is the binary representation of $n$ written backwards. For example, $r(2023)=$ $r\left(11111100111_{2}\right)=11100111111_{2}=1855$. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression $a_{1}, \ldots, a_{8}$ of odd positive integers such that $r\left(a_{1}\right), \ldots$, $r\left(a_{8}\right)$ is an arithmetic progression in that order.

Let $A B C$ be a triangle. Point $D$ lies on segment $B C$ such that $\angle B A D=\angle D A C$. Point $X$ lies on the opposite side of line $B C$ as $A$ and satisfies $X B=X D$ and $\angle B X D=\angle A C B$. Analogously, point $Y$ lies on the opposite side of line $B C$ as $A$ and satisfies $Y C=Y D$ and $\angle C Y D=\angle A B C$. Prove that lines $X Y$ and $A D$ are perpendicular.

Find, with proof, all nonconstant polynomials $P(x)$ with real coefficients such that, for all nonzero real numbers $z$ with $P(z) \neq 0$ and $P\left(\frac{1}{z}\right) \neq 0$, we have

$$\frac{1}{P(z)}+\frac{1}{P\left(\frac{1}{z}\right)}=z+\frac{1}{z}$$

Let $A B C$ be a triangle with $A B<A C$. The incircle of triangle $A B C$ is tangent to side $B C$ at $D$ and intersects the perpendicular bisector of segment $B C$ at distinct points $X$ and $Y$. Lines $A X$ and $A Y$ intersect line $B C$ at $P$ and $Q$, respectively. Prove that, if $D P \cdot D Q=(A C-A B)^{2}$, then $A B+A C=3 B C$.

One thousand people are in a tennis tournament where each person plays against each other person exactly once, and there are no ties. Prove that it is possible to put all the competitors in a line so that each of the 998 people who are not at an end of the line either defeated both their neighbors or lost to both their neighbors

Suppose $a$ and $b$ are positive integers such that $a^{b}=2^{2023}$. Compute the smallest possible value of $b^{a}$.

Let $n$ be a positive integer, and let $s$ be the sum of the digits of the base-four representation of $2^{n}-1$. If $s=2023$ (in base ten), compute $n$ (in base ten).

Let $A B C D$ be a convex quadrilateral such that $\angle A B D=\angle B C D=90^{\circ}$, and let $M$ be the midpoint of segment $B D$. Suppose that $C M=2$ and $A M=3$. Compute $A D$.

A standard $n$-sided die has $n$ sides labeled 1 to $n$. Luis, Luke, and Sean play a game in which they roll a fair standard 4 -sided die, a fair standard 6 -sided die, and a fair standard 8 -sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

If $a$ and $b$ are positive real numbers such that $a \cdot 2^{b}=8$ and $a^{b}=2$, compute $a^{\log _{2} a} 2^{b^{2}}$.

Let $A, E, H, L, T$, and $V$ be chosen independently and at random from the set $\left\{0, \frac{1}{2}, 1\right\}$. Compute the probability that $\lfloor T \cdot H \cdot E\rfloor=L \cdot A \cdot V \cdot A$.

Let $\Omega$ be a sphere of radius 4 and $\Gamma$ be a sphere of radius 2 . Suppose that the center of $\Gamma$ lies on the surface of $\Omega$. The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumference.

Suppose $a, b$, and $c$ are distinct positive integers such that $\sqrt{a \sqrt{b \sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c$.

One hundred points labeled 1 to 100 are arranged in a $10 \times 10$ grid such that adjacent points are one unit apart. The labels are increasing left to right, top to bottom (so the first row has labels 1 to 10 , the second row has labels 11 to 20 , and so on). Convex polygon $\mathcal{P}$ has the property that every point with a label divisible by 7 is either on the boundary or in the interior of $\mathcal{P}$. Compute the smallest possible area of $\mathcal{P}$.

The number

$$316990099009901=\frac{32016000000000001}{101}$$

is the product of two distinct prime numbers. Compute the smaller of these two primes.

The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 15 wooden blocks of weights $F_{2}, F_{3}, \ldots, F_{16}$, compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.

The number 770 is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either 40 or 41 from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N=a \cdot 2^{b}$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100 a+b$.

Suppose $a, b, c$, and $d$ are pairwise distinct positive perfect squares such that $a^{b}=c^{d}$. Compute the smallest possible value of $a+b+c+d$.

Acute triangle $A B C$ has circumcenter $O$. The bisector of $\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\angle B A C=n^{\circ}$, where $n$ is a positive integer, compute the largest possible value of $n$.

Let $A$ and $B$ be points in space for which $A B=1$. Let $\mathcal{R}$ be the region of points $P$ for which $A P \leq 1$ and $B P \leq 1$. Compute the largest possible side length of a cube contained within $\mathcal{R}$.

The graph of the equation $x+y=\left\lfloor x^{2}+y^{2}\right\rfloor$ consists of several line segments. Compute the sum of their lengths.

An equilateral triangle lies in the Cartesian plane such that the $x$-coordinates of its vertices are pairwise distinct and all satisfy the equation $x^{3}-9 x^{2}+10 x+5=0$. Compute the side length of the triangle.

Elisenda has a piece of paper in the shape of a triangle with vertices $A, B$, and $C$ such that $A B=42$. She chooses a point $D$ on segment $A C$, and she folds the paper along line $B D$ so that $A$ lands at a point $E$ on segment $B C$. Then, she folds the paper along line $D E$. When she does this, $B$ lands at the midpoint of segment $D C$. Compute the perimeter of the original unfolded triangle.

Compute the number of ways to select 99 cells of a $19 \times 19$ square grid such that no two selected cells share an edge or vertex.

Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Suppose that $p=\frac{a}{2^{b}}$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100 a+b$.

Let $x, y$, and $N$ be real numbers, with $y$ nonzero, such that the sets $\left\{(x+y)^{2},(x-y)^{2}, x y, x / y\right\}$ and $\{4,12.8,28.8, N\}$ are equal. Compute the sum of the possible values of $N$.

Let $a_{0}, a_{1}, a_{2}, \ldots$ be an infinite sequence where each term is independently and uniformly random in the set $\{1,2,3,4\}$. Define an infinite sequence $b_{0}, b_{1}, b_{2}, \ldots$ recursively by $b_{0}=1$ and $b_{i+1}=a_{i}^{b_{i}}$. Compute the expected value of the smallest positive integer $k$ such that $b_{k} \equiv 1(\bmod 5)$.

A subset $S$ of the set $\{1,2, \ldots, 10\}$ is chosen randomly, with all possible subsets being equally likely. Compute the expected number of positive integers which divide the product of the elements of $S$. (By convention, the product of the elements of the empty set is 1.)

Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$. Suppose $A X \cdot A Y=6, B X \cdot B Y=5$, and $C X \cdot C Y=4$. Compute $A B^{2}$.

The spikiness of a sequence $a_{1}, a_{2}, \ldots, a_{n}$ of at least two real numbers is the sum $\sum_{i=1}^{n-1}\left|a_{i+1}-a_{i}\right|$. Suppose $x_{1}, x_{2}, \ldots, x_{9}$ are chosen uniformly and randomly from the interval $[0,1]$. Let $M$ be the largest possible value of the spikiness of a permutation of $x_{1}, x_{2}, \ldots, x_{9}$. Compute the expected value of $M$.

Let $P A B C$ be a tetrahedron such that $\angle A P B=\angle A P C=\angle B P C=90^{\circ}, \angle A B C=30^{\circ}$, and $A P^{2}$ equals the area of triangle $A B C$. Compute $\tan \angle A C B$.

Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_{1}, x_{2}, \ldots, x_{n}$ for which

• $x_{1}^{k}+x_{2}^{k}+\cdots+x_{n}^{k}=1$ for $k=1,2, \ldots, n-1$;
• $x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n}=2$; and
• $x_{1}^{m}+x_{2}^{m}+\cdots+x_{n}^{m}=4$.

Compute the smallest possible value of $m+n$.

Suppose $A B C D$ is a convex quadrilateral with $\angle A B D=105^{\circ}, \angle A D B=15^{\circ}, A C=7$, and $B C=C D=5$. Compute the sum of all possible values of $B D$.

Let $P_{1}(x), P_{2}(x), \ldots, P_{k}(x)$ be monic polynomials of degree 13 with integer coefficients. Suppose there are pairwise distinct positive integers $n_{1}, n_{2}, \ldots, n_{k}$ for which, for all positive integers $i$ and $j$ less than or equal to $k$, the statement " $n_{i}$ divides $P_{j}(m)$ for every integer $m$ " holds if and only if $i=j$. Compute the largest possible value of $k$.

Five pairs of twins are randomly arranged around a circle. Then they perform zero or more swaps, where each swap switches the positions of two adjacent people. They want to reach a state where no one is adjacent to their twin. Compute the expected value of the smallest number of swaps needed to reach such a state.

Let

$$P=\prod_{i=0}^{2016}\left(i^{3}-i-1\right)^{2}$$

The remainder when $P$ is divided by the prime 2017 is not zero. Compute this remainder.

Let $A B C$ be a triangle with $\angle B A C>90^{\circ}$. Let $D$ be the foot of the perpendicular from $A$ to side $B C$. Let $M$ and $N$ be the midpoints of segments $B C$ and $B D$, respectively. Suppose that $A C=2$, $\angle B A N=\angle M A C$, and $A B \cdot B C=A M$. Compute the distance from $B$ to line $A M$.

Given a function $f$, let $\pi(f)=f \circ f \circ f \circ f \circ f$. The attached sheet has the graphs of ten smooth functions from the interval $(0,1)$ to itself. The left-hand side consists of five functions:

• $F_{1}(x)=0.005+\frac{1}{2} \sin 2 x+\frac{1}{4} \sin 4 x+\frac{1}{8} \sin 8 x+\frac{1}{16} \sin 16 x+\frac{1}{32} \sin 32 x ;$
• $F_{2}(x)=F_{1}\left(F_{1}(x+0.25)\right)$;
• $F_{3}(x)=F_{1}\left((1-x) F_{1}\left((1-x)^{2}\right)\right)$;
• $F_{4}(x)=F_{1}(x)+0.05 \sin (2 \pi x)$;
• $F_{5}(x)=F_{1}(x+1.45)+0.65$.

The right-hand side consists of the five functions $A, B, C, D$, and $E$, which are $\pi\left(F_{1}\right), \ldots, \pi\left(F_{5}\right)$ in some order. Compute which of the functions $\{A, B, C, D, E\}$ correspond to $\pi\left(F_{k}\right)$ for $k=1,2,3,4,5$. Your answer should be a five-character string containing $A, B, C, D, E$, or $X$ for blank. For instance, if you think $\pi\left(F_{1}\right)=A$ and $\pi\left(F_{5}\right)=E$, then you would answer $A X X X E$. If you attempt to identify $n$ functions and get them all correct, then you will receive $n^{2}$ points. Otherwise, you will receive 0 points.

The number 2027 is prime. For $i=1,2, \ldots, 2026$, let $p_{i}$ be the smallest prime number such that $p_{i} \equiv i(\bmod 2027)$. Estimate $\max \left(p_{1}, \ldots, p_{2026}\right)$. Submit a positive integer $E$. If the correct answer is $A$, you will receive $\left\lfloor 25 \min \left((E / A)^{8},(A / E)^{8}\right)\right\rfloor$ points. (If you do not submit a positive integer, you will receive zero points for this question.)

The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 30 wooden blocks of weights $\sqrt[3]{F_{2}}, \sqrt[3]{F_{3}}, \ldots, \sqrt[3]{F_{31}}$, estimate the number of ways to paint each block either red or blue such that the total weight of the red blocks and the total weight of the blue blocks differ by at most 1 .

Submit a positive integer $E$. If the correct answer is $A$, you will receive $\left\lfloor 25 \min \left((E / A)^{8},(A / E)^{8}\right)\right\rfloor$ points. (If you do not submit a positive integer, you will receive zero points for this question.)

After the Guts round ends, the HMMT organizers will calculate $A$, the total number of points earned over all participating teams on questions 33,34 , and 35 of this round (that is, the other estimation questions). Estimate $A$. Submit a positive integer $E$. You will receive $\max (0,25-3 \cdot|E-A|)$ points. (If you do not submit a positive integer, you will receive zero points for this question.) For your information, there are about 70 teams competing.